A Positivity-preserving Mickens-typeDiscretization of an Epidemic Model
S.M. MOGHADASa,*, M.E. ALEXANDERb, B.D. CORBETTa and A.B. GUMELa
aDepartment of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2;bInstitute for Biodiagnostics, National Research Council Canada, Winnipeg, Manitoba,Canada, R3B 1Y6
(Received 4 February 2003; In final form 30 April 2003)
Dedicated to Professor Ronald E. Mickens on the occasion of his 60th Birthday
A deterministic model for the transmission dynamics of two strains of an epidemic in the presence of a preventivevaccine is considered. Theoretical results on the existence and stability of the associated equilibria of the model aregiven. A robust, positivity-preserving, non-standard finite-difference scheme, having the same qualitative features asthe continuous model, is constructed. The theoretical and numerical analyses of the model enable the determinationof a threshold level of vaccination coverage needed for community-wide eradication of the epidemic.
Keywords: Basic reproductive number; Mickens-type discretization; Epidemic model; Preventive vaccine
INTRODUCTION
Over the last few decades, scientists and public health officials have often resorted to the use
of mathematical models to gain deeper insights into the mechanisms of disease transmission
and to evaluate control strategies (see Refs. [1,7] and the references therein). The resulting
compartmental models, mostly deterministic and non-linear in nature, are generally solved
(simulated) using standard numerical integrators (such as explicit Runge-Kutta methods
[4,6,8]). These methods, however, are known to exhibit scheme-dependent instabilities
and/or converge to spurious solutions for certain values of the discretization and model
parameters [4,6,8].
In this paper, a robust Mickens-type finite-difference method will be constructed and used
to solve a deterministic model for the transmission dynamics of two strains of an epidemic in
the presence of a preventive vaccine. The model, which is a very slight modification of the
model in Ref. [7], consists of the following equations:
dS
dt¼ ð1 2 pÞP2 bwIwS 2 brIrS 2 mS; ð1Þ
dV
dt¼ pP2 bwð1 2 fwÞIwV 2 brð1 2 frÞIrV 2 mV; ð2Þ
ISSN 1023-6198 print/ISSN 1563-5120 online q 2003 Taylor & Francis Ltd
DOI: 10.1080/1023619031000146913
*Corresponding author. E-mail: [email protected]
Journal of Difference Equations and Applications,
Vol. 9, No. 11, November 2003, pp. 1037–1051
dIw
dt¼ ð1 2 qÞbwIwS þ bwð1 2 fwÞIwV 2 ðmþ gÞIw; ð3Þ
dIr
dt¼ qbwIwS þ brIrS þ brð1 2 frÞIrV 2 ðmþ gÞIr; ð4Þ
dR
dt¼ gIw þ gIr 2 mR; ð5Þ
where S, V, Iw, Ir and R represent the populations of susceptible, vaccinated, wild-type strain,
vaccine-resistant strain and recovered individuals, respectively. The model presented in
Ref. [7] monitors a constant total population size N ¼ S þ V þ Iw þ Ir þ R by assuming
equal net birth and death rates (mN). In this paper, the model Eqs. (1)–(5) considers a varying
total population containing a susceptible class generated via recruitment (either by birth or
immigration). Furthermore, unlike in Ref. [7] (where a proportion of susceptible individuals
were vaccinated), vaccination is administered to a fraction of recruited individuals (see Ref.
[3] and the references therein). The quantity P represents the recruitment of individuals
(considered susceptible) into the population; p is the coverage level of preventive vaccine;
bw and br are the transmission probabilities of infection from wild-type and vaccine-resistant
strains, respectively; fw and fr are the fractions of vaccinated individuals in whom vaccine
induces protection against wild-type and vaccine-resistant strains, respectively; m is the
natural death rate; and g is the recovery rate of infected individuals. The quantity q represents
the fraction of individuals infected by the wild-type strain who develop vaccine resistance
(mutated wild-type infections). Further details on the description of the model and its
associated parameters are given in Ref. [7]. Since the model monitors human populations, it is
assumed that all the state variables and parameters of the model are non-negative. Therefore,
we consider the model in the following space of parameter values and state variables:
D :S $ 0; V $ 0; Iw $ 0; Ir $ 0; R $ 0;
P [ Zþ; 0 # p;bw;br;m;fw;fr; q; g # 1:
(
Since the aim of this study is to construct a non-standard finite-difference scheme which
preserves the qualitative features of the model (e.g. stability, positivity, etc.), we shall start by,
first of all, investigating the existence and stability of the associated equilibria of the
continuous model. This is done in the second, third and fourth sections. A positivity-
preserving scheme is then constructed, analyzed and simulated in the fifth, sixth and seventh
sections. It should be emphasized that no theoretical analysis or numerical method for solving
Eqs. (1)–(5) is given in Ref. [7].
DISEASE-FREE EQUILIBRIUM (DFE)
In the absence of infection (where Iw ¼ Ir ¼ R ¼ 0Þ; the model has a unique disease-free
equilibrium (DFE) given by
DFE ¼ð1 2 pÞP
m;pP
m; 0; 0; 0
� �:
S.M. MOGHADAS et al.1038
The local stability of the DFE is established by evaluating the Jacobian of the model at the
DFE. This leads to the following eigenvalues l* ¼ 2m , 0 (of multiplicity 3) and
l1 ¼bw½ð1 2 qÞð1 2 pÞ þ ð1 2 fwÞp�P
m2 ðmþ gÞ; l2 ¼
brð1 2 pfrÞP
m2 ðmþ gÞ:
Thus, the DFE is locally asymptotically stable if l1 , 0 and l2 , 0: Letting,
Rw ¼bw½ð1 2 qÞð1 2 pÞ þ ð1 2 fwÞp�P
mðmþ gÞ; Rr ¼
brð1 2 pfrÞP
mðmþ gÞ;
and defining R0 ¼ max{Rw;Rr}; it follows that l1 and l2 are both negative if and only if
R0 , 1: This result is summarized below.
Lemma 2.1 The DFE is locally asymptotically stable if R0 , 1 and unstable if R0 . 1:
The threshold quantity R0 is defined as the basic reproductive number of infection [1].
Lemma 2.1 determines the long-term asymptotic behavior of the solutions of the model that
initiate in the basin of attraction of the DFE when R0 , 1: Consequently, it is instructive to
investigate the long-term behavior of solutions that initiate outside the basin of attraction of
the DFE when R0 , 1 (as follows):
Theorem 2.1 If R0 , 1; then the DFE of Eqs. (1)–(5) is globally asymptotically stable
on D.
Proof Adding Eqs. (1)–(5) gives ðdN=dtÞ ¼ P2 mN: Thus, N ! ðP=mÞ as t !1: Here,
V ¼ ðS;V ; Iw; Ir;RÞ : S þ V þ Iw þ Ir þ R #P
m
� �;
is a positively invariant region for the model (1)–(5). Consequently, every solution of the
model that initiates in the positive region {ðS;V ; Iw; Ir;RÞ : S;V ; Iw; Ir;R $ 0} remains
there and eventually enters V.
It follows from Eq. (1) that dS=dt # ð1 2 pÞP2 mS ( for S; Iw; Ir $ 0). Defining
S1 ¼t!1lim
u$tsup SðuÞ;
it is easy to see, using the Comparison Theorem (see Ref. [9], that a solution of the equation
ðds=dtÞ ¼ ð1 2 pÞP2 ms; with sð0Þ ¼ Sð0Þ $ 0 is an upper solution of S(t) (that is, sðtÞ $
SðtÞ for all t $ 0). Since sðtÞ! ð1 2 pÞP=m as t !1; it follows that for a given e . 0; there
exists a t0 . 0 such that SðtÞ # sðtÞ # ðð1 2 pÞPÞ=mþ e for t $ t0: Thus, S1 #
ðð1 2 pÞPÞ=mþ e : Letting e ! 0, we have
S1 #ð1 2 pÞP
m: ð6Þ
Similarly from Eq. (2), it can be seen that V 1 # ð pPÞ=m; where V 1 ¼ limt!1 supu$tVðuÞ:
Using the upper bounds of S(t) and V(t) for t $ t0 in Eq. (3) gives
dIw
dt#
bw½ð1 2 qÞð1 2 pÞ þ ð1 2 fwÞp�P
m2 ðmþ gÞ þ bw½ð1 2 qÞ þ ð1 2 fwÞ�e
� �Iw
¼ ðmþ gÞ ðRw 2 1Þ þbw½ð1 2 qÞ þ ð1 2 fwÞe�
ðmþ gÞ
� �Iw: ð7Þ
MICKENS-TYPE DISCRETIZATION 1039
The assumption R0 , 1 implies that Rw , 1 and Rr , 1: Thus, for sufficiently small e,
it follows that the coefficient Iw in inequality (7) is negative (since Rw , 1). Consequently,
Iw ! 0 as t !1:
Similarly, using the upper bounds of S(t), V(t) and Iw(t) ( for t $ t0 sufficiently large) in
Eq. (4) gives
dIr
dt#
bwqð1 2 pÞPe
mþ
brð1 2 pfrÞP
m2 ðmþ gÞ
� �Ir
¼bwqð1 2 pÞPe
mþ ðmþ gÞðRr 2 1ÞIr: ð8Þ
Since Rr , 1; it follows that for sufficiently small e, the coefficient of Ir in Eq. (8) is negative.
Hence, Ir ! 0 as t !1: Using Iw, Ir ! 0 in Eq. (5) implies that R ! 0 as t !1:
Furthermore, it follows from Eq. (1) that ( for t $ t0)
dS
dt$ ð1 2 pÞP2 ebwS 2 ebrS 2 mS:
Thus, using the Comparison Theorem, it can be shown that
S1 $ð1 2 pÞP
m; ð9Þ
where S1 ¼ limt!1 infu$t SðuÞ: Consequently, from Eqs. (6) and (9), it follows that
SðtÞ! ðð1 2 pÞPÞ=m as t !1: By using Iw, Ir ! 0 in Eq. (2), we have V1 $ ð pPÞ=m;
where V1 ¼ limt!1 infu$tVðuÞ: Hence VðtÞ! ð pPÞ=m; as t !1: In summary, we have
shown that if R0 , 1; then (S, V, Iw, Ir, R) ! DFE as t !1: Thus, the DFE is globally
asymptotically stable on D. A
Remark It should be noted, from the expression for Rw and Rr (written as functions of p)
that
R0
wð pÞ ¼bwðq 2 fwÞP
mðmþ gÞ; R
0
rð pÞ ¼2brfrP
mðmþ gÞ:
It follows that if q 2 fw , 0; then Rw is a decreasing function of p. Furthermore, it is clear
that Rr is always a decreasing function of p. It is easy to see that there are critical vaccination
thresholds
pw ¼bwð1 2 qÞP2 mðmþ gÞ
bwðfw 2 qÞP; pr ¼
brP2 mðmþ gÞ
brfrP;
for which Rwð pwÞ ¼ 1 and Rrð prÞ ¼ 1: Let pc ¼ max{pw; pr}: Therefore, for the case
q 2 fw , 0; it can be seen that if p . pc; then R0 , 1: Thus, the disease will be eradicated
from the community whenever the vaccination coverage level exceeds this threshold (that is,
p . pc).
BOUNDARY EQUILIBRIUM (BE)
In order to find the condition(s) for the existence of possible boundary equilibria (BE) of the
model Eqs. (1)–(5) where either Iw ¼ 0 or Ir ¼ 0; we consider G ¼ bwIw and H ¼ brIr as
S.M. MOGHADAS et al.1040
the expressions for the force of infection of the model (see Ref. [1]). Thus, at equilibrium,
Eqs. (1)–(5) become
S* ¼ð1 2 pÞP
mþ G* þ H*; ð10Þ
V* ¼pP
ð1 2 fwÞG* þ ð1 2 frÞH* þ m; ð11Þ
I*w ¼
ð1 2 qÞð1 2 pÞPG*
ðmþ gÞðmþ G* þ H* Þþ
ð1 2 fwÞpPG*
ðmþ gÞ ð1 2 fwÞG* þ ð1 2 frÞH* þ m� ; ð12Þ
I*r ¼
qð1 2 pÞG* þ ð1 2 pÞH*�
P
ðmþ gÞðmþ G* þ H* Þþ
ð1 2 frÞpPH*
ðmþ gÞ ð1 2 fwÞG* þ ð1 2 frÞH* þ m� ; ð13Þ
R* ¼1
m
g
bw
G* þg
br
H*
� �: ð14Þ
Substituting Eqs. (12) and (13) into the expressions for G and H, respectively, gives
G* ¼bwð1 2 qÞð1 2 pÞPG*
ðmþ gÞðmþ G* þ H* Þþ
bwð1 2 fwÞpPG*
ðmþ gÞ ð1 2 fwÞG* þ ð1 2 frÞH* þ m� ; ð15Þ
H* ¼br qð12 pÞG* þ ð12 pÞH*�
P
ðmþ gÞðmþG* þH*Þþ
brð12frÞpPH*
ðmþ gÞ ð12fwÞG* þ ð12frÞH* þm� : ð16Þ
The fixed points of Eqs. (15) and (16) give the equilibria of the model. Clearly,
ðG* ;H* Þ ¼ ð0; 0Þ is a fixed point of Eqs. (15) and (16) which corresponds to the DFE.
Substituting G* ¼ 0 (as a fixed point of Eq. (15)) into Eq. (16) gives
H* ¼brð1 2 pÞPH*
ðmþ gÞðmþ H* Þþ
brð1 2 frÞpPH*
ðmþ gÞ ð1 2 frÞH* þ m� : ð17Þ
When H* – 0; a positive fixed point of Eq. (17) is a positive solution of the equation
FðHÞ2 1 ¼ 0 where
FðHÞ ¼brð1 2 pÞP
ðmþ gÞðmþ HÞþ
brð1 2 frÞpP
ðmþ gÞ ð1 2 frÞH þ m� :
Let FðHÞ ¼ FðHÞ2 1: It is easy to see that Fð0Þ ¼ Rr 2 1: Thus, if Rr . 1 (so that
R0 . 1 and, consequently, DFE is unstable), then Fð0Þ . 0: Differentiating F(H) gives
F0ðHÞ ¼ 2brð1 2 pÞP
ðmþ gÞðmþ HÞ22
brð1 2 frÞ2pP
ðmþ gÞ ð1 2 frÞH þ m� 2
, 0:
This implies that F(H) is a decreasing function on (0,1). It can also be
seen that limH!1 FðHÞ ¼ 21: Therefore, F(H) has a unique positive root, namely H0.
MICKENS-TYPE DISCRETIZATION 1041
Thus, (0, H0) is a positive fixed point of Eq. (17). Substituting this fixed point into Eqs. (10)–
(14) gives a unique BE point of the model (in which Iw ¼ 0 but Ir – 0). This result is
summarized below.
Theorem 3.1 The model has a unique boundary equilibrium if Rr . 1 ð p , prÞ and no
boundary equilibrium if Rr # 1 ð p $ prÞ:
Remark It should be mentioned that since the BE cannot be expressed in closed form, its
stability using traditional methods (such as considering the characteristic equation of the
corresponding Jacobian) is generally difficult to establish.
ENDEMIC EQUILIBRIUM (EE)
Since, like the BE, the endemic equilibrium (EE) of the model (where all the state variables
of the model are positive) cannot be expressed in closed form, the determination of
conditions for its existence is generally very tedious. However, in this section, conditions for
the existence of a unique endemic equilibrium for a special case where bwð1 2 fwÞ ¼
brð1 2 frÞ are derived.
Multiplying Eq. (15) by H* and Eq. (16) by G* (and equating the respective right hand
sides) gives (for G* – 0 and H* – 0Þ :
G* ¼½bwð1 2 qÞ2 br�H*
qbr
ð18Þ
Thus, if bwð1 2 qÞ2 br . 0; then substituting Eq. (18) into Eq. (16) gives the following
quadratic equation:
ð1 2 qÞð1 2 fwÞðmþ gÞðbw 2 brÞ2
ðqbrÞ2
ðH* Þ2
2bw 2 br
qbr
nð1 2 qÞð1 2 fwÞbwP2 mðmþ gÞ ð1 2 fwÞ þ ð1 2 qÞ
� oH*
2 m2ðmþ gÞðRw 2 1Þ ¼ 0: ð19Þ
This equation has a unique positive solution (for H*) if Rw . 1 and no positive solution if
Rw # 1: Substituting the positive solution of Eq. (19) into Eq. (18) gives a unique fixed point
of Eqs. (15) and (16). It can also be shown that if bwð1 2 qÞ2 br , 0; then no positive fixed
point of Eqs. (15) and (16) exists. Therefore, we have the following theorem.
Theorem 4.1 Suppose bwð1 2 fwÞ ¼ brð1 2 frÞ and bwð1 2 qÞ2 br . 0: Then the
model has a unique EE if Rw . 1:
Remark It should be noted that if bwð1 2 fwÞ ¼ brð1 2 frÞ, then Rw . Rr: Since in this
case, the basic reproductive number of infection of the wild-type strain (Rw) exceeds that of
the vaccine resistant strain (Rr), the wild-type strain will always dominate the vaccine-
resistant strain. Furthermore, if Rr . 1; then it follows from Theorems 3.1 and 4.1 that the
model has a unique boundary equilibrium and a unique endemic equilibrium. Biologically-
speaking, this means (at the very least) that an epidemic of vaccine-resistant strain emerges.
S.M. MOGHADAS et al.1042
It is important to note that if Rw . 1 . Rr; then the model has no boundary equilibrium and
an epidemic of both strains occurs (see Theorem 4.1). Numerical simulations in the Section
“Simulation”, together with some of the theoretical results of this paper, allow us to propose
some conjectures on the existence and stability of the boundary and endemic equilibria of the
model (see Section “Conclusion”).
Remark Note that in the case R0 ¼ 1 ðRw ¼ Rr ¼ 1Þ; the DFE is the only equilibrium of the
model. Since V is a positively invariant region for the model, the global stability of the DFE
can be established by proving the nonexistence of certain types of solutions such as periodic
orbits, homoclinic orbits or polygons. The non-existence of such solutions may not be easy to
show, especially in high dimensional models such as Eqs. (1)–(5). For models of dimension
less than three, the Dulac criterion or Lemma 3.1 in Ref. [2] may be applied to prove the
nonexistence of these types of solutions.
MICKENS-TYPE NUMERICAL METHOD (NSFD)
To construct a Mickens-type numerical scheme that preserves the positivity property of the
model (see Refs. [6,8]), the derivatives in Eqs. (1)–(5) are approximated by their first-order
forward-difference approximations and the right-hand side functions are approximated in
ways which lead to the following implicit methods:
Snþ1 2 Sn
‘¼ ð1 2 pÞP2 bwIn
wSnþ1 2 brInr Snþ1 2 mSnþ1; ð20Þ
V nþ1 2 V n
‘¼ pP2 bwð1 2 fwÞI
nwV nþ1 2 brð1 2 frÞI
nr V nþ1 2 mV nþ1; ð21Þ
Inþ1w 2 In
w
‘¼ ð1 2 qÞbw 2In
w 2 Inþ1w
� Snþ1 þ bwð1 2 fwÞ 2In
w 2 Inþ1w
� V nþ1
2 ðmþ gÞInþ1w ; ð22Þ
Inþ1r 2 In
r
‘¼ qbwInþ1
w þ brð2Inr 2 Inþ1
r Þ�
Snþ1 þ brð1 2 frÞð2Inr 2 Inþ1
r ÞV nþ1
2 ðmþ gÞInþ1r ; ð23Þ
Rnþ1 2 Rn
‘¼ g Inþ1
w þ Inþ1r
� 2 mRnþ1; ð24Þ
where ‘ . 0 is the time-step. Note that, to ensure positivity, the Mickens non-local
approximations Inw ! 2In
w 2 Inþ1w and In
r ! 2Inr 2 Inþ1
r have been used for Inw and In
r in
Eqs. (22) and (23), respectively. Rearranging the implicit formulations in Eqs. (20)–(24)
gives, respectively,
Snþ1 ¼‘ð1 2 pÞPþ Sn
1 þ ‘ bwInw þ brI
nr þ m
� ; ð25Þ
V nþ1 ¼‘pPþ V n
1 þ ‘ bwð1 2 fwÞInw þ brð1 2 frÞI
nr þ m
� ; ð26Þ
MICKENS-TYPE DISCRETIZATION 1043
Inþ1w ¼
2‘bwð1 2 qÞSnþ1 þ 2‘bwð1 2 fwÞVnþ1 þ 1
� In
w
1 þ ‘ bwð1 2 qÞSnþ1 þ bwð1 2 fwÞV nþ1 þ mþ g� ; ð27Þ
Inþ1r ¼
Inr þ ‘bwqSnþ1Inþ1
w þ 2‘brSnþ1In
r þ 2‘brð1 2 frÞVnþ1In
r
1 þ ‘½brSnþ1 þ brð1 2 frÞV nþ1 þ mþ g�; ð28Þ
Rnþ1 ¼‘g Inþ1
w þ Inþ1r
� þ Rn
1 þ ‘m: ð29Þ
Since the right hand sides of Eqs. (25)–(29) are positive (for 0 # p; q, fw, fr # 1), it is clear
that for any positive initial data, the above non-standard scheme Eqs. (25)–(29) will give positive
solutions which are located in the feasible region. In other words, the scheme Eqs. (25)–(29)
satisfies the positivity property of Eqs. (1)–(5). Furthermore, it is worth mentioning
that although this numerical scheme is implicit by construction, Eqs. (25)–(29) enable the
solution of Eqs. (1)–(5) to be computed explicitly via a Gauss–Seidel-type sequential process
(by substituting Eqs. (25) and (26) into Eq. (27) to compute Inþ1w ; followed by the computation of
Inþ1r using Eqs. (25)–(27) and finally substituting Eqs. (27) and (28) into Eq. (29) to determine
R nþ1). It should be noted that integrating Eqs. (1)–(5) with explicit Runge-Kutta methods will
lead to discrete models that admit negative terms which may, consequently, give solution profiles
that exhibit scheme-dependent numerical instabilities (such asconvergence to spurious solutions)
for large values of step-size. However, these instabilities may generally be removed by using
sufficiently small step-sizes [3,4,6].
FIXED-POINT ANALYSIS OF THE NUMERICAL METHOD
The aim here is to check whether the numerical scheme Eqs. (25)–(29) has the same stability
property as the original model Eqs. (1)–(5). Since we have only the DFE in closed form, we
shall restrict the fixed point analysis to the DFE accordingly.
It is easy to see that the
DFE ¼ð1 2 pÞP
m;pP
m; 0; 0; 0
� �is a fixed point of the numerical scheme Eqs. (25)–(29). Furthermore, it can be seen, after
some tedious manipulations, that the Jacobian of Eqs. (25)–(29) at the DFE has eigenvalues
h1 ¼ h2 ¼ h3 ¼ ð1=ð1 þ ‘mÞÞ and
h 4 ¼mþ 2‘brð1 2 pfrÞP
mþ ‘ brð1 2 pfrÞPþ mðmþ gÞ� ;
h 5 ¼mþ 2‘bw ð1 2 pfwÞ2 qð1 2 pÞ
� P
mþ ‘bw ð1 2 pfwÞ2 qð1 2 pÞ�
Pþ ‘mðmþ gÞ:
Clearly, for ‘ . 0; jhij , 1 for i ¼ 1; 2; 3: Here, we shall show that if R0 , 1; then
jh4j , 1 and jh5j , 1: Consider h4 ¼ h4ð‘Þ as a function of ‘ for ‘ [ ð0;1Þ: It is easy to
see that lim‘!0h4ð‘Þ ¼ 1: Differentiating h4(‘) gives
dh4ð‘Þ
d‘¼
m brð1 2 pfrÞP2 mðmþ gÞ�
mþ ‘ brð1 2 pfrÞPþ mðmþ gÞ� � 2
:
S.M. MOGHADAS et al.1044
Since R0 , 1 (and, consequently, Rr , 1), it follows that dh4ð‘Þ=d‘Þ , 0 for all ‘ . 0:
Thus, h4(‘) is a decreasing function on (0,1). Noting that h4ð‘Þ . 0 for all ‘ . 0; it follows
then that h4ð‘Þ [ ð0; 1Þ for ‘ . 0: Similarly, it can be shown that since R0 , 1 ðRw , 1Þ;
jh5ð‘Þj , 1 for all ‘ . 0: Therefore, we have established the following theorem.
Theorem 6.1 If R0 , 1; then the numerical scheme (25)–(29) will converge to the DFE
for every ‘ . 0:
This theorem is of public health significance since epidemiological models, such as
Eqs. (1)–(5), are generally monitored over a long-term period. Therefore, it is desirable to
construct numerical methods, such as Eqs. (25)–(29), that preserve the physical properties of
the model for large step-sizes. Theorem 6.1 shows that the numerical method Eqs. (25)–(29)
will always converge to the correct steady-state solution (DFE), irrespective of the step-size
used in the simulations, whenever R0 , 1:
It is worth mentioning that in order to remove the scheme-dependent instabilities that often
arise when standard numerical integrators, such as the RK methods (whether adaptive or
otherwise), are used to solve non-linear initial-value problems (such as Eqs. (1)–(5)), these
schemes must be simulated with sufficiently small step-sizes [3,4,6,8]. Consequently, the
extra computing cost associated with examining the long-term behaviour of the dynamical
system (model) being investigated may be substantial.
SIMULATIONS
Experiment 1. In order to illustrate the theoretical results of the paper, the model Eqs. (1)–(5)
was simulated using the numerical scheme Eqs. (25)–(29) constructed in Section “Mickens-
type numerical method”. In the first part of the simulations, the following parameter values
(estimated in Ref. [7]) were used: P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009; m ¼ 0:02; g ¼ 26;
fw ¼ 0:95; fr ¼ 0:5; q ¼ 0:0001: With these parameter values, the critical vaccination rates
are pw ¼ 0:863 and pr ¼ 0:843 (note that pw . pr). In this case, the critical vaccination rate
needed for community-wide eradication of the disease is pc ¼ max{pw; pr} ¼ 0:863:
The numerical method was then simulated using an arbitrarily-chosen initial condition
X0 ¼ ð1000; 500; 5; 5; 10Þ [ D with ‘ ¼ 0:1 and values of vaccination coverage level ( p)
chosen for three scenarios namely: p , pr; p . pw and pr , p , pw: The simulation results
obtained are tabulated in Table I. This table shows which equilibrium point is reached
for different values of p. Figures 1 and 2 depict the profiles of the model variables for
p ¼ 0:8 , pc and p ¼ 0:87 . pc; respectively. In the case p ¼ 0:8 , pr; the unique endemic
TABLE I Numerical results generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009;m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:5; q ¼ 0:0001; X0 ¼ ð1000; 500; 5; 5; 10Þ and various values of p
P Rw Rr Equilibrium solutions Comments
0.8 1.34 1.04 DFE ¼ ð10000; 40000; 0; 0; 0Þ EE is stableBE ¼ ð9463; 38896; 0; 1; 1639ÞEE ¼ ð7015; 39164; 3; 0:003; 3818Þ
0.844 1.10 0.99 DFE ¼ ð7800; 42200; 0; 0; 0Þ EE is stableEE ¼ ð6788; 41917; 1; 0:002; 1205Þ
0.87 0.97 0.98 DFE ¼ ð6500; 43500; 0; 0; 0Þ DFE is stable
MICKENS-TYPE DISCRETIZATION 1045
FIG
UR
E1
Pro
file
sof
the
model
var
iable
sg
ener
ated
usi
ng
NS
FD
wit
hl¼
0:1;P
¼1
00
0;b
w¼
0:0
02
9;b
r¼
0:0
00
9;m¼
0:0
2;g¼
26;f
w¼
0:9
5;f
r¼
0:5;
q¼
0:0
00
1;
p¼
0:8
ðp,
prÞ
and
X0¼
ð10
00;5
00;5;5;1
0Þ.
S.M. MOGHADAS et al.1046
equilibrium is stable (boundary equilibrium exists but is unstable). Further simulations for
the case pr , p ¼ 0:844 , pw show similar profiles to those obtained in Fig. 1. In this case
ð p ¼ 0:844Þ; the model has no boundary equilibrium. However, when p was increased to
p ¼ 0:87 . pc; the model has only the DFE which is stable (see Fig. 2). Thus, with this
choice of parameter values, community-wide eradication of the epidemic is feasible if at least
87% of the recruited individuals are vaccinated. This is in line with Theorem 2.1.
Experiment 2. Here, the parameter values in Experiment 1 were used with exception of fr
which is now decreased to fr ¼ 0:44: In this case, pw ¼ 0:863 , pr ¼ 0:958 and the critical
FIGURE 2 Profiles of the model variables generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029;br ¼ 0:0009; m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:5; q ¼ 0:0001; p ¼ 0:87 ð pw , pÞ and X0 ¼ð1000; 500; 5; 5; 10Þ:
MICKENS-TYPE DISCRETIZATION 1047
vaccination rate for disease eradication is pc ¼ 0:958: Numerical results for various values of p
are tabulated in Table II and Figs. 3 and 4 (for p ¼ 0:8 and p ¼ 0:9; respectively). Figures 3 and 4
show that if p , pw ð pw , p , prÞ; the unique EE (the unique BE) is stable. However, by
increasing p to p ¼ 0:96 . pc; the DFE becomes stable (see also Fig. 2). Note that, in this case,
the DFE is the only existing equilibrium of the model. Thus, here, the vaccination coverage level
must be at least 96% to ensure community-wide eradication. In other words, the reduced value of
fr (vaccine protection) has, expectedly, led to an increase in the threshold vaccination coverage
level needed for eradication (from 87% whenfr ¼ 0:5 to 96% whenfr ¼ 0:44). Overall, these
simulations are consistent with the theoretical results in Theorems 2.1 and 3.1.
The effect of step-size (‘) on the convergence of the Mickens-type numerical method
was monitored by simulating the method with large step-sizes (such as ‘ ¼ 10; 103, 106)
for the case R0 , 1 ð p . pcÞ: It was observed that, in this case, the method always
converges to the DFE (consistent with Theorem 6.1). It should be mentioned that although
the use of large step-size in numerical computations is desirable (for monitoring long-term
dynamics), the transient behaviour of the dynamical systems may not always be captured
for large ‘.
For comparison purposes, the model was simulated using RK45 method (the Dormand-
Prince pair formula, relative error tolerance 1023 and absolute error tolerance 1026 [5])
with the parameter values used in Experiments 1 and 2 and various values of the
vaccination coverage ( p). With the parameter values used in Experiment 1, the RK45
method gave profiles that converge to the correct steady-state solution for p ¼ 0:8; 0.844,
0.87. Thus, for this choice of parameter values, the RK45 method gave results that are
consistent with those obtained using the Mickens-type numerical method (tabulated in
Table I). However, the RK45 method fails to converge to the correct stable equilibrium
(BE) when the parameter values in Experiment 2 were used. The simulation results show
that the RK45, unlike the Mickens-type numerical method, does not converge to the BE
when 0:863 ¼ pw , p , pr ¼ 0:958: For example, the profiles of Iw, Ir and R for p ¼ 0:9;
depicted in Fig. 5, reveal that the RK45 fails not only to converge to the BE, but also to
preserve the positivity property of the model. Thus, for this choice of parameter values, the
RK45, unlike the Mickens-type method (see Table II), gave profiles that exhibit scheme-
dependent instabilities involving negative values (see Fig. 5). For population models, such
as Eqs. (1)–(5), these negative profiles are, of course, biologically unrealistic. It should be
noted that the theoretical results in section “Disease-free Equilibrium” show that
TABLE II Numerical results generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009;m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; X0 ¼ ð1000; 500; 5; 5; 10Þ and various values of p
P Rw Rr Equilibrium solutions Comments
0.8 1.33 1.12 DFE ¼ ð10000; 40000; 0; 0; 0Þ EE is stableBE ¼ ð8511; 36430; 0; 4; 5056ÞEE ¼ ð7022; 39020; 3; 0:17; 3955Þ
0.85 1.07 1.08 DFE ¼ ð7500; 42500; 0; 0; 0Þ BE is stableBE ¼ ð6667; 39721; 0; 1; 3609Þ
0.9 0.8 1.04 DFE ¼ ð5000; 45000; 0; 0; 0Þ BE is stableBE ¼ ð4670; 43288; 0; 2; 2041Þ
0.96 0.49 0.99 DFE ¼ ð2000; 48000; 0; 0; 0Þ DFE is stable
S.M. MOGHADAS et al.1048
the feasible region V is positively invariant. Thus, any solution with initial condition in V
(such as X0) must remain there for all t . 0:
CONCLUSION
A Mickens-type finite-difference method was constructed and used to solve an epidemic
model for the transmission dynamics of two strains of a childhood disease. Critical
FIGURE 3 Profiles of the model variables generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029;br ¼ 0:0009; m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; p ¼ 0:8 ðp , pwÞ andX0 ¼ ð1000; 500; 5; 5; 10Þ:
MICKENS-TYPE DISCRETIZATION 1049
vaccination rates needed for community-wide eradication of the disease were obtained
for different scenarios. Theoretical results, together with numerical simulations of
the Mickens-type numerical method, confirm that:
(1) this method always converges to the correct steady-state solution;
(2) this method always preserves the positivity property of the model.
FIGURE 4 Profiles of the model variables generated using NSFD with l ¼ 0:1; P ¼ 1000; bw ¼ 0:0029;br ¼ 0:0009; m ¼ 0:02; g ¼ 26; fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; p ¼ 0:9 ð pw , p , prÞ andX0 ¼ ð1000; 500; 5; 5; 10Þ:
S.M. MOGHADAS et al.1050
Simulation results using the RK45 method reveal that:
(1) the RK45 method does not always converge to the correct steady-state solution.
(2) the RK45 method does not always preserve the positivity property of the model.
Based on the theoretical and numerical results of this paper, the following conjectures are
proposed:
(i) The model has an unique EE if Rw . Rr and Rw . 1; and no endemic equilibrium if
Rw # Rr: If such an equilibrium exists, then it is globally asymptotically stable.
(ii) The unique boundary equilibrium is globally asymptotically stable if Rw # Rr and
Rr . 1:
Acknowledgements
This work was supported in part by the Natural Sciences and Engineering Research Council
of Canada (NSERC). The authors are grateful to the referees for their comments which have
improved the paper.
References
[1] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, London/New York,1991.
[2] S. Busenburg and P. Van den Driessche, Analysis of a disease transmission model in a population with varyingsize, J. Math. Biol., 28 (1990), 257–270.
[3] A.B. Gumel, R.E. Mickens, B.D. Corbett, A non-standard finite-difference scheme for a model of HIVtransmission and control, J. Comput. Methods Sci. Eng., To appear.
[4] J. D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, Wiley,Chichester, UK, 1991.
[5] Matlab, The language of Technical Computing, by The MathWorks, Inc., 1997.[6] R. E. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comp.
Appl. Math., 110 (1999), 181–185.[7] A. Scherer and A. R. McLean, Mathematical models of vaccination, Brit. Med. Bull., 62 (2002), 187–199.[8] A. Serfaty de Markus and R. E. Mickens, Suppression of numerically induced chaos with nonstandard finite
difference schemes, J. Comp. Appl. Math., 106 (1999), 317–324.[9] G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd Ed., McGraw-Hill 1991.
FIGURE 5 Profiles of Ir and R generated by RK45 with P ¼ 1000; bw ¼ 0:0029; br ¼ 0:0009; m ¼ 0:02; g ¼ 26;fw ¼ 0:95; fr ¼ 0:44; q ¼ 0:0001; p ¼ 0:9 ð pw , p , prÞ and X0 ¼ ð1000; 500; 5; 5; 10Þ:
MICKENS-TYPE DISCRETIZATION 1051