research article disease control of delay seir model with...
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Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 830237 11 pageshttpdxdoiorg1011552013830237
Research ArticleDisease Control of Delay SEIR Model with Nonlinear IncidenceRate and Vertical Transmission
Yan Cheng12 Qiuhui Pan3 and Mingfeng He1
1 School of Mathematical Sciences Dalian University of Technology Dalian 116024 China2Department of Mathematics Tonghua Normal University Tonghua 136000 China3 School of Innovation Experiment Dalian University of Technology Dalian 116024 China
Correspondence should be addressed to Mingfeng He mfhedluteducn
Received 19 July 2013 Revised 30 September 2013 Accepted 3 October 2013
Academic Editor Seiya Imoto
Copyright copy 2013 Yan Cheng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate continuous treatment andimpulsive vaccination for a class of epidemic with latent period and vertical transition For continuous treatment we obtain a basicreproductive number R
0and prove the global stability by using the Lyapunov functional method We obtain two thresholds Rlowast
and Rlowastfor impulsive vaccination and prove that if Rlowast lt 1 then the disease-free periodic solution is globally attractive and if
Rlowastgt 1 then the disease is permanent by using the comparison theorem of impulsive differential equation Numerical simulations
indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease
1 Introduction
Mathematical models describing the population dynamics ofinfectious diseases have been playing an important role inunderstanding epidemiological patterns and disease controlResearchers have studied the epidemic models by ordinarydifferential equations [1ndash3] and the references cited thereinA customarily epidemic model is susceptible infectious andrecovered model (SIR for short) [4ndash7] But in real life manydiseases have a period of incubation time inside the hostsbefore the hosts become infectious if we include incubationperiod of the hosts the model is described as SEIRmodel Astuberculosis (TB) measles and so on a susceptible individualbecomes exposed (infected but not infective) by adequatecontact with an infectious individual SEIR infections diseasemodel has been studied by many authors for its importantbiological meaning [8ndash13] In [13] the authors considered thefollowing delayed SEIR epidemic model
1198781015840(119905) = Λ minus 120583119878 (119905) minus
120573119878 (119905) 119868 (119905)
1 + 120572119868 (119905)
+ 120575119877 (119905)
1198641015840(119905) =
120573119878 (119905) 119868 (119905)
1 + 120572119868 (119905)
minus
120573119890minus120583120591119878 (119905 minus 120591) 119868 (119905 minus 120591)
1 + 120572119868 (119905 minus 120591)
minus 120583119864
1198681015840(119905) =
120573119890minus120583120591119878 (119905 minus 120591) 119868 (119905 minus 120591)
1 + 120572119868 (119905 minus 120591)
minus (120583 + 120574) 119868
1198771015840(119905) = 120574119868 minus (120583 + 120575) 119877 (119905)
(1)where 119878(119905) 119864(119905) 119868(119905) and 119877(119905) represent the number of indi-viduals who are susceptible exposed infected and removedrespectively The parameters Λ 120573 120574 and 120583 are positiveconstants and hereΛ is the constant recruitment rate into thepopulation 120573 is the contact rate 120583 is the birth and death rate120574 is the removal rate 120591 gt 0 represents a time delay describingthe latent period of the disease and the term (120573119890
minus120583120591119878(119905minus120591)119868(119905minus
120591))(1 + 120572119868(119905 minus 120591)) represents the individuals surviving in thelatent period 120591 and becoming infective at time 119905The sufficientconditions are obtained for the global asymptotic stability ofthe endemic equilibrium
In the study of epidemic model the spread of an infec-tious disease is a crucial issue which depends on both thepopulation behavior and the infectivity of the disease Thesetwo aspects are captured in the incidence rate of a disease Inmany epidemiological models the incidence rate is describedas mass action incidence with bilinear interactions givenby 120573119878119868 where 120573 is the probability of transmission per
2 Computational and Mathematical Methods in Medicine
contact and 119878 and 119868 represent the susceptible and infectedpopulations respectively This contact law is more appro-priate for a few of infected individuals when the size ofinfected individuals is increasing the underlying assumptionof homogeneous mixing may not be valid In fact with theincrease of infected populations the susceptible individualwill takemeasure to prevent unbounded contact rates In [14]Anderson andMay proposed a saturated incidence rate of theform 120573119878119868(1 + 120572119878) in which 120573119878119868measures the infection forceof the disease and 1(1 + 120572119878) measures the inhibition effectfrom the behavioral change of the susceptible individualsThesame as the nonlinear incidence rates of the form 119896119868
119901119878119902 were
investigated by Liu et al [15 16]In real life some diseases may be transferred through
horizontal transmission and vertical transmission (disease isthe passing of an infection to offspring of infected parents)The offspring of infected parentsmay already be infectedwiththe disease at birth so many infections in nature transmitthrough both horizontal and vertical modes such as tuber-culosis (TB) rubella hepatitis B and AIDS [17ndash21]
Vaccination and treatment are important strategy for theelimination of infectious diseases Recently pulse vaccinationhas been confirmed as an effective method to prevent thespread of the disease [22ndash24] Theoretical results show thatthe pulse vaccination strategy can be distinguished fromthe conventional strategies in leading to disease eradicationat relatively low values of vaccination [25] The study ofvaccination treatment and associated behavioral changesrelated to disease transmission has been the subject of intensetheoretical analysis
The literature on SEIR model with nonlinear incidenceconstant infectious period impulsive vaccination dealingwith the analysis of disease that is vertically and horizontallytransmitted is not extensive [17 19] But in fact under thesituation of disease with vertical transmission the contin-uous treatment should be considered for the infected andimpulsive vaccination to the susceptible newborns of thesusceptible exposed and the removed and newborns ofinfected which not be vertical infected
Motivated by the literature above we introduce delayepidemic models with nonlinear incidence rates of the form120573119878119901119868 and we also considered the constant latency period
and vertically and horizontally in (2) The purpose of thispaper is to study the nonlinear dynamics of system andwe consider two different strategies to the model which areconstant treatment and pulse vaccination to the newbornsand susceptible
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
(2)
The basic assumptions are as follows
(i) The total population size at time 119905 (day) is denotedby 119873 = 119878 + 119864 + 119868 + 119877 For = 0 this shows thatthe total population has a constant size Without lossof generality we assume in this paper 119873 = 1 Thenewborns of 119878 119864 and 119877 are susceptible individualsand the newborns of 119868 who are not vertically infectedare also susceptible individuals
(ii) The positive constant 119887 (per day) denotes the deathrate and birth rate of susceptible exposed and recov-ered individuals The positive constant 120575 (per day)denotes the death rate and birth rate of infectiveindividuals The positive constant 120574 (per day) is thenatural recovery rate of infective individuals Thepositive constant 119902 (119902 le 1) (per day) is the verticaltransmission rate and note 1199021015840 = 1 minus 119902 1199021015840 lt 119902
(per day) and then 0 lt 1199021015840lt 1 Fraction 1198981015840 of all
newborns with mothers in the susceptible exposedand recovered classes are vaccinated and appearedin the recovered class while the remaining fraction119898 = 1 minus 119898
1015840 appears in the susceptible class suppose119887119898 gt 119902
1015840120575 (119889119879) is the proportion of those cured
successfully(iii) The incidence rate is described by a nonlinear func-
tion 120573119878119901119868 where 120573 (per day) is a positive constant
describing the infection rate 120591 gt 0 is the length ofthe latent period and the term 120573119890
minus119887120591119878119901(119905 minus 120591)119868(119905 minus 120591)
reflects the fact that an individual is surviving in thelatent period 120591 and becoming infective at time 119905
The remaining part of this paper is organized as followsIn Section 2 we investigate the global stability of the endemicequilibrium of (2) by using Rouches theorem and Lyapunov-LaSalle type theorem The global asymptotic stability ofdisease-free periodic solution and the conditions for thepermanence of the disease by comparison techniques aredescribed in Section 3 Numerical simulations are presentedin Section 4 In Section 5 we conclude this paper with someremarks
2 Continuous Treatment Strategy ofthe SEIR Model
In this section we consider a continuous treatment of SEIRmodel with constant latent period and nonlinear incidencerate By using 119878 + 119877 + 119864 = 1 minus 119868 notice that first andthird equations of system (2) do not contain the variables 119864and 119877 therefore system (2) is equivalent to the following 2-dimensional system
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
(3)
21 Disease-Free Equilibrium and Its Stability First we define
R0=
120573119898119901119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
(4)
Computational and Mathematical Methods in Medicine 3
Theorem 1 If R0lt 1 the disease-free equilibrium 119864
0(119898 0)
of system (3) is locally asymptotically stable for all 120591 ge 0 ifR0gt 1 the disease-free equilibrium 119864
0(119898 0) is unstable
Proof Steady states of system satisfy the following system ofequations
119887119898 (1 minus 119868) minus 120573119878119901119868 minus 119887119878 + 119902
1015840120575119868 = 0
120573119890minus119887120591119878119901119868 + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868 = 0
(5)
Obviously 1198640(119898 0) is the disease-free equilibrium of (3)
In order to analyze the behavior of the system (3) near 1198640 we
linearize the system about the equilibrium point let 119878(119905) =119883(119905) + 119898 119868(119905) = 119884(119905)
1198831015840(119905) = minus119887119883 (119905) minus (120573119898
119901+ 119887119898 minus 119902
1015840120575)119884 (119905)
1198841015840(119905) = minus(120575 minus 119902120575 + 120574 +
119889
119879
)119884 (119905) + 120573119890minus119887120591119898119901119884 (119905 minus 120591)
(6)
1205821= minus119887 lt 0 is one of the eigenvalues of the linearization of
system (6) near the steady state 1198640 and the other eigenvalue
1205822is determined by equation
120582 minus 120573119898119901119890minus(119887+120582)120591
+ 120575 minus 119902120575 + 120574 +
119889
119879
= 0 (7)
Let
119891 (120582) = 120582 minus 120573119898119901119890minus(119887+120582)120591
+ 120575 minus 119902120575 + 120574 +
119889
119879
(8)
ifR0gt 1 it is easy to show that for 120582 real
119891 (0) = (120575 minus 119902120575 + 120574 +
119889
119879
) (1 minusR0) lt 0
lim120582rarr+infin
119891 (120582) = +infin
(9)
hence 119891(120582) = 0 has a positive real root Therefore ifR0gt 1
the disease-free equilibrium 1198640(119898 0) is unstable
If R0lt 1 we prove that the disease-free equilibrium
1198640(119898 0) is locally stable Otherwise Re 120582 ge 0 We note that
Re 120582 = (120575 minus 119902120575 + 120574 + 119889
119879
) (R0119890minusRe120582120591 cos (119897119898120582120591) minus 1)
le (120575 minus 119902120575 + 120574 +
119889
119879
) (R0minus 1)
(10)
a contradiction Hence the disease-free equilibrium 1198640is
locally asymptotically stable ifR0lt 1
Theorem 2 If R0lt 1 the disease-free equilibrium 119864
0(119898 0)
of system (3) is globally asymptotically stable for all 120591 ge 0
To proof the global stability of the disease-free equilib-rium 119864
0(119898 0) we choose Lyapunov function
119881 (119905) = 119883 (119905) + 119884 (119905) + 120573119890minus119887120591119898119901int
119905
119905minus120591
119884 (120585) 119889120585 (11)
and it is easy to prove 1198811015840(119905) lt 0 lim119905rarrinfin
119881(119905) = 0 it followsthat lim
119905rarrinfin119883(119905) = 0 lim
119905rarrinfin119884(119905) = 0
22 Endemic Equilibrium and Its Stability If R0gt 1 then
system (3) has a unique positive equilibrium 1198641(119878lowast 119868lowast)
where
119878lowast= (
(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591
120573
)
1119901
119868lowast=
119887
(119887119898 minus 1199021015840120575) + (120575 minus 119902120575 + 120574 + (119889119879)) 119890
119887120591
times (119898 minus (
(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591
120573
)
1119901
)
(12)
Theorem 3 If R0gt 1 conditions (17) and (22) are satisfied
then for 120591 ge 0 the endemic equilibrium 1198641(119878lowast 119868lowast) of system (3)
is locally asymptotically stable
Proof Let 119878(119905) = 119883(119905) + 119878lowast 119868(119905) = 119884(119905) + 119868
lowast the linearizedsystem is obtained
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
minus (120573119878lowast119901+ 119887119898 minus 119902
1015840120575)119884 (119905)
1198841015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119884 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119884 (119905)
(13)
From the linearized system we obtain the characteristic equ-ation
1205822+ 119901120582 + 119903 + (119892120582 + 119902) 119890
minus120582120591= 0 (14)
where
119901 = 120575 minus 119902120575 + 120574 +
119889
119879
+ 120573119901119878lowast(119901minus1)
119868lowast+ 119887
119903 = (119887 + 120573119901119868lowast119878lowast(119901minus1)
) (120575 minus 119902120575 + 120574 +
119889
119879
)
119892 = minus120573119878lowast119901119890minus119887120591
119902 = minus (119887 + 120573119901119868lowast119878lowast(119901minus1)
) 120573119890minus119887120591119878lowast119901
+ (120573119878lowast119901+ 119887119898 minus 119902
1015840120575) 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
(15)
For 120591 = 0 the characteristic equation becomes
1205822+ (119901 + 119892) 120582 + (119903 + 119902) = 0 (16)
and we can see that both roots are negative and real if andonly if
119901 + 119892 gt 0 119903 + 119902 gt 0 (17)
Now for 120591 = 0 if 120582 = 120596119894 is a root of (16) we have
minus1205962+ 119902119890minus120596120591119894
+ 119901120596119894 + 119903 + 119892120596119890minus120596120591119894
= 0 (18)
4 Computational and Mathematical Methods in Medicine
Separating the real and imaginary parts we have
119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0
119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)
Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial
1205964+ (1199012minus 1198922minus 2119903) 120596
2+ 1199032minus 1199022= 0 (20)
from which we have
1205962=
1198922minus 1199012+ 2119903 plusmn radic(119892
2minus 1199012+ 2119903)2minus 4 (119903
2minus 1199022)
2
(21)
It follows that if
1199012minus 1198922minus 2119903 gt 0 119903
2minus 1199022gt 0 (22)
are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved
Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877
4is equivalent
to that of the following system
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
(23)
The initial conditions for system (23) take the form
119878 (120585) = 1205931(120585) 119864 (120585) = 120593
2(120585) 119868 (120585) = 120593
3(120585)
1205931 (0) gt 0 120593
2 (0) gt 0 1205933 (0) gt 0
(24)
where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877
3
+0) the space of
continuous functions mapping the interval [minus120591 0] into 1198773+0
where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3
For continuity of the initial conditions we require
119864 (0) = int
0
minus120591
120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)
It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0
Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872
Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR
0gt 1 then for any infectious period 120591 satisfying
120591 gt max
1
119887
ln120573119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)
120573119901119868lowast119872119901minus1
minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)
1
119887
ln120573119872119901minus1
(120588 + 1) (119868lowast119901 +119872)
4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1
1
119887
ln120573119872119901minus1
(119868lowast119901 + 2119872119901 + 3119872)
2(120575 minus 119902120575 + 120574 +
119889
119879
) + 120588 (119887119898 minus 1199021015840120575)
(26)
where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable
Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
+ (1199021015840120575 minus 120573119878
lowast119901minus 119887119898)119885 (119905)
1198841015840(119905) = 120573119901119868
lowast119878lowast(119901minus1)
119883(119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)
1198851015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)
(27)
The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878
lowast 119864lowast 119868lowast) of system (23) is globally asymptot-
ically stable We will employ Lyapunov functional techniqueto prove it
Now let us introduce the following functions
1198811 (119905) =
1
2
120588(119883 (119905) + 119884 (119905))2+
1
2
(1198842(119905) + 119885
2(119905))
1198812(119905) = (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
int
119905
119905minus120591
1198832(120585) 119889120585
+ (120588 + 1) 120573119890minus119887120591119872119901int
119905
119905minus120591
1198852(120585) 119889120585
(28)
Computational and Mathematical Methods in Medicine 5
where 120588 gt 0 is an arbitrary real constant Choosing 120588 =
120573119901119878lowast119901(119887119898 minus 119902
1015840120575) the derivative of 119881
1(119905) is
1198811015840
1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883
1015840(119905) + 119884
1015840(119905))
+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885
1015840(119905)
= 120588 (119883 (119905) + 119884 (119905))
times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]
+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)
119883 (119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]
+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)
minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)]
= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
]1198852(119905)
+ [120573119901119868lowast119878lowast(119901minus1)
minus 2120588119887]119883 (119905) 119884 (119905)
+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905)119883 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)119885 (119905)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)
(29)
and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression
1198811015840
1(119905) le minus [2119887120588 minus
1
2
120573119901119868lowast119878lowast(119901minus1)
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573120588119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119883
2(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119885
2(119905)
+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)
1198832(119905 minus 120591)
+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)
(30)
We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +
1198812(119905) and we get
1198811015840(119905) = 119881
1015840
1(119905) + (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
times (1198832(119905) minus 119883
2(119905 minus 120591)) + (120588 + 1) 120573119890
minus119887120591119872119901
times (1198852(119905) minus 119885
2(119905 minus 120591))
(31)
Substituting this in the inequality for 1198811(119905) we get
1198811015840(119905) le minus [2119887120588 +
1
2
120588 (119887119898 minus 1199021015840120575) minus
1
2
120573119901119868lowast119872lowast(119901minus1)
minus
1
2
120573119890minus119887120591119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)]1198832(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 + 2119872120588 + 3119872)]119885
2(119905)
(32)
The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim
119905rarrinfin119883(119905) rarr
0 lim119905rarrinfin
119884(119905) rarr 0 lim119905rarrinfin
119885(119905) rarr 0 The proof iscomplete
3 Continuous Treatment andPulse Vaccination Strategies
When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear
6 Computational and Mathematical Methods in Medicine
infectious force and vertical transmission it can be written asfollows
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905
+) = 119864 (119905)
119868 (119905+) = 119868 (119905) 119877 (119905
+) = 119877 (119905) + 120579119878 (119905)
119905 = 119896119879 119896 isin 119885+
(33)
and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate
We also consider the following reduced systems
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905
+) = 119868 (119905)
119905 = 119896119879 119896 isin 119885+
(34)
Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877
2
+|119878 ge
0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)
We first state two important lemmas which are useful inour following discussions
Lemma 6 (see [12]) Consider the following impulsive differ-ential equation
1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879
119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879
(35)
where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by
119906lowast(119905) =
119886
119887
+ (119906 minus
119886
119887
) 119890minus119887(119905minus119896119879)
119896119879 lt 119905 le (119896 + 1) 119879 (36)
which is globally asymptotically stable where
119906 =
119886
119887
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
(37)
Lemma 7 (see [27 28]) Consider the following equation
1199061015840(119905) = 119886
1119906 (119905 minus 120591) minus 119886
2119906 (119905) (38)
where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has
(1) if 1198861lt 1198862 then lim
119905rarrinfin119906(119905) = 0
(2) if 1198861gt 1198862 then lim
119905rarrinfin119906(119905) = +infin
31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy
1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885
+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885
+
(39)
By Lemma 6 we obtain the periodic solution of system (39)
119878lowast(119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
119896119879 lt 119905 le (119896 + 1) 119879 (40)
and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)
Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that
Rlowast=
119898119901119890minus119887120591120573
120575 minus 119902120575 + 120574 + (119889119879)
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
lt 1 (41)
Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small
such that
119890minus119887120591120573(
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760)
119901
lt 120575 minus 119902120575 + 120574 +
119889
119879
(42)
From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus
119887119878 and then we consider the following comparison systemwith pulse
1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879
119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879
119908 (0+) = 119878 (0
+)
(43)
In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and
lim119905rarrinfin
119908 (119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
= 119878lowast(119905)
119896119879 lt 119905 le (119896 + 1) 119879
(44)
There exists an integer 1198961 such that
119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
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2 Computational and Mathematical Methods in Medicine
contact and 119878 and 119868 represent the susceptible and infectedpopulations respectively This contact law is more appro-priate for a few of infected individuals when the size ofinfected individuals is increasing the underlying assumptionof homogeneous mixing may not be valid In fact with theincrease of infected populations the susceptible individualwill takemeasure to prevent unbounded contact rates In [14]Anderson andMay proposed a saturated incidence rate of theform 120573119878119868(1 + 120572119878) in which 120573119878119868measures the infection forceof the disease and 1(1 + 120572119878) measures the inhibition effectfrom the behavioral change of the susceptible individualsThesame as the nonlinear incidence rates of the form 119896119868
119901119878119902 were
investigated by Liu et al [15 16]In real life some diseases may be transferred through
horizontal transmission and vertical transmission (disease isthe passing of an infection to offspring of infected parents)The offspring of infected parentsmay already be infectedwiththe disease at birth so many infections in nature transmitthrough both horizontal and vertical modes such as tuber-culosis (TB) rubella hepatitis B and AIDS [17ndash21]
Vaccination and treatment are important strategy for theelimination of infectious diseases Recently pulse vaccinationhas been confirmed as an effective method to prevent thespread of the disease [22ndash24] Theoretical results show thatthe pulse vaccination strategy can be distinguished fromthe conventional strategies in leading to disease eradicationat relatively low values of vaccination [25] The study ofvaccination treatment and associated behavioral changesrelated to disease transmission has been the subject of intensetheoretical analysis
The literature on SEIR model with nonlinear incidenceconstant infectious period impulsive vaccination dealingwith the analysis of disease that is vertically and horizontallytransmitted is not extensive [17 19] But in fact under thesituation of disease with vertical transmission the contin-uous treatment should be considered for the infected andimpulsive vaccination to the susceptible newborns of thesusceptible exposed and the removed and newborns ofinfected which not be vertical infected
Motivated by the literature above we introduce delayepidemic models with nonlinear incidence rates of the form120573119878119901119868 and we also considered the constant latency period
and vertically and horizontally in (2) The purpose of thispaper is to study the nonlinear dynamics of system andwe consider two different strategies to the model which areconstant treatment and pulse vaccination to the newbornsand susceptible
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
(2)
The basic assumptions are as follows
(i) The total population size at time 119905 (day) is denotedby 119873 = 119878 + 119864 + 119868 + 119877 For = 0 this shows thatthe total population has a constant size Without lossof generality we assume in this paper 119873 = 1 Thenewborns of 119878 119864 and 119877 are susceptible individualsand the newborns of 119868 who are not vertically infectedare also susceptible individuals
(ii) The positive constant 119887 (per day) denotes the deathrate and birth rate of susceptible exposed and recov-ered individuals The positive constant 120575 (per day)denotes the death rate and birth rate of infectiveindividuals The positive constant 120574 (per day) is thenatural recovery rate of infective individuals Thepositive constant 119902 (119902 le 1) (per day) is the verticaltransmission rate and note 1199021015840 = 1 minus 119902 1199021015840 lt 119902
(per day) and then 0 lt 1199021015840lt 1 Fraction 1198981015840 of all
newborns with mothers in the susceptible exposedand recovered classes are vaccinated and appearedin the recovered class while the remaining fraction119898 = 1 minus 119898
1015840 appears in the susceptible class suppose119887119898 gt 119902
1015840120575 (119889119879) is the proportion of those cured
successfully(iii) The incidence rate is described by a nonlinear func-
tion 120573119878119901119868 where 120573 (per day) is a positive constant
describing the infection rate 120591 gt 0 is the length ofthe latent period and the term 120573119890
minus119887120591119878119901(119905 minus 120591)119868(119905 minus 120591)
reflects the fact that an individual is surviving in thelatent period 120591 and becoming infective at time 119905
The remaining part of this paper is organized as followsIn Section 2 we investigate the global stability of the endemicequilibrium of (2) by using Rouches theorem and Lyapunov-LaSalle type theorem The global asymptotic stability ofdisease-free periodic solution and the conditions for thepermanence of the disease by comparison techniques aredescribed in Section 3 Numerical simulations are presentedin Section 4 In Section 5 we conclude this paper with someremarks
2 Continuous Treatment Strategy ofthe SEIR Model
In this section we consider a continuous treatment of SEIRmodel with constant latent period and nonlinear incidencerate By using 119878 + 119877 + 119864 = 1 minus 119868 notice that first andthird equations of system (2) do not contain the variables 119864and 119877 therefore system (2) is equivalent to the following 2-dimensional system
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
(3)
21 Disease-Free Equilibrium and Its Stability First we define
R0=
120573119898119901119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
(4)
Computational and Mathematical Methods in Medicine 3
Theorem 1 If R0lt 1 the disease-free equilibrium 119864
0(119898 0)
of system (3) is locally asymptotically stable for all 120591 ge 0 ifR0gt 1 the disease-free equilibrium 119864
0(119898 0) is unstable
Proof Steady states of system satisfy the following system ofequations
119887119898 (1 minus 119868) minus 120573119878119901119868 minus 119887119878 + 119902
1015840120575119868 = 0
120573119890minus119887120591119878119901119868 + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868 = 0
(5)
Obviously 1198640(119898 0) is the disease-free equilibrium of (3)
In order to analyze the behavior of the system (3) near 1198640 we
linearize the system about the equilibrium point let 119878(119905) =119883(119905) + 119898 119868(119905) = 119884(119905)
1198831015840(119905) = minus119887119883 (119905) minus (120573119898
119901+ 119887119898 minus 119902
1015840120575)119884 (119905)
1198841015840(119905) = minus(120575 minus 119902120575 + 120574 +
119889
119879
)119884 (119905) + 120573119890minus119887120591119898119901119884 (119905 minus 120591)
(6)
1205821= minus119887 lt 0 is one of the eigenvalues of the linearization of
system (6) near the steady state 1198640 and the other eigenvalue
1205822is determined by equation
120582 minus 120573119898119901119890minus(119887+120582)120591
+ 120575 minus 119902120575 + 120574 +
119889
119879
= 0 (7)
Let
119891 (120582) = 120582 minus 120573119898119901119890minus(119887+120582)120591
+ 120575 minus 119902120575 + 120574 +
119889
119879
(8)
ifR0gt 1 it is easy to show that for 120582 real
119891 (0) = (120575 minus 119902120575 + 120574 +
119889
119879
) (1 minusR0) lt 0
lim120582rarr+infin
119891 (120582) = +infin
(9)
hence 119891(120582) = 0 has a positive real root Therefore ifR0gt 1
the disease-free equilibrium 1198640(119898 0) is unstable
If R0lt 1 we prove that the disease-free equilibrium
1198640(119898 0) is locally stable Otherwise Re 120582 ge 0 We note that
Re 120582 = (120575 minus 119902120575 + 120574 + 119889
119879
) (R0119890minusRe120582120591 cos (119897119898120582120591) minus 1)
le (120575 minus 119902120575 + 120574 +
119889
119879
) (R0minus 1)
(10)
a contradiction Hence the disease-free equilibrium 1198640is
locally asymptotically stable ifR0lt 1
Theorem 2 If R0lt 1 the disease-free equilibrium 119864
0(119898 0)
of system (3) is globally asymptotically stable for all 120591 ge 0
To proof the global stability of the disease-free equilib-rium 119864
0(119898 0) we choose Lyapunov function
119881 (119905) = 119883 (119905) + 119884 (119905) + 120573119890minus119887120591119898119901int
119905
119905minus120591
119884 (120585) 119889120585 (11)
and it is easy to prove 1198811015840(119905) lt 0 lim119905rarrinfin
119881(119905) = 0 it followsthat lim
119905rarrinfin119883(119905) = 0 lim
119905rarrinfin119884(119905) = 0
22 Endemic Equilibrium and Its Stability If R0gt 1 then
system (3) has a unique positive equilibrium 1198641(119878lowast 119868lowast)
where
119878lowast= (
(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591
120573
)
1119901
119868lowast=
119887
(119887119898 minus 1199021015840120575) + (120575 minus 119902120575 + 120574 + (119889119879)) 119890
119887120591
times (119898 minus (
(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591
120573
)
1119901
)
(12)
Theorem 3 If R0gt 1 conditions (17) and (22) are satisfied
then for 120591 ge 0 the endemic equilibrium 1198641(119878lowast 119868lowast) of system (3)
is locally asymptotically stable
Proof Let 119878(119905) = 119883(119905) + 119878lowast 119868(119905) = 119884(119905) + 119868
lowast the linearizedsystem is obtained
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
minus (120573119878lowast119901+ 119887119898 minus 119902
1015840120575)119884 (119905)
1198841015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119884 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119884 (119905)
(13)
From the linearized system we obtain the characteristic equ-ation
1205822+ 119901120582 + 119903 + (119892120582 + 119902) 119890
minus120582120591= 0 (14)
where
119901 = 120575 minus 119902120575 + 120574 +
119889
119879
+ 120573119901119878lowast(119901minus1)
119868lowast+ 119887
119903 = (119887 + 120573119901119868lowast119878lowast(119901minus1)
) (120575 minus 119902120575 + 120574 +
119889
119879
)
119892 = minus120573119878lowast119901119890minus119887120591
119902 = minus (119887 + 120573119901119868lowast119878lowast(119901minus1)
) 120573119890minus119887120591119878lowast119901
+ (120573119878lowast119901+ 119887119898 minus 119902
1015840120575) 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
(15)
For 120591 = 0 the characteristic equation becomes
1205822+ (119901 + 119892) 120582 + (119903 + 119902) = 0 (16)
and we can see that both roots are negative and real if andonly if
119901 + 119892 gt 0 119903 + 119902 gt 0 (17)
Now for 120591 = 0 if 120582 = 120596119894 is a root of (16) we have
minus1205962+ 119902119890minus120596120591119894
+ 119901120596119894 + 119903 + 119892120596119890minus120596120591119894
= 0 (18)
4 Computational and Mathematical Methods in Medicine
Separating the real and imaginary parts we have
119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0
119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)
Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial
1205964+ (1199012minus 1198922minus 2119903) 120596
2+ 1199032minus 1199022= 0 (20)
from which we have
1205962=
1198922minus 1199012+ 2119903 plusmn radic(119892
2minus 1199012+ 2119903)2minus 4 (119903
2minus 1199022)
2
(21)
It follows that if
1199012minus 1198922minus 2119903 gt 0 119903
2minus 1199022gt 0 (22)
are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved
Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877
4is equivalent
to that of the following system
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
(23)
The initial conditions for system (23) take the form
119878 (120585) = 1205931(120585) 119864 (120585) = 120593
2(120585) 119868 (120585) = 120593
3(120585)
1205931 (0) gt 0 120593
2 (0) gt 0 1205933 (0) gt 0
(24)
where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877
3
+0) the space of
continuous functions mapping the interval [minus120591 0] into 1198773+0
where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3
For continuity of the initial conditions we require
119864 (0) = int
0
minus120591
120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)
It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0
Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872
Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR
0gt 1 then for any infectious period 120591 satisfying
120591 gt max
1
119887
ln120573119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)
120573119901119868lowast119872119901minus1
minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)
1
119887
ln120573119872119901minus1
(120588 + 1) (119868lowast119901 +119872)
4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1
1
119887
ln120573119872119901minus1
(119868lowast119901 + 2119872119901 + 3119872)
2(120575 minus 119902120575 + 120574 +
119889
119879
) + 120588 (119887119898 minus 1199021015840120575)
(26)
where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable
Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
+ (1199021015840120575 minus 120573119878
lowast119901minus 119887119898)119885 (119905)
1198841015840(119905) = 120573119901119868
lowast119878lowast(119901minus1)
119883(119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)
1198851015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)
(27)
The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878
lowast 119864lowast 119868lowast) of system (23) is globally asymptot-
ically stable We will employ Lyapunov functional techniqueto prove it
Now let us introduce the following functions
1198811 (119905) =
1
2
120588(119883 (119905) + 119884 (119905))2+
1
2
(1198842(119905) + 119885
2(119905))
1198812(119905) = (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
int
119905
119905minus120591
1198832(120585) 119889120585
+ (120588 + 1) 120573119890minus119887120591119872119901int
119905
119905minus120591
1198852(120585) 119889120585
(28)
Computational and Mathematical Methods in Medicine 5
where 120588 gt 0 is an arbitrary real constant Choosing 120588 =
120573119901119878lowast119901(119887119898 minus 119902
1015840120575) the derivative of 119881
1(119905) is
1198811015840
1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883
1015840(119905) + 119884
1015840(119905))
+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885
1015840(119905)
= 120588 (119883 (119905) + 119884 (119905))
times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]
+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)
119883 (119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]
+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)
minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)]
= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
]1198852(119905)
+ [120573119901119868lowast119878lowast(119901minus1)
minus 2120588119887]119883 (119905) 119884 (119905)
+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905)119883 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)119885 (119905)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)
(29)
and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression
1198811015840
1(119905) le minus [2119887120588 minus
1
2
120573119901119868lowast119878lowast(119901minus1)
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573120588119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119883
2(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119885
2(119905)
+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)
1198832(119905 minus 120591)
+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)
(30)
We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +
1198812(119905) and we get
1198811015840(119905) = 119881
1015840
1(119905) + (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
times (1198832(119905) minus 119883
2(119905 minus 120591)) + (120588 + 1) 120573119890
minus119887120591119872119901
times (1198852(119905) minus 119885
2(119905 minus 120591))
(31)
Substituting this in the inequality for 1198811(119905) we get
1198811015840(119905) le minus [2119887120588 +
1
2
120588 (119887119898 minus 1199021015840120575) minus
1
2
120573119901119868lowast119872lowast(119901minus1)
minus
1
2
120573119890minus119887120591119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)]1198832(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 + 2119872120588 + 3119872)]119885
2(119905)
(32)
The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim
119905rarrinfin119883(119905) rarr
0 lim119905rarrinfin
119884(119905) rarr 0 lim119905rarrinfin
119885(119905) rarr 0 The proof iscomplete
3 Continuous Treatment andPulse Vaccination Strategies
When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear
6 Computational and Mathematical Methods in Medicine
infectious force and vertical transmission it can be written asfollows
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905
+) = 119864 (119905)
119868 (119905+) = 119868 (119905) 119877 (119905
+) = 119877 (119905) + 120579119878 (119905)
119905 = 119896119879 119896 isin 119885+
(33)
and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate
We also consider the following reduced systems
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905
+) = 119868 (119905)
119905 = 119896119879 119896 isin 119885+
(34)
Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877
2
+|119878 ge
0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)
We first state two important lemmas which are useful inour following discussions
Lemma 6 (see [12]) Consider the following impulsive differ-ential equation
1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879
119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879
(35)
where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by
119906lowast(119905) =
119886
119887
+ (119906 minus
119886
119887
) 119890minus119887(119905minus119896119879)
119896119879 lt 119905 le (119896 + 1) 119879 (36)
which is globally asymptotically stable where
119906 =
119886
119887
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
(37)
Lemma 7 (see [27 28]) Consider the following equation
1199061015840(119905) = 119886
1119906 (119905 minus 120591) minus 119886
2119906 (119905) (38)
where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has
(1) if 1198861lt 1198862 then lim
119905rarrinfin119906(119905) = 0
(2) if 1198861gt 1198862 then lim
119905rarrinfin119906(119905) = +infin
31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy
1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885
+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885
+
(39)
By Lemma 6 we obtain the periodic solution of system (39)
119878lowast(119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
119896119879 lt 119905 le (119896 + 1) 119879 (40)
and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)
Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that
Rlowast=
119898119901119890minus119887120591120573
120575 minus 119902120575 + 120574 + (119889119879)
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
lt 1 (41)
Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small
such that
119890minus119887120591120573(
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760)
119901
lt 120575 minus 119902120575 + 120574 +
119889
119879
(42)
From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus
119887119878 and then we consider the following comparison systemwith pulse
1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879
119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879
119908 (0+) = 119878 (0
+)
(43)
In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and
lim119905rarrinfin
119908 (119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
= 119878lowast(119905)
119896119879 lt 119905 le (119896 + 1) 119879
(44)
There exists an integer 1198961 such that
119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
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Parkinsonrsquos Disease
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Computational and Mathematical Methods in Medicine 3
Theorem 1 If R0lt 1 the disease-free equilibrium 119864
0(119898 0)
of system (3) is locally asymptotically stable for all 120591 ge 0 ifR0gt 1 the disease-free equilibrium 119864
0(119898 0) is unstable
Proof Steady states of system satisfy the following system ofequations
119887119898 (1 minus 119868) minus 120573119878119901119868 minus 119887119878 + 119902
1015840120575119868 = 0
120573119890minus119887120591119878119901119868 + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868 = 0
(5)
Obviously 1198640(119898 0) is the disease-free equilibrium of (3)
In order to analyze the behavior of the system (3) near 1198640 we
linearize the system about the equilibrium point let 119878(119905) =119883(119905) + 119898 119868(119905) = 119884(119905)
1198831015840(119905) = minus119887119883 (119905) minus (120573119898
119901+ 119887119898 minus 119902
1015840120575)119884 (119905)
1198841015840(119905) = minus(120575 minus 119902120575 + 120574 +
119889
119879
)119884 (119905) + 120573119890minus119887120591119898119901119884 (119905 minus 120591)
(6)
1205821= minus119887 lt 0 is one of the eigenvalues of the linearization of
system (6) near the steady state 1198640 and the other eigenvalue
1205822is determined by equation
120582 minus 120573119898119901119890minus(119887+120582)120591
+ 120575 minus 119902120575 + 120574 +
119889
119879
= 0 (7)
Let
119891 (120582) = 120582 minus 120573119898119901119890minus(119887+120582)120591
+ 120575 minus 119902120575 + 120574 +
119889
119879
(8)
ifR0gt 1 it is easy to show that for 120582 real
119891 (0) = (120575 minus 119902120575 + 120574 +
119889
119879
) (1 minusR0) lt 0
lim120582rarr+infin
119891 (120582) = +infin
(9)
hence 119891(120582) = 0 has a positive real root Therefore ifR0gt 1
the disease-free equilibrium 1198640(119898 0) is unstable
If R0lt 1 we prove that the disease-free equilibrium
1198640(119898 0) is locally stable Otherwise Re 120582 ge 0 We note that
Re 120582 = (120575 minus 119902120575 + 120574 + 119889
119879
) (R0119890minusRe120582120591 cos (119897119898120582120591) minus 1)
le (120575 minus 119902120575 + 120574 +
119889
119879
) (R0minus 1)
(10)
a contradiction Hence the disease-free equilibrium 1198640is
locally asymptotically stable ifR0lt 1
Theorem 2 If R0lt 1 the disease-free equilibrium 119864
0(119898 0)
of system (3) is globally asymptotically stable for all 120591 ge 0
To proof the global stability of the disease-free equilib-rium 119864
0(119898 0) we choose Lyapunov function
119881 (119905) = 119883 (119905) + 119884 (119905) + 120573119890minus119887120591119898119901int
119905
119905minus120591
119884 (120585) 119889120585 (11)
and it is easy to prove 1198811015840(119905) lt 0 lim119905rarrinfin
119881(119905) = 0 it followsthat lim
119905rarrinfin119883(119905) = 0 lim
119905rarrinfin119884(119905) = 0
22 Endemic Equilibrium and Its Stability If R0gt 1 then
system (3) has a unique positive equilibrium 1198641(119878lowast 119868lowast)
where
119878lowast= (
(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591
120573
)
1119901
119868lowast=
119887
(119887119898 minus 1199021015840120575) + (120575 minus 119902120575 + 120574 + (119889119879)) 119890
119887120591
times (119898 minus (
(120575 minus 119902120575 + 120574 + (119889119879)) 119890119887120591
120573
)
1119901
)
(12)
Theorem 3 If R0gt 1 conditions (17) and (22) are satisfied
then for 120591 ge 0 the endemic equilibrium 1198641(119878lowast 119868lowast) of system (3)
is locally asymptotically stable
Proof Let 119878(119905) = 119883(119905) + 119878lowast 119868(119905) = 119884(119905) + 119868
lowast the linearizedsystem is obtained
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
minus (120573119878lowast119901+ 119887119898 minus 119902
1015840120575)119884 (119905)
1198841015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119884 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119884 (119905)
(13)
From the linearized system we obtain the characteristic equ-ation
1205822+ 119901120582 + 119903 + (119892120582 + 119902) 119890
minus120582120591= 0 (14)
where
119901 = 120575 minus 119902120575 + 120574 +
119889
119879
+ 120573119901119878lowast(119901minus1)
119868lowast+ 119887
119903 = (119887 + 120573119901119868lowast119878lowast(119901minus1)
) (120575 minus 119902120575 + 120574 +
119889
119879
)
119892 = minus120573119878lowast119901119890minus119887120591
119902 = minus (119887 + 120573119901119868lowast119878lowast(119901minus1)
) 120573119890minus119887120591119878lowast119901
+ (120573119878lowast119901+ 119887119898 minus 119902
1015840120575) 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
(15)
For 120591 = 0 the characteristic equation becomes
1205822+ (119901 + 119892) 120582 + (119903 + 119902) = 0 (16)
and we can see that both roots are negative and real if andonly if
119901 + 119892 gt 0 119903 + 119902 gt 0 (17)
Now for 120591 = 0 if 120582 = 120596119894 is a root of (16) we have
minus1205962+ 119902119890minus120596120591119894
+ 119901120596119894 + 119903 + 119892120596119890minus120596120591119894
= 0 (18)
4 Computational and Mathematical Methods in Medicine
Separating the real and imaginary parts we have
119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0
119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)
Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial
1205964+ (1199012minus 1198922minus 2119903) 120596
2+ 1199032minus 1199022= 0 (20)
from which we have
1205962=
1198922minus 1199012+ 2119903 plusmn radic(119892
2minus 1199012+ 2119903)2minus 4 (119903
2minus 1199022)
2
(21)
It follows that if
1199012minus 1198922minus 2119903 gt 0 119903
2minus 1199022gt 0 (22)
are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved
Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877
4is equivalent
to that of the following system
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
(23)
The initial conditions for system (23) take the form
119878 (120585) = 1205931(120585) 119864 (120585) = 120593
2(120585) 119868 (120585) = 120593
3(120585)
1205931 (0) gt 0 120593
2 (0) gt 0 1205933 (0) gt 0
(24)
where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877
3
+0) the space of
continuous functions mapping the interval [minus120591 0] into 1198773+0
where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3
For continuity of the initial conditions we require
119864 (0) = int
0
minus120591
120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)
It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0
Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872
Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR
0gt 1 then for any infectious period 120591 satisfying
120591 gt max
1
119887
ln120573119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)
120573119901119868lowast119872119901minus1
minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)
1
119887
ln120573119872119901minus1
(120588 + 1) (119868lowast119901 +119872)
4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1
1
119887
ln120573119872119901minus1
(119868lowast119901 + 2119872119901 + 3119872)
2(120575 minus 119902120575 + 120574 +
119889
119879
) + 120588 (119887119898 minus 1199021015840120575)
(26)
where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable
Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
+ (1199021015840120575 minus 120573119878
lowast119901minus 119887119898)119885 (119905)
1198841015840(119905) = 120573119901119868
lowast119878lowast(119901minus1)
119883(119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)
1198851015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)
(27)
The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878
lowast 119864lowast 119868lowast) of system (23) is globally asymptot-
ically stable We will employ Lyapunov functional techniqueto prove it
Now let us introduce the following functions
1198811 (119905) =
1
2
120588(119883 (119905) + 119884 (119905))2+
1
2
(1198842(119905) + 119885
2(119905))
1198812(119905) = (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
int
119905
119905minus120591
1198832(120585) 119889120585
+ (120588 + 1) 120573119890minus119887120591119872119901int
119905
119905minus120591
1198852(120585) 119889120585
(28)
Computational and Mathematical Methods in Medicine 5
where 120588 gt 0 is an arbitrary real constant Choosing 120588 =
120573119901119878lowast119901(119887119898 minus 119902
1015840120575) the derivative of 119881
1(119905) is
1198811015840
1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883
1015840(119905) + 119884
1015840(119905))
+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885
1015840(119905)
= 120588 (119883 (119905) + 119884 (119905))
times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]
+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)
119883 (119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]
+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)
minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)]
= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
]1198852(119905)
+ [120573119901119868lowast119878lowast(119901minus1)
minus 2120588119887]119883 (119905) 119884 (119905)
+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905)119883 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)119885 (119905)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)
(29)
and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression
1198811015840
1(119905) le minus [2119887120588 minus
1
2
120573119901119868lowast119878lowast(119901minus1)
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573120588119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119883
2(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119885
2(119905)
+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)
1198832(119905 minus 120591)
+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)
(30)
We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +
1198812(119905) and we get
1198811015840(119905) = 119881
1015840
1(119905) + (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
times (1198832(119905) minus 119883
2(119905 minus 120591)) + (120588 + 1) 120573119890
minus119887120591119872119901
times (1198852(119905) minus 119885
2(119905 minus 120591))
(31)
Substituting this in the inequality for 1198811(119905) we get
1198811015840(119905) le minus [2119887120588 +
1
2
120588 (119887119898 minus 1199021015840120575) minus
1
2
120573119901119868lowast119872lowast(119901minus1)
minus
1
2
120573119890minus119887120591119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)]1198832(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 + 2119872120588 + 3119872)]119885
2(119905)
(32)
The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim
119905rarrinfin119883(119905) rarr
0 lim119905rarrinfin
119884(119905) rarr 0 lim119905rarrinfin
119885(119905) rarr 0 The proof iscomplete
3 Continuous Treatment andPulse Vaccination Strategies
When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear
6 Computational and Mathematical Methods in Medicine
infectious force and vertical transmission it can be written asfollows
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905
+) = 119864 (119905)
119868 (119905+) = 119868 (119905) 119877 (119905
+) = 119877 (119905) + 120579119878 (119905)
119905 = 119896119879 119896 isin 119885+
(33)
and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate
We also consider the following reduced systems
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905
+) = 119868 (119905)
119905 = 119896119879 119896 isin 119885+
(34)
Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877
2
+|119878 ge
0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)
We first state two important lemmas which are useful inour following discussions
Lemma 6 (see [12]) Consider the following impulsive differ-ential equation
1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879
119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879
(35)
where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by
119906lowast(119905) =
119886
119887
+ (119906 minus
119886
119887
) 119890minus119887(119905minus119896119879)
119896119879 lt 119905 le (119896 + 1) 119879 (36)
which is globally asymptotically stable where
119906 =
119886
119887
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
(37)
Lemma 7 (see [27 28]) Consider the following equation
1199061015840(119905) = 119886
1119906 (119905 minus 120591) minus 119886
2119906 (119905) (38)
where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has
(1) if 1198861lt 1198862 then lim
119905rarrinfin119906(119905) = 0
(2) if 1198861gt 1198862 then lim
119905rarrinfin119906(119905) = +infin
31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy
1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885
+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885
+
(39)
By Lemma 6 we obtain the periodic solution of system (39)
119878lowast(119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
119896119879 lt 119905 le (119896 + 1) 119879 (40)
and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)
Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that
Rlowast=
119898119901119890minus119887120591120573
120575 minus 119902120575 + 120574 + (119889119879)
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
lt 1 (41)
Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small
such that
119890minus119887120591120573(
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760)
119901
lt 120575 minus 119902120575 + 120574 +
119889
119879
(42)
From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus
119887119878 and then we consider the following comparison systemwith pulse
1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879
119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879
119908 (0+) = 119878 (0
+)
(43)
In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and
lim119905rarrinfin
119908 (119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
= 119878lowast(119905)
119896119879 lt 119905 le (119896 + 1) 119879
(44)
There exists an integer 1198961 such that
119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
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Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
4 Computational and Mathematical Methods in Medicine
Separating the real and imaginary parts we have
119903 minus 1205962+ 119892120596 sin (120596120591) + 119902 cos120596120591 = 0
119901120596 + 119892120596 cos120596120591 minus 119902 sin120596120591 = 0(19)
Adding both equations and regrouping by powers of 120596 weobtain the following fourth degree polynomial
1205964+ (1199012minus 1198922minus 2119903) 120596
2+ 1199032minus 1199022= 0 (20)
from which we have
1205962=
1198922minus 1199012+ 2119903 plusmn radic(119892
2minus 1199012+ 2119903)2minus 4 (119903
2minus 1199022)
2
(21)
It follows that if
1199012minus 1198922minus 2119903 gt 0 119903
2minus 1199022gt 0 (22)
are satisfied (20) does not have positive solutions and thecharacteristic equation (14) does not have purely imaginaryroots Inequalities in (17) and (22) guarantee that all roots of(14) have no positive roots According to Rouchersquos theoremTheorem 3 is proved
Subsequently we discuss the sufficient conditions underwhich the endemic equilibrium is globally asymptoticallystable for the system (3) For 119878 + 119864 + 119868 + 119877 = 1 hence thedynamics of system (3) in the first octant of 119877
4is equivalent
to that of the following system
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
(23)
The initial conditions for system (23) take the form
119878 (120585) = 1205931(120585) 119864 (120585) = 120593
2(120585) 119868 (120585) = 120593
3(120585)
1205931 (0) gt 0 120593
2 (0) gt 0 1205933 (0) gt 0
(24)
where (1205931(120585) 1205932(120585) 1205933(120585)) isin 119862([minus120591 0] 119877
3
+0) the space of
continuous functions mapping the interval [minus120591 0] into 1198773+0
where 1198773+0= (1199091 1199092 1199093) 119909119894ge 0 119894 = 1 2 3
For continuity of the initial conditions we require
119864 (0) = int
0
minus120591
120573119890119887120585119878(120585)119901119868 (120585) 119889120585 (25)
It is well known by the fundamental theory of functionaldifferential equations [26] the system (23) has a uniquesolution (119878(119905) 119864(119905) 119868(119905)) satisfying the initial conditions Itis easy to show that all solutions of system (23) with initialconditions are defined on [0 +infin) and remain positive for all119905 ge 0
Lemma 4 (see [25]) Let the initial condition be 119878(120585) = 119878(0) gt0 119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) Then119878(119905) le max1 119878(0) + 119864(0) + 119868(0) = 119872
Theorem 5 Let the initial condition be 119878(120585) = 119878(0) gt 0119864(120585) = 119864(0) gt 0 and 119868(0) gt 0 for all 120585 isin [minus120591 0) FurthersupposeR
0gt 1 then for any infectious period 120591 satisfying
120591 gt max
1
119887
ln120573119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)
120573119901119868lowast119872119901minus1
minus 4119887120588 minus 120588 (119887119898 minus 1199021015840120575)
1
119887
ln120573119872119901minus1
(120588 + 1) (119868lowast119901 +119872)
4120588119887 + 2119887 minus 120573119901119868lowast119872119901minus1
1
119887
ln120573119872119901minus1
(119868lowast119901 + 2119872119901 + 3119872)
2(120575 minus 119902120575 + 120574 +
119889
119879
) + 120588 (119887119898 minus 1199021015840120575)
(26)
where119872 = max1 119878(0)+119864(0)+119868(0) the endemic equilibriumis globally asymptotically stable
Proof Let 119878(119905) = 119883(119905)+119878lowast 119864(119905) = 119884(119905)+119864lowast 119868(119905) = 119885(119905)+ 119868lowastthe linearized system is
1198831015840(119905) = minus (119887 + 120573119901119868
lowast119878lowast(119901minus1)
)119883 (119905)
+ (1199021015840120575 minus 120573119878
lowast119901minus 119887119898)119885 (119905)
1198841015840(119905) = 120573119901119868
lowast119878lowast(119901minus1)
119883(119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)
1198851015840(119905) = 120573119901119890
minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)
(27)
The trivial solution of system (27) is globally asymptot-ically stable and is equivalent to the fact that the endemicequilibrium (119878
lowast 119864lowast 119868lowast) of system (23) is globally asymptot-
ically stable We will employ Lyapunov functional techniqueto prove it
Now let us introduce the following functions
1198811 (119905) =
1
2
120588(119883 (119905) + 119884 (119905))2+
1
2
(1198842(119905) + 119885
2(119905))
1198812(119905) = (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
int
119905
119905minus120591
1198832(120585) 119889120585
+ (120588 + 1) 120573119890minus119887120591119872119901int
119905
119905minus120591
1198852(120585) 119889120585
(28)
Computational and Mathematical Methods in Medicine 5
where 120588 gt 0 is an arbitrary real constant Choosing 120588 =
120573119901119878lowast119901(119887119898 minus 119902
1015840120575) the derivative of 119881
1(119905) is
1198811015840
1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883
1015840(119905) + 119884
1015840(119905))
+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885
1015840(119905)
= 120588 (119883 (119905) + 119884 (119905))
times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]
+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)
119883 (119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]
+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)
minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)]
= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
]1198852(119905)
+ [120573119901119868lowast119878lowast(119901minus1)
minus 2120588119887]119883 (119905) 119884 (119905)
+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905)119883 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)119885 (119905)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)
(29)
and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression
1198811015840
1(119905) le minus [2119887120588 minus
1
2
120573119901119868lowast119878lowast(119901minus1)
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573120588119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119883
2(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119885
2(119905)
+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)
1198832(119905 minus 120591)
+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)
(30)
We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +
1198812(119905) and we get
1198811015840(119905) = 119881
1015840
1(119905) + (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
times (1198832(119905) minus 119883
2(119905 minus 120591)) + (120588 + 1) 120573119890
minus119887120591119872119901
times (1198852(119905) minus 119885
2(119905 minus 120591))
(31)
Substituting this in the inequality for 1198811(119905) we get
1198811015840(119905) le minus [2119887120588 +
1
2
120588 (119887119898 minus 1199021015840120575) minus
1
2
120573119901119868lowast119872lowast(119901minus1)
minus
1
2
120573119890minus119887120591119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)]1198832(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 + 2119872120588 + 3119872)]119885
2(119905)
(32)
The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim
119905rarrinfin119883(119905) rarr
0 lim119905rarrinfin
119884(119905) rarr 0 lim119905rarrinfin
119885(119905) rarr 0 The proof iscomplete
3 Continuous Treatment andPulse Vaccination Strategies
When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear
6 Computational and Mathematical Methods in Medicine
infectious force and vertical transmission it can be written asfollows
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905
+) = 119864 (119905)
119868 (119905+) = 119868 (119905) 119877 (119905
+) = 119877 (119905) + 120579119878 (119905)
119905 = 119896119879 119896 isin 119885+
(33)
and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate
We also consider the following reduced systems
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905
+) = 119868 (119905)
119905 = 119896119879 119896 isin 119885+
(34)
Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877
2
+|119878 ge
0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)
We first state two important lemmas which are useful inour following discussions
Lemma 6 (see [12]) Consider the following impulsive differ-ential equation
1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879
119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879
(35)
where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by
119906lowast(119905) =
119886
119887
+ (119906 minus
119886
119887
) 119890minus119887(119905minus119896119879)
119896119879 lt 119905 le (119896 + 1) 119879 (36)
which is globally asymptotically stable where
119906 =
119886
119887
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
(37)
Lemma 7 (see [27 28]) Consider the following equation
1199061015840(119905) = 119886
1119906 (119905 minus 120591) minus 119886
2119906 (119905) (38)
where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has
(1) if 1198861lt 1198862 then lim
119905rarrinfin119906(119905) = 0
(2) if 1198861gt 1198862 then lim
119905rarrinfin119906(119905) = +infin
31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy
1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885
+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885
+
(39)
By Lemma 6 we obtain the periodic solution of system (39)
119878lowast(119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
119896119879 lt 119905 le (119896 + 1) 119879 (40)
and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)
Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that
Rlowast=
119898119901119890minus119887120591120573
120575 minus 119902120575 + 120574 + (119889119879)
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
lt 1 (41)
Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small
such that
119890minus119887120591120573(
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760)
119901
lt 120575 minus 119902120575 + 120574 +
119889
119879
(42)
From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus
119887119878 and then we consider the following comparison systemwith pulse
1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879
119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879
119908 (0+) = 119878 (0
+)
(43)
In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and
lim119905rarrinfin
119908 (119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
= 119878lowast(119905)
119896119879 lt 119905 le (119896 + 1) 119879
(44)
There exists an integer 1198961 such that
119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Research and TreatmentAIDS
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Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 5
where 120588 gt 0 is an arbitrary real constant Choosing 120588 =
120573119901119878lowast119901(119887119898 minus 119902
1015840120575) the derivative of 119881
1(119905) is
1198811015840
1(119905) = 120588 (119883 (119905) + 119884 (119905)) (119883
1015840(119905) + 119884
1015840(119905))
+ 119884 (119905) 1198841015840(119905) + 119885 (119905) 119885
1015840(119905)
= 120588 (119883 (119905) + 119884 (119905))
times [minus119887119883 (119905) + (1199021015840120575 minus 119887119898)119885 (119905) minus 119887119884 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591) minus 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) ]
+ 119884 (119905) [120573119901119868lowast119878lowast(119901minus1)
119883 (119905) + 120573119878lowast119901119885 (119905)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905 minus 120591)
minus120573119890minus119887120591119878lowast119901119885 (119905 minus 120591) minus 119887119884 (119905)]
+ 119885 (119905) [120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)
minus (120575 minus 119902120575 + 120574 +
119889
119879
)119885 (119905)]
= minus 1198871205881198832(119905) minus (119887120588 + 119887) 119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
]1198852(119905)
+ [120573119901119868lowast119878lowast(119901minus1)
minus 2120588119887]119883 (119905) 119884 (119905)
+ 120588 (1199021015840120575 minus 119887119898)119885 (119905)119883 (119905)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883 (119905)119883 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119883 (119905) 119885 (119905 minus 120591)
minus 120573120588119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
minus 120588120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119884 (119905)119883 (119905 minus 120591)
minus 120573119890minus119887120591119878lowast119901119884 (119905) 119885 (119905 minus 120591)
+ 120573119901119890minus119887120591119868lowast119878lowast(119901minus1)
119883(119905 minus 120591)119885 (119905)
+ 120573119890minus119887120591119878lowast119901119885 (119905 minus 120591)119885 (119905)
(29)
and applying Cauchy-Chwartz inequality to all productterms we obtain the following expression
1198811015840
1(119905) le minus [2119887120588 minus
1
2
120573119901119868lowast119878lowast(119901minus1)
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573120588119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119883
2(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119885
2(119905)
+ (120588 + 1) 120573119901119890minus119887120591119868lowast119872(119901minus1)
1198832(119905 minus 120591)
+ (120588 + 1) 120573119890minus1198871205911198721199011198852(119905 minus 120591)
(30)
We choose Lyapunov function to be the form 119881(119905) = 1198811(119905) +
1198812(119905) and we get
1198811015840(119905) = 119881
1015840
1(119905) + (120588 + 1) 120573119901119890
minus119887120591119868lowast119872(119901minus1)
times (1198832(119905) minus 119883
2(119905 minus 120591)) + (120588 + 1) 120573119890
minus119887120591119872119901
times (1198852(119905) minus 119885
2(119905 minus 120591))
(31)
Substituting this in the inequality for 1198811(119905) we get
1198811015840(119905) le minus [2119887120588 +
1
2
120588 (119887119898 minus 1199021015840120575) minus
1
2
120573119901119868lowast119872lowast(119901minus1)
minus
1
2
120573119890minus119887120591119872119901minus1
(120588119872 + 2119901119868lowast+ 3120588119901119868
lowast)]1198832(119905)
minus [2120588119887 + 119887 minus
1
2
119901120573119868lowast119872119901minus1
minus
1
2
(120588 + 1) 120573119890minus119887120591119872119901minus1
(119868lowast119901 +119872)]119884
2(119905)
minus [120575 minus 119902120575 + 120574 +
119889
119879
+
1
2
120588 (119887119898 minus 1199021015840120575)
minus
1
2
120573119890minus119887120591119872119901minus1
(119868lowast119901 + 2119872120588 + 3119872)]119885
2(119905)
(32)
The right-hand expression of the above inequality is alwaysnegative provided that (26) holds A direct application of theLyapunov-LaSalle type theorem shows that lim
119905rarrinfin119883(119905) rarr
0 lim119905rarrinfin
119884(119905) rarr 0 lim119905rarrinfin
119885(119905) rarr 0 The proof iscomplete
3 Continuous Treatment andPulse Vaccination Strategies
When continuous treatment and pulse vaccination strategiesare included in the SEIR epidemic model with the nonlinear
6 Computational and Mathematical Methods in Medicine
infectious force and vertical transmission it can be written asfollows
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905
+) = 119864 (119905)
119868 (119905+) = 119868 (119905) 119877 (119905
+) = 119877 (119905) + 120579119878 (119905)
119905 = 119896119879 119896 isin 119885+
(33)
and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate
We also consider the following reduced systems
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905
+) = 119868 (119905)
119905 = 119896119879 119896 isin 119885+
(34)
Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877
2
+|119878 ge
0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)
We first state two important lemmas which are useful inour following discussions
Lemma 6 (see [12]) Consider the following impulsive differ-ential equation
1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879
119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879
(35)
where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by
119906lowast(119905) =
119886
119887
+ (119906 minus
119886
119887
) 119890minus119887(119905minus119896119879)
119896119879 lt 119905 le (119896 + 1) 119879 (36)
which is globally asymptotically stable where
119906 =
119886
119887
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
(37)
Lemma 7 (see [27 28]) Consider the following equation
1199061015840(119905) = 119886
1119906 (119905 minus 120591) minus 119886
2119906 (119905) (38)
where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has
(1) if 1198861lt 1198862 then lim
119905rarrinfin119906(119905) = 0
(2) if 1198861gt 1198862 then lim
119905rarrinfin119906(119905) = +infin
31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy
1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885
+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885
+
(39)
By Lemma 6 we obtain the periodic solution of system (39)
119878lowast(119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
119896119879 lt 119905 le (119896 + 1) 119879 (40)
and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)
Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that
Rlowast=
119898119901119890minus119887120591120573
120575 minus 119902120575 + 120574 + (119889119879)
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
lt 1 (41)
Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small
such that
119890minus119887120591120573(
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760)
119901
lt 120575 minus 119902120575 + 120574 +
119889
119879
(42)
From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus
119887119878 and then we consider the following comparison systemwith pulse
1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879
119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879
119908 (0+) = 119878 (0
+)
(43)
In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and
lim119905rarrinfin
119908 (119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
= 119878lowast(119905)
119896119879 lt 119905 le (119896 + 1) 119879
(44)
There exists an integer 1198961 such that
119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
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6 Computational and Mathematical Methods in Medicine
infectious force and vertical transmission it can be written asfollows
1198781015840(119905) = 119887119898 (119878 + 119877 + 119864) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198641015840(119905) = 120573119878
119901119868 minus 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) minus 119887119864
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
1198771015840(119905) = 120574119868 minus 119887119877 + 119887119898
1015840(119878 + 119877 + 119864) +
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119864 (119905
+) = 119864 (119905)
119868 (119905+) = 119868 (119905) 119877 (119905
+) = 119877 (119905) + 120579119878 (119905)
119905 = 119896119879 119896 isin 119885+
(33)
and 119878(119905) 119864(119905) 119868(119905) and 119877(119905) are the number of susceptibleexposed infectious and recovered at time 119905 respectively 120579 isthe proportion of those vaccinated successfully at 119896119879 whichis called pulse vaccination rate
We also consider the following reduced systems
1198781015840(119905) = 119887119898 (1 minus 119868) minus 120573119878
119901119868 minus 119887119878 + 119902
1015840120575119868
1198681015840(119905) = 120573119890
minus119887120591119878119901(119905 minus 120591) 119868 (119905 minus 120591) + 119902120575119868 minus 120575119868 minus 120574119868 minus
119889
119879
119868
119905 = 119896119879 119896 isin 119885+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119868 (119905
+) = 119868 (119905)
119905 = 119896119879 119896 isin 119885+
(34)
Let Ω be the following subset of 1198772+ Ω = (119878 119868) isin 119877
2
+|119878 ge
0 119868 ge 0 119878+119868 le 1 From biological considerations we discusssystem (34) in the closed set Ω It can be verified that Ω ispositively invariant with respect to system (34)
We first state two important lemmas which are useful inour following discussions
Lemma 6 (see [12]) Consider the following impulsive differ-ential equation
1199061015840(119905) = 119886 minus 119887119906 (119905) 119905 = 119896119879
119906 (119905+) = (1 minus 120579) 119906 (119905) 119905 = 119896119879
(35)
where 119886 gt 0 119887 gt 0 0 lt 120579 lt 1 Then the above system has aunique positive periodic solution given by
119906lowast(119905) =
119886
119887
+ (119906 minus
119886
119887
) 119890minus119887(119905minus119896119879)
119896119879 lt 119905 le (119896 + 1) 119879 (36)
which is globally asymptotically stable where
119906 =
119886
119887
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
(37)
Lemma 7 (see [27 28]) Consider the following equation
1199061015840(119905) = 119886
1119906 (119905 minus 120591) minus 119886
2119906 (119905) (38)
where 1198861 1198862 120591 gt 0 119906(119905) gt 0 for minus120591 le 119905 le 0 One has
(1) if 1198861lt 1198862 then lim
119905rarrinfin119906(119905) = 0
(2) if 1198861gt 1198862 then lim
119905rarrinfin119906(119905) = +infin
31 Global Stability of theDisease-Free Periodic Solution Nowwe will prove the disease-free periodic solution (119878(119905) 0) isglobal attractively We first demonstrate the existence of thedisease-free periodic solution inwhich infectious individualsare entirely absent from the population permanently that is119868(119905) equiv 0 for all 119905 gt 0 Under this condition the growth ofsusceptible individuals must satisfy
1198781015840(119905) = 119887119898 minus 119887119878 119905 = 119896119879 119896 isin 119885
+
119878 (119905+) = (1 minus 120579) 119878 (119905) 119905 = 119896119879 119896 isin 119885
+
(39)
By Lemma 6 we obtain the periodic solution of system (39)
119878lowast(119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
119896119879 lt 119905 le (119896 + 1) 119879 (40)
and this solution is globally asymptotically stable Hence thesystem (34) has a disease-free periodic solution (119878lowast(119905) 0)
Theorem 8 Let (119878(119905) 119868(119905)) be any solution of (34) then thedisease-free periodic solution (119878lowast(119905) 0) is globally asymptoti-cally stable provided that
Rlowast=
119898119901119890minus119887120591120573
120575 minus 119902120575 + 120574 + (119889119879)
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
lt 1 (41)
Proof SinceRlowast lt 1 we can choose 1205760gt 0 sufficiently small
such that
119890minus119887120591120573(
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760)
119901
lt 120575 minus 119902120575 + 120574 +
119889
119879
(42)
From the first equation of system (34) we have 1198781015840(119905) lt 119887119898 minus
119887119878 and then we consider the following comparison systemwith pulse
1199081015840(119905) = 119887119898 minus 119887119908 (119905) 119905 = 119896119879
119908 (119905+) = (1 minus 120579)119908 (119905) 119905 = 119896119879
119908 (0+) = 119878 (0
+)
(43)
In view of Lemma 6 we obtain 119878(119905) le 119908(119905) and
lim119905rarrinfin
119908 (119905) = 119898 minus
119898120579119890minus119887(119905minus119899119879)
1 minus (1 minus 120579) 119890minus119887119879
= 119878lowast(119905)
119896119879 lt 119905 le (119896 + 1) 119879
(44)
There exists an integer 1198961 such that
119878 (119905) le 119908 (119905) lt 119878lowast(119905) + 1205760 (45)
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
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Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 7
That is
119878 (119905) lt 119878lowast(119905) + 120576
0le
119898(1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
+ 1205760= 119878
119896119879 lt 119905 le (119896 + 1) 119879 119896 gt 1198961
(46)
Furthermore from the second equation we have
1198681015840(119905) le 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(47)
and then we consider the following comparison equation
1199081015840
1(119905) = 120573119890
minus119887120591119878
119901
119868 (119905 minus 120591) minus (120575 minus 119902120575 + 120574 +
119889
119879
) 119868 (119905)
119905 = 119896119879 119896 gt 1198961
(48)
then from (41) we have 120573119890minus119887120591119878119901 lt 120575 minus 119901120575 + 120574 + (119889119879) Inview of Lemma 7 we have 1199081015840
1(119905) lt 0 lim
119896rarrinfin119868(119905) = 0 So
there must exist an integer 1198962gt 1198961 such that 119868(119905) lt 120576
1for all
119905 gt 1198962119879
When 119905 gt 1198962119879 from the first equation of system (34) we
have
1198781015840(119905) gt (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761) minus (120573119878
119901minus1
1205761+ 119887) 119878 (119905) (49)
Consider the following comparison impulsive differentialequation for all 119905 gt 119896
2119879
1199081015840
2(119905) = (119887119898 minus (119887119898 minus 119902
1015840120575) 1205761)
minus (120573119878
119901minus1
1205761+ 119887)119908
2 (119905) 119905 = 119896119879
1199082(119905+) = (1 minus 120579)119908
2(119905) 119905 = 119896119879
1199082(0+) = 119878 (0
+)
(50)
By Lemma 6 we have the unique periodic solution of system(50) given by
119908lowast
2=
119887119898 minus (119887119898 minus 1199021015840120575) 1205761
120573119878
119901minus1
1205761+ 119887
(1 minus
120579119890minus(1205761120573119878119901minus1
+119887)(119905minus119899119879)
1 minus (1 minus 120579) 119890minus(1205761120573119878119901minus1
+119887)119879
)
119896119879 lt 119905 le (119896 + 1) 119879
(51)
By the comparison theorem there exists an integer 1198963gt 1198962
such that
119878 (119905) gt 1199082(119905) gt 119908
lowast
2minus 1205761 119896119879 lt 119905 le (119896 + 1) 119879 (52)
Because 1205760and 1205761are sufficiently small it follows from (46)
and (52) that lim119905rarrinfin
119878(119905) = 119878lowast(119905) Therefore the disease-free
solution (119878lowast(119905) 0) of system (34) is globally attractive The
proof is completed
Denote
120579lowast= (119890119887119879minus 1)((
120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
)
1119901
minus 1)
120591lowast=
1
119887
ln120573119898119901
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
1 minus 119890minus119887119879
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(53)
Corollary 9 (i) If 120573119898119901 le 119890119887120591(120575 minus 119902120575 + 120574 + (119889119879)) then the
infection-free periodic solution is globally attractive(ii) If 120573119898119901 gt 119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free
periodic solution is globally attractive provided that 120579 gt 120579lowast
Corollary 10 (i) If 120573119898119901((1 minus 119890minus119887119879
)(1 minus (1 minus 120579)119890minus119887119879
))119901le
119890119887120591(120575minus119902120575+120574+(119889119879)) then the infection-free periodic solution
is globally attractive(ii) If 120573119898119901((1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 119890
119887120591(120575 minus 119902120575 +
120574 + (119889119879)) then the disease will be endemic and system (34) ispermanent provided that 120591 gt 120591lowast
Theorem 8 determines the global attractivity of (34) inΩfor the caseRlowast lt 1 Its epidemiological implication is that theinfectious population vanishes in time so the disease dies outCorollaries 9 and 10 imply that the diseasewill disappear if thevaccination rate or the length of latent period of the diseaseis large enough
32 Persistent In this section we say the disease is endemicif the infectious population persists above a certain positivelevel for sufficiently large time The endemicity of the diseasecan be well captured and studied through the notion ofuniform persistence
Definition 11 System (34) is said to be uniformly persistent ifthere exist positive constants 119872
119894ge 119898119894 119894 = 1 2 (both are
independent of the initial values) such that every solution(119878(119905) 119868(119905)) with positive initial conditions of system (34)satisfies
1198981le 119878 (119905) le 119872
1 119898
2le 119868 (119905) le 119872
2 (54)
Denote
Rlowast=
120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(55)
Theorem 12 If Rlowastgt 1 then there is a positive constant
119898119868such that each positive solution (119878(119905) 119868(119905)) of system (34)
satisfies 119868(119905) ge 119898119868for all t sufficiently large
Proof Let (119878(119905) 119868(119905)) be any solution with initial values ofsystem (34) and then it is obvious that 119878(119905) le 1 119868(119905) le 1
for all 119905 gt 0 We are left to prove there exist positive constants119898119878 119898119868and 1199050(1199050is sufficiently large) such that 119878(119905) ge 119898
119878
119868(119905) ge 119898119868for all 119905 gt 119905
0
Firstly from the first equation of system (34) we have
1198781015840(119905) gt 119902
1015840120575 minus (120573 + 119887) 119878 (56)
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
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Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
8 Computational and Mathematical Methods in Medicine
Consider the following comparison equations
1198831015840(119905) = 119902
1015840120575 minus (120573 + 119887)119883 (119905) 119905 = 119896119879
119883 (119905+) = (1 minus 120579)119883 (119905) 119905 = 119896119879
(57)
By Lemma 6 and the comparison theorem [29] we knowthat for any sufficiently small 120576 gt 0 there exists a 119905
0(1199050is
sufficiently large) such that
119878 (119905) ge 119883 (119905) gt 119883lowast(119905) minus 120576
ge
1199021015840120575
120573 + 119887
(
(1 minus 120579) (1 minus 119890minus(120573+119887)119879
)
1 minus (1 minus 120579) 119890minus(120573+119887)119879
) minus 120576 = 119898119878gt 0
(58)
Now we will prove that there exist 119898119868gt 0 and a suff-
iciently large 1199050such that 119868(119905) ge 119898
119868for all 119905 gt 119905
0 Since the
proof is rather long it will be convenient to divide it into twosteps
Step 1 Since Rlowastgt 1 there exist 119898lowast
119868gt 0 120576 gt 0 sufficiently
small such that
120573120578119901119890minus119887120591
minus (120575 + 120574 +
119889
119879
minus 119902120575) gt 0 (59)
where 120578 = (1199021015840120575(120573119898
lowast
119868+ 119887))((1 minus 120579)(1 minus 119890
minus119887119879)(1 minus (1 minus
120579)119890minus119887119879
)) minus 120576We claim that for any 119905
0gt 0 it is impossible that 119868(119905) lt
119898lowast
119868for all 119905 ge 119905
0 Suppose that the claim is not valid There
exists a 1199050gt 0 such that 119868(119905) lt 119898
lowast
119868for all 119905 ge 119905
0 and then
follows from the first equation of system (34) that for 119905 ge 1199050
1198781015840(119905) gt 119902
1015840120575 minus (120573119898
lowast
119868+ 119887) 119878 (119905) (60)
Consider the comparison impulsive system for 119905 ge 1199050
1198831015840
1(119905) = 119902
1015840120575 minus (120573119898
lowast
119868+ 119887)119883 (119905) 119905 = 119896119879
1198831(119905+) = (1 minus 120579)1198831 (119905) 119905 = 119896119879
(61)
According to Lemma 6 there exists 1198791gt 1199050such that
119878 (119905) gt 119883lowast
1(119905) minus 120576
ge
1199021015840120575
120573119898lowast
119868+ 119887
(
(1 minus 120579) (1 minus 119890minus120573(119898
lowast
119868+119887)119879
)
1 minus (1 minus 120579) 119890minus120573(119898
lowast
119868+119887)119879
) minus 120576 = 120578
(62)
for all 1198791gt 1199050 The second equation of system (34) can be
translated into the following form
1198681015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
minus120573119890minus119887120591 119889
119889119905
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585
(63)
Define a function 119881(119905) such that
119881 (119905) = 119868 (119905) + 120573119890minus119887120591
int
119905
119905minus120591
119878119901(120585) 119868 (120585) 119889120585 (64)
then the derivative of 119881(119905) along the solution of system (34)is
1198811015840(119905) = (120573119878
119901(119905) 119890minus119887120591
minus 120575 + 119902120575 minus 120574 minus
119889
119879
) 119868 (119905)
= (120575 minus 119902120575 + 120574 +
119889
119879
)(
120573119878119901(119905) 119890minus119887120591
120575 minus 119902120575 + 120574 + (119889119879)
minus 1) 119868 (119905)
119905 gt 1198791
(65)
From (55) we obtain 1198811015840(119905) gt 0 119905 gt 1198791 which implies that
119881(119905) rarr infin 119905 rarr infin This is contrary to 119881(119905) lt 1 + 120573120591119890minus119887120591
Hence there exists a 1199051gt 0 such that 119868(119905
1) ge 119898
lowast
119868
Step 2 According to Step 1 for any positive solution(119878(119905) 119868(119905)) of system (34) we are left to consider two casesFirst if 119868(119905) gt 119898
lowast
119868for all 119905 gt 119905
1 then our aim is obtained
Second 119868(119905) oscillates about 119898lowast119868for all large 119905 In this case
setting 119905lowast = inf119905gt1199051
119868(119905) le 119898lowast
119868 there are two possible cases for
119905lowast
Define
119898119868= min
119898lowast
119868
2
1199021 119902
1= 119898lowast
119868119890minus(120575minus119902120575+120574+(119889119879))120591
(66)
We hope to show that 119868(119905) ge 119898119868for all large 119905 The conclusion
is evident in the first case For the second case let 119905lowast gt 0 and120588 gt 0 satisfy 119868(119905lowast) = 119868(119905lowast + 120588) = 119898lowast
119868 and 119868(119905) lt 119868lowast 119878(119905) gt 120578
for 119905lowast lt 119905 lt 119905lowast + 120588 Therefore it is certain that there exists a 119892(0 lt 119892 lt 120591) such that
119868 (119905) ge
119898lowast
119868
2
for 119905lowast lt 119905 lt 119905lowast + 119892 (67)
In this case we will discuss three possible cases in terms ofthe sizes of 119892 120588 and 120591
Case 1 If 120588 le 119892 lt 120591 then 119868(119905) ge (119898lowast1198682) for 119905lowast lt 119905 lt 119905lowast + 120588
Case 2 If 119892 le 120588 le 120591 then from the second equation of system(34) we can deduce 1198681015840(119905) gt minus(120575 minus 119902120575 + 120574 + (119889119879))119868(119905) for119905 isin [119905lowast 119905lowast+120591] and 119868(119905lowast) = 119898lowast
119868 and it is obvious that 119868(119905) ge 119902
1
for 119905lowast lt 119905 lt 119905lowast + 119892
Case 3 If 119892 le 119879 le 120588 we will consider the following two casesrespectivelySubcase 31 For 119905lowast lt 119905 lt 119905lowast + 120591 it is easy to obtain 119868(119905) gt 119902
1
Subcase 32For 119905lowast+120591 lt 119905 lt 119905lowast+120588 it is easy to obtain 119868(119905) gt 1199021
Then proceeding exactly as the proof for the above claim wesee that 119868(119905) ge 119898
119868for 119905lowast + 120591 lt 119905 lt 119905
lowast+ 120588 Since this kind of
interval [119905lowast 119905lowast+120588] is chosen in an arbitraryway (we only need119905lowast to be large) we conclude that 119868(119905) ge 119898
119868for all large 119905 in
the second case In view of our above discussions the choicesof 119898119868are independent of the positive solution and we have
proved that any positive solution of (34) satisfies 119868(119905) ge 119898119868
for all large 119905 The proof is completed
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 9
Set
120579lowast=
(119890119887119879minus 1) (1 minus ((119890
119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
)
((119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))) 120573)
1119901
+ (119890119887119879minus 1)
120591lowast=
1
119887
ln120573
119890119887120591(120575 minus 119902120575 + 120574 + (119889119879))
(
(1 minus 120579) (1 minus 119890minus119887119879
)
1 minus (1 minus 120579) 119890minus119887119879
)
119901
(68)
From Theorems 12 we also easily obtain the followingresults
Corollary 13 (i) If 120573119890minus119887120591 gt 120575minus119902120575+120574+(119889119879) then the diseasewill be endemic and system (34) is permanent provided that120579 lt 120579lowast
(ii) If 120573((1 minus 120579)(1 minus 119890minus119887119879)(1 minus (1 minus 120579)119890minus119887119879))119901 gt 120575 minus 119902120575 +
120574 + (119889119879) then the disease will be endemic and system (34) ispermanent provided that 120591 lt 120591
lowast
4 Numerical Simulations
In this section we present some numerical simulations todemonstrate our theoretical results established in this paper
Example 1 Letting 119887 = 05 119898 = 085 1198981015840 = 015 120591 = 05120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01 119879 = 2 119901 = 3we consider the pulse vaccination strategy (see Figure 1) theresult shows that the disease fades away when the proportionof those vaccinated successfully 120579 = 05 but the diseasewill exist everlasting when the proportion of those vaccinatedsuccessfully 120579 = 01 So this verifies the results in Corollaries9 and 13 for the epidemic disease with vertical transitionperiodical vaccination is an effective method to prevent thedisease Also it can be seen that with the increase of 120579 whichtypically causes oscillation bigger of the susceptible
Example 2 We use the same parameters as in Example 1except choosing 120579 = 01 and 120591 = 01 08 respectivelyCompare 120591 = 01 with 120591 = 08 is obviously that the longerof the latent period the lower of the infective number (seeFigure 2) and this shows disease with long latent perioddisadvantage to the spread of the disease
Example 3 For the same parameters as in Example 1 and take120579 isin [0 01] and 120591 isin [0 08] we obtain the number of infectedindividual (see Figure 3) when 119905 = 100 With the increaseof 120579 and 120591 the number of infected individual is decreasingso these results indicate that it will be helpful to control thedisease with vertical transition for bigger 120579 and 120591
5 Conclusion
In this paper for a class of epidemic disease with latentperiod and vertical transition we present two delayed SEIRepidemic models with nonlinear incidence rate based on thespread characters of the disease (such as tuberculosis) Ourmodel is more approach to the realistic problem which isdifferent from [17 19] Moreover the methods in our model
0 20 40 60 80 100t
120579 = 01
120579 = 05
02
03
04
05
06
07
08
09
S(t)
(a)
0 20 40 60 80 100t
120579 = 01
120579 = 05
0
005
01
015
02
025
03
035
04
045
05
I(t)
(b)
Figure 1 Dynamical behavior of the system (34) with 119887 = 05 119898 =
0851198981015840 = 015 120591 = 05 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120579 = 01 05 respectively (b) Time-series of the infective populationwith 120579 = 01 05
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
10 Computational and Mathematical Methods in Medicine
0 20 40 60 80 100t
03
035
04
045
05
055
06
065
07
075S(t)
120591 = 01
120591 = 08
(a)
0 20 40 60 80 100t
005
01
015
02
025
03
035
04
045
05
I(t)
120591 = 01
120591 = 08
(b)
Figure 2 Dynamical behavior of the system (34) with 119887 = 05119898 =
0851198981015840 = 015 120579 = 01 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 and 119901 = 3 (a) Time-series of the susceptible population with120591 = 01 08 respectively (b) Time-series of the infective populationwith 120591 = 01 08
006008
04
08
12
16
20
002004
01
0
005
01
015
02
025
03
035
120579
120591
I
Figure 3 The value of infectious individual with 119905 = 100 119887 = 05119898 = 085 1198981015840 = 015 120573 = 07 1199021015840 = 05 119902 = 05 120574 = 005 120575 = 01119879 = 2 119901 = 3 120579 isin [0 01] 120591 isin [0 08]
are different from the existing results becausemore factors areconsidered When only considering constant treatment weobtain basic reproductive number R
0and prove the global
stability by using the Lyapunov functional method For theSEIRmodel with pulse vaccination we also get the theoreticalresult ifRlowast lt 1 the disease-free periodic solution is globallyattractive and if R
lowastgt 1 the disease is permanent by using
the comparison theorem of impulsive differential equationBy some simulation experiments it clearly shows that thelarger of the proportion of those vaccinated successfully thelower of the infective individuals and the longer of the latentperiod the lower of the infective individuals So these resultsdemonstrate that it will be helpful to control the disease withvertical transition for bigger 120579 and 120591
Acknowledgment
Thework is supported by the National Natural Science Foun-dation of China (no 1124319)
References
[1] M E Alexander and S M Moghadas ldquoPeriodicity in an epid-emic model with a generalized non-linear incidencerdquo Mathe-matical Biosciences vol 189 no 1 pp 75ndash96 2004
[2] R M Anderson and R M May Eds Infectious Diseases ofHumans Dynamics and Control Oxford University Press NewYork NY USA 1991
[3] Y Jin W Wang and S Xiao ldquoAn SIRS model with a nonlinearincidence raterdquo Chaos Solitons and Fractals vol 34 no 5 pp1482ndash1497 2007
[4] A Korobeinikov ldquoLyapunov functions and global stability forSIR and SIRS epidemiological models with non-linear tran-smissionrdquo Bulletin of Mathematical Biology vol 68 no 3 pp615ndash626 2006
[5] L Nie Z Teng and A Torres ldquoDynamic analysis of anSIR epidemic model with state dependent pulse vaccinationrdquo
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 11
Nonlinear Analysis Real World Applications vol 13 no 4 pp1621ndash1629 2012
[6] X Meng and L S Chen ldquoThe dynamics of a new SIR epidemicmodel concerning pulse vaccination strategyrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 582ndash597 2008
[7] D Xiao and S Ruan ldquoGlobal analysis of an epidemic modelwith nonmonotone incidence raterdquo Mathematical Biosciencesvol 208 no 2 pp 419ndash429 2007
[8] A drsquoOnofrio ldquoStability properties of pulse vaccination strategyin SEIR epidemicmodelrdquoMathematical Biosciences vol 179 no1 pp 57ndash72 2002
[9] M A Safi and S M Garba ldquoGlobal stability analysis of SEIRmodel with holling type II incidence functionrdquo Computationaland Mathematical Methods in Medicine vol 2012 Article ID826052 8 pages 2012
[10] S Gao Z Teng and D Xie ldquoThe effects of pulse vaccinationon SEIRmodel with two time delaysrdquo Applied Mathematics andComputation vol 201 no 1-2 pp 282ndash292 2008
[11] G Li and Z Jin ldquoGlobal stability of a SEIR epidemicmodel withinfectious force in latent infected and immune periodrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1177ndash1184 2005
[12] S Gao L S Chen and Z Teng ldquoImpulsive vaccination of anSEIRSmodel with time delay and varying total population sizerdquoBulletin ofMathematical Biology vol 69 no 2 pp 731ndash745 2007
[13] R Xu and Z Ma ldquoGlobal stability of a delayed SEIRS epidemicmodel with saturation incidence raterdquoNonlinear Dynamics vol61 no 1-2 pp 229ndash239 2010
[14] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978
[15] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987
[16] W M Liu S A Levin and Y Iwasa ldquoInfluence of nonlinearincidence rates upon the behavior of SIRS epidemiologicalmodelsrdquo Journal of Mathematical Biology vol 23 no 2 pp 187ndash204 1985
[17] M Herrera et al ldquoModeling the spread of tuberculosis in semi-closed communitiesrdquo Computational and Mathematical Meth-ods in Medicine vol 2013 Article ID 648291 19 pages 2013
[18] A drsquoOnofrio ldquoOn pulse vaccination strategy in the SIR epi-demic model with vertical transmissionrdquo Applied MathematicsLetters vol 18 no 7 pp 729ndash732 2005
[19] Z Lu X Chi and L S Chen ldquoThe effect of constant and pulsevaccination on SIR epidemicmodel with horizontal and verticaltransmissionrdquo Mathematical and Computer Modelling vol 36no 9-10 pp 1039ndash1057 2002
[20] M Kgosimore and E M Lungu ldquoThe effects of vaccination andtreatment on the spread of HIVAIDSrdquo Journal of BiologicalSystems vol 12 no 4 pp 399ndash417 2004
[21] M C Boily and R M Anderson ldquoSexual contact patterns bet-weenmen and women and the spread of HIV-1 in urban centresin Africardquo IMA Journal of Mathematics Applied inMedicine andBiology vol 8 no 4 pp 221ndash247 1991
[22] A B Sabin ldquoMeasles killer of millions in developing countriesstrategy for rapid elimination and continuing controlrdquo Euro-pean Journal of Epidemiology vol 7 no 1 pp 1ndash22 1991
[23] C A de Quadros J K Andrus J Olive et al ldquoEradicationof poliomyelitis progress in the Americasrdquo Pediatric InfectiousDisease Journal vol 10 no 3 pp 222ndash229 1991
[24] M Ramsay N Gay E Miller et al ldquoThe epidemiology ofmeasles in England and Wales rationale for the 1994 nationalvaccination campaignrdquo Communicable Disease Report vol 4no 12 pp R141ndashR146 1994
[25] P Yongzhen L Shuping L Changguo and S Chen ldquoThe effectof constant and pulse vaccination on an SIR epidemic modelwith infectious periodrdquo Applied Mathematical Modelling vol35 no 8 pp 3866ndash3878 2011
[26] J Hale Theory of Functional Differential Equations SpringerHeidelberg Germany 1977
[27] Y Kuang Delay Differential Equation with Application in Popu-lation Dynamics Academic Press NewYork NY USA 1993
[28] Y N Xiao and L S Chen ldquoModeling and analysis of a predator-prey model with disease in the preyrdquoMathematical Biosciencesvol 171 no 1 pp 59ndash82 2001
[29] V Lakshmikantham D Bainov and P S Simeonov Theory ofImpulsive Differential Equations World Scientific Singapore1989
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom