fractional order sir model with constant population · authors fto, ska, ikd supervised the study...

British Journal of Mathematics & Computer Science

14(2): 1-12, 2016, Article no.BJMCS.23017

ISSN: 2231-0851

SCIENCEDOMAIN internationalwww.sciencedomain.org

Fractional Order SIR Model with Constant Population

Eric Okyere1∗, Francis Tabi Oduro2, Samuel Kwame Amponsah2,Isaac Kwame Dontwi2 and Nana Kena Frempong2

1Department of Basic Sceinces, University of Health and Allied Sciences, Ho, Ghana.2Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi,

Ghana.

Authors’ contributions

This work was carried out in collaboration between all authors. Author EO designed the study,wrote the Matlab codes, performed the numerical simulations and wrote the first and revised draft

of the manuscript. Authors FTO, SKA, IKD supervised the study and author NKF internallyreviewed the first draft of the manuscript. All authors read and approved the final manuscript.

Article Information

DOI: 10.9734/BJMCS/2016/23017Editor(s):

(1) Kai-Long Hsiao, Taiwan Shoufu University, Taiwan.Reviewers:

(1) Bhupesh K. Tripathi, University of Allahabad, India.(2) Sergio Adriani David, University of Sao Paulo, Brazil.

(3) Grienggrai Rajchakit, Maejo University, Thailand.(4) Bharat Raj Singh, Dr. A.P.J. Abdul Kalam Technical University, Lucknow, India.

Complete Peer review History: http://sciencedomain.org/review-history/13115

Received: 10th November 2015

Accepted: 14th January 2016

Short Research Article Published: 28th January 2016

Abstract

The main objective of this paper is to formulate an epidemiological model using fractional order

derivatives which has an advantage over the classical integer order models due to its memory

effect property. Our mathematical formulation of the non-integer order initial value problem

will be based on the famous fractional order Caputo derivative. We discuss and show the

existence of non-negative solutions of the mathematical model. We further investigate local

asymptotic stability analysis of model equilibria. Finally, numerical solutions are presented using

Adams-type predictor-corrector method to illustrate fractional model trajectories.

*Corresponding author: E-mail: [email protected];

www.sciencedomain.org

http://sciencedomain.org/review-history/13115

Okyere et al.; BJMCS 14(2), 1-12, 2016; Article no.BJMCS.23017

Keywords: Initial value problem; caputo derivatives; predictor-corrector method; asymptotic stability;model equilibria.

2010 Mathematics Subject Classification: 53C25; 83C05; 57N16.

1 Introduction

Mathematical modeling of dynamical systems or phenomena in science, economics, finance, engineer-ing and in particular infectious disease modeling using the classical integer order system of differentialequations has gained much attention over the past several years (see, e.g, [1]-[23]). However due tothe effective nature of fractional derivatives and integrals, several epidemiological models and othermodels in science and engineering have successful being formulated and analyzed (see, e.g, [24]-[44]).

Fractional order derivatives has an important characteristics called memory effect and this specialproperty do not exist in the classical derivatives. These derivatives are nonlocal opposed to thelocal behavior of integer derivatives. It implies the next state of a fractional system depends notonly upon its current state but also upon all of its historical states. According to Petras and Magin[45] ”it is well known that the state of many systems (biological, electrochemical, viscoelastic, etc.)at a given time depends on their configuration at previous times”.

The main objective of this paper is to formulate an epidemiological model using fractional orderderivatives which has an advantage over the classical integer order models due to its memory effectproperty. We will consider qualitative stability analysis for the model and finally perform numericalsimulations. This paper is organized as follows. In section 2, the initial value fractional orderproblem is formulated and discussed. We show the non-negativeness of model solutions in section3. In section 4, we establish existence of model equilibria and give a detailed local asymptoticstability analysis. Numerical solutions of the mathematical model are given in section 5. In section6, results and discussion are considered. We then end our paper with conclusion in the last section.

2 Model Derivation

This section deals with mathematical formulation of a non-integer order initial value problem usingfractional order derivatives. Fractional calculus is a powerful tool for mathematical modeling andindustrial applications. It has been applied in many areas of research such as science, economics andengineering. In fractional calculus there are many interesting definitions of fractional derivatives[see, e.g, 44, 46], but for this purpose we will apply the famous Caputo derivatives due its advantageon initial value problems.

Definition 2.1 (44, 46). Fractional integral of order α is defined by

Iαh(t) =1

Γ(α)

∫ t

0

h(x)

(t− x)1−αdx

for 0 < α < 1, t > 0.

Definition 2.2 ( 44, 46 ). Caputo fractional derivative is defined by

Dαh(t) =1

Γ(q − α)

∫ t

0

hq(x)

(t− x)α+1−qdx.

for q − 1 < α < q.

2


In 2000, Hethcote [47] conducted a comprehensive survey and reviews on disease modeling. Heformulated and analysed an SIR (Susceptible-Infected-Recovered) endemic model based on theassumption of constant population N with equal birth and death rates, µ. Dividing his initial valueproblem by N , he obtained the following scaled system.

ds

dt= µ− µs− βsi;

di

dt= βsi− (γ + µ)i;

dr

dt= γi− µr;

(2.1)

with non-negative initial conditions, the constant γ represent recovery rate, µ is the death rate, βis the contact rate, 1/µ is the average lifetime, 1/γ represent average infectious period, where s, iand r represent the fractions in the classes and s+ i+ r = 1, Hethcote [47].

Now if replace the integer derivatives in system (2.1) by Caputo derivatives of order α, then themodified system can be written as

Dαs = µ− µs− βsi;

Dαi = βsi− (γ + µ)i;

Dαr = γi− µr;

(2.2)

Since r(t) = 1− s(t)− i(t), we can neglect the third differential equation in system (2.2) to obtainthe following reduced fractional order model (2.3).

Dαs = µ− µs− βsi;

Dαi = βsi− (γ + µ)i;

(2.3)

with r(t) = 1− s(t)− i(t).

This type of mathematical formulation has been considered by many researchers [see, e.g, 25, 27,35,36, 37, 38, 48]. It is very clear that, there is mismatch of model dimensions in the fractional ordermodel (2.3), but this drawback of fractionalization has been addressed by Diethelm [49]. In hisresearch paper, he formulated a fractional order compartmental model for dengue fever infection.He demonstrated the effectiveness of the method using dengue fever epidemic data from Cape VerdeIslands. Sardar et al [50] proposed and analysed a similar model using the method by [49]. Theyfound out that for epidemic models with memory in both vector and host populations, simulationsresults with data gives better representation than integer order models.

Following the method by Diethelm [49], system (2.3) becomes

Dαs = µα − µαs− βαsi;

Dαi = βαsi− (γα + µα)i;

(2.4)

with r(t) = 1− s(t)− i(t)

3


It is now very obvious that the dimension on both sides of the system correspond to (time)−α.It is important to note that when the fractional order α → 1, the model problem (2.4) withr(t) = 1− s(t)− i(t), becomes the classical endemic model (2.1).

3 Non-negative Solutions

Let R2+ = {Y ∈ R2 : Y ≥ 0}, where Y = (s, i)T . We will apply the following Lemma in [51] to

show the theorem about the non-negative solutions of the model.

Lemma 3.1 (51). (Generalized Mean Value Theorem).Assume that h(x) ∈ C[a, b] and Dαh(x) ∈ C(a, b] for 0 < α ≤ 1, then we have

h(x) = h(a) +1

Γ(α)Dαh(ξ)(x− a)α

with a ≤ ξ ≤ x, ∀x ∈ (a, b].

Remark 3.1. Assume that h(x) ∈ C[a, b] and Dαh(x) ∈ C(a, b], for 0 < α ≤ 1. It follows fromLemma 3.1 that if Dαh(x) ≥ 0,∀x ∈ (a, b) then h(x) is non-decreasing ∀x ∈ [a, b] and if Dαh(x) ≤0,∀x ∈ (a, b), then h(x) is non-increasing ∀x ∈ [a, b].

Theorem 3.2. The fractional order initial value problem (2.4) has a unique solution and it remainsin R2

+.

Proof. By applying Theorem 3.1 and Remark 3.2 in [52], the uniqueness and existence of thesolution of model problem (2.4) in (0,∞) follows. The domain R2

+ for the model problem ispositively invariant, because

Dαs|s=0 = µα ≥ 0,

Dαi|i=0 = 0,

(3.1)

on each hyperplane bounding the non-negative orthant, the vector filed points into R2+.

4 Model Equilibria and Analysis

To determine model equilibria of the fractional order model (2.4), let{Dαs = 0,

Dαi = 0.(4.1)

Then P o = (1, 0) is the disease free equilibrium and P ∗ =(

1σ, µα(σ−1)

βα

)is the endemic equilibrium,

where σ = βα

γα+µα .

It is important to remark that, the disease free equilibrium point is where the infectives in themodel equals zero (i = 0) and the endemic point is the one with non-zero infectives (i = 0).

The Jacobian matrix J(P o) for the fractional order model (2.4) computed at P ois given by

J(P o) =

−µα −βα

0 βα − (γα + µα)

. (4.2)

4


Theorem 4.1. The equilibrium point P o of system (2.4) is locally asymptotically stable ifσ < 1 and unstable if σ > 1.

Proof. The equilibrium point P o is locally asymptotically if all the eigenvalues λi, i = 1, 2 of J(P o)satisfy the following condition ([25], [53]):

∣∣arg(λi)∣∣ > απ

2.

From the Jacobian matrix J(P o), it is clear that λ1 = −µα is negative and therefore if λ2 =βα − (γα + µα) < 0 ( βα

γα+µα < 1), then all the eigenvalues λi satisfy the condition∣∣arg(λi)

∣∣ > απ2.

Hence the disease-free equilibrium point is locally asymptotically stable if σ = βα

γα+µα < 1. On theother hand, if βα − (γα + µα) > 0, then P o becomes unstable.

Theorem 4.2. The equilibrium point P ∗ of the fractional order mode (2.4) is locally asymptoticallystable if σ > 1.

Proof. The Jacobian matrix J(P ∗) for the fractional order model (2.4) computed at P ∗ yields

J(P ∗) =

−µασ −βα

σ

µα(σ − 1) 0

(4.3)

Since the trace of J(P ∗) is less than zero (−σµα < 0) and the determinant, βαµα(σ−1)σ

> 0it follows that all the eigenvalues λi satisfy the condition

∣∣arg(λi)∣∣ > απ

2. Hence the endemic

equilibrium point is locally asymptotically stable. This conclusion is true since βα

γα+µα > 1 in the

endemic situation. Conversely, it becomes unstable when βα

γα+µα < 1.

5 Numerical Simulations

We solve the initial value fractional order problem using an efficient predictor-corrector numericalscheme for both linear and nonlinear fractional order differential equations developed by Diethelm[54, 55]. Several authors have applied this powerful discretization method to compute numericalsolutions of their mathematical models [see, e.g, , 33, 48, 49, 56].

The first step in this discretization scheme is to convert the fractional order endemic model (2.4)into system of integral equations:

s(t) = s(0) + Iα(µα − µαs− βαsi),

i(t) = i(0) + Iα(βαsi− (γα + µα)i).

(5.1)

and then apply the PECE (Predict, Evaluate, Correct, Evaluate) method.

5


6 Results and Discussion

The numerical solutions s(t) and i(t) are shown in Figs. 1 and 2 with fixed parameter values anddifferent values of α. Phase portraits are displayed in Figs. 3 and 4 with basic reproduction

(a) (b)

Fig. 1. Solutions of fractional order model with α = 1, 0.99, 0.95, 0.90, γ = 1/3, µ = 1/60,σ = 3.0

(a) (b)

(c) (d)

Fig. 2. Trajectories of fractional order model with α = 1, 0.99, 0.95, 0.90,γ = 1/3, µ = 1/60 σ = 3.0

6


numbers σ = 3 and σ = 0.5 respectively. When α = 1, the model problem (2.4) with r(t) =1− s(t)− i(t) is equivalent to the classical initial value problem (2.1). In all the plots generated, weconsidered four different values of α, α = 1, 0.99, 0.95 0.90. One can easily see that as the fractionalorder α increases the behaviour of the fractional order model (2.4) trajectories approaches that ofthe classical integer order model (2.1). The simulated results can be compared to those obtainedby Hethcote [47] and we strongly believe that, we have improved the dynamics of the SIR endemicmodel by using fractional derivatives of order α.

(a) (b)

(c) (d)

Fig. 3. Phase portrait for fractional order model with α = 1, 0.99, 0.95, 0.90,γ = 1/3, µ = 1/60, σ = 3.0

7


(a) (b)

(c) (d)

Fig. 4. Phase portrait for fractional order model with α = 1, 0.99, 0.95, 0.90,γ = 1/3, µ = 1/60, σ = 0.50

7 Conclusion

This research work considered fractional order generalization of an endemic model that was presentedand investigated by Hethcote [47] in an extensive review of disease modeling using deterministicdifferential equations. In this paper, we have established the existence of non-negative solutions ofthe mathematical model. We have again shown that the disease free equilibrium point is locallyasymptotically stable when the basic reproduction number σ < 1, otherwise, unstable. The endemicequilibrium was found to be locally asymptotically stable when σ > 1, otherwise unstable. We haveused Adams-type predictor-corrector to numerically demonstrate that fractional order models ofbiological systems can yield interesting results. The simulated results can be compared to thoseobtained by Hethcote [47] and we strongly believe that, we have improved the dynamics of the SIRendemic model by using nonlinear fractional derivatives of order α.

Acknowledgement

The authors are grateful to the referees for their careful reading, constructive criticisms, commentsand suggestions, which have helped us to improve this work significantly.

8


Competing Interests

The authors declare that no competing interests exist.

References

[1] Magal P, Ruan S. Susceptible-infectious-recovered models revisited: From the individual levelto the population level. Mathematical Biosciences. 2014;250:26-40.

[2] Duan X, Yuan S, Li X. Global stability of an SVIR model with age of vaccination. AppliedMathematics and Computation. 2014;226:528-540.

[3] Xue Y, Yuan X, Liu M. Global stability of a multi-group SEI model. Applied Mathematics andComputation. 2014;226:51-60.

[4] Hethcote HW, van den Driessche P. Two SIS epidemiologic models with delays. J. Math. Biol.2000;40:3-26.

[5] Huang W, Cooke KL, Castillo-Chavez C. Stability and bifurcation for a multiple-group model forthe dynamics of HIV/AIDS transmission. SIAM Journal on Applied Mathematics. 1992;52:835-854.

[6] J. Dushoff J, Huang W, Castillo-Chavez C. Backwards bifurcations and catastrophe in simplemodels of fatal diseases. Journal of Mathematical Biology. 1998;36:227-248.

[7] Castillo-Chavez C, Thieme HR. Asymptotically autonomous epidemic models. MathematicalPopulation Dynamics: Analysis of Heterogeneity. 1995;1:33-50.

[8] Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman JM. The basicreproductive number of Ebola and the effects of public health measures: The cases of Congoand Uganda. Journal of Theoretical Biology. 2004;229:119-126.

[9] Chitnis N, Cushing JM, Hyman JM. Bifurcation analysis of a mathematical model for malariatransmission. SIAM Journal on Applied Mathematics. 2006;67:24-45.

[10] Lawi GO, Mugisha JYT, Omolo-Ongati N. Mathematical model for malaria and meningitisco-infection among children. Applied Mathematical Sciences. 2011;5:2337-2359.

[11] Pongtavornpinyo W, Yeung S, Hastings IM, Dondorp AM, Day NPJ, White NJ. Spread of anti-malarial drug resistance: Mathematical model with implications for ACT drug policies. MalariaJournal. 2008;7:229.

[12] Okosun KO, Makinde OD. Optimal control analysis of malaria in the presence of non-linearincidence rate. Appl. Comput. Math. 2013;12:20-32.

[13] Koella JC, Anita R. Epidemiological models for the spread of anti-malarial resistance. MalariaJournal. 2003;2:3.

[14] Tumwiine J, Mugisha JYN, Luboobi LS. A host-vector model for malaria with infectiveimmigrants. J. Math. Anal. Appl. 2010;361:139-149.

[15] Tumwiine J, Mugisha JYN, Luboobi LS. A mathematical model for the dynamics of malariain a human host and mosquito vector with temporary immunity. Applied Mathematics andComputation. 2007;189:1953-1965.

9


[16] Mukandavire Z, Gumel AB, Garira W, Tchuenche JM. Mathematical analysis of a model forHIV-malaria co-infection. Math Biosci Eng. 2009;6:333-362.

[17] Barley K, Murillo D, Roudenko S, Tameru AM, Tatum S. A mathematical model of HIV andmalaria co-infection in Sub-Saharan Africa. Journal of AIDS Clinical Research. 2012;3:173. doi:10.4172/2155-6113.1000173.

[18] Brauer F, Sanchez DA. Constant rate population harvesting: Equilibrium and stability.Theoretical Population Biology. 1975;8(1):12-30.

[19] Chitnis N, Cushing JM, Hyman JM. Determining important parameters in the spread of malariathrough the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology.2008;70(5):1272-1296.

[20] Feng Z, Velasco-Hernandez JX. Competitive exclusion in a vector-host model for the denguefever. Journal of Mathematical Biology. 1997;35(5):523-544.

[21] Hethcote HW. Note on determining the limiting susceptible population in an epidemic model.Mathematical Biosciences. 1970;9:161163.

[22] Hethcote HW. Qualitative analyses of communicable disease models. Mathematical Biosciences.1976;28(3-4):335-356.

[23] Ngwa GA, Shu WS. A mathematical model for endemic malaria with variable human andmosquito populations. Mathematical and Computer Modelling. 2000;32(7-8):747-763.

[24] Gonzalez-Parra G, Arenas AJ, Chen-Charpentier B. A fractional order epidemic model for thesimulation of outbreaks of influenza A(H1N1). Mathematical Methods in the Applied Sciences.2014;37:2218-2226.

[25] Ahmed A, El-Sayed AMA, El-Saka HAA. Equilibrium points, stability and numerical solutionsof fractional-order predator-prey and rabies models. Journal of Mathematical Analysis andApplications. 2007;325:542-553.

[26] Liu W, Chen W. Chaotic behavior in a new fractional-order love triangle system withcompetition. Chaos, Solitons and Fractals. 2015;33:103-113.

[27] Area I, Batarfi H, Losada J, Nieto JJ, Shammakh M, Torres A. On a fractional order ebolaepidemic model. Advances in Difference Equations. 2015;278. DOI: 10.1186/s13662-015-0613-5.

[28] Arafa AAM, Rida SZ, Khalil M. Solution of fractional order model of childhood diseases withconstant vaccination strategy. Mathematical Sciences Letters. 2012;1:17-23.

[29] Liu Z, Lu P. Stability analysis for HIV infection of CD4+ T-cells by a fractional differentialtime-delay model with cure rate. Advances in Difference Equations 2014. 2014;298.

[30] El-Shahed M, Alsaedi A. The fractional SIRC model and influenza A. Mathematical Problemsin Engineering. 2011;9. Article ID 480378.

[31] Javidi M, Ahmad B. A study of fractional-order cholera mode. Appl. Math. Inf. Sci. 2014;8:2195-2206.

[32] Rocco A, West BJ. Fractional calculus and the evolution of fractional phenomena. Physica A:Statistical Mechanics and Its Applications. 1999;265(3-4):535-546.

10


[33] Ding Y, Ye H. A fractional-order differential equation model of HIV infection of CD4+T-cells.Mathematical and Computer Modelling. 2009;50(3-4):386-392.

[34] Ding Y, Wang Z, Ye H. Optimal control of a fractional-order HIV-immune system with memory.IEEE Trans. Contr. Sys. Techn. 2012;20(3):763-769.

[35] Ahmed E, El-Saka HA. On fractional order models for hepatitis C. Nonlinear BiomedicalPhysics. 2010;4:1-3.

[36] Arafa AAM, Rida SZ, Khalil M. A fractional-order model of HIV infection with drug therapyeffect. Journal of the Egyptian Mathematical Society. 2014;22:538-543.

[37] El-Saka HAA. Backward bifurcations in fractional-order vaccination models. Journal of theEgyptian Mathematical Society. 2015;23(1):49-55.

[38] Pinto CMA, Machado JAT. Fractional model for malaria transmission under control strategies.Computers and Mathematics with Applications. 2013;66(5):908-916.

[39] Ahmad WM, El-Khazali R. Fractional-order dynamical models of love. Chaos, Solitons andFractals. 2007;33(4):1367-1375.

[40] Liu Y, Lu P, Szanton I. Numerical analysis for a fractional differential time-delay model ofHIV infection of CD4 + T-cell proliferation under antiretroviral therapy. Abstract and AppliedAnalysis. 2014;Vol. 2014, Article ID 291614, 13 pages, doi:10.1155/2014/291614.

[41] Rihan FA, Baleanu D, Lakshmanan S, Rakkiyappan R. On fractional SIRC model withsalmonella bacterial infection. Computers and Mathematics with Applications. 2014;vol. 2014,Article ID 136263, 9 pages, doi:10.1155/2014/136263.

[42] Ye H, Ding Y. Nonlinear dynamics and chaos in a fractional-order HIV model. MathematicalProblems in Engineering. 2009; vol. 2009, Article ID 378614, 12 pages. doi:10.1155/2009/378614

[43] Gokdogan A, Yildirim A, Merdan M. Solving a fractional order model of HIV infection of CD4+T cells. Mathematical and Computer Modelling. 2011;54:21322138.

[44] Petras I. Fractional-order nonlinear systems: Modeling, analysis and simulation. Springer-Verlag; 2011.

[45] Petras I, Magin IR. Simulation of drug uptake in a two compartmental fractional model for abiological system. Commun Nonlinear Sci Numer Simul. 2011;16(12):4588-4595.

[46] Podlubny I. Fractional differential equations. Academic Press, New York; 1999.

[47] Hethcote HW. The mathematics of infectious diseases. Siam Review. 2000;42(4):599-653.

[48] Ozalp N, Demirci EV. Nonlinear dynamics and chaos in a fractional-order HIV model.Mathematical and Computer Modelling. 2011;54(1-2):1-6.

[49] Diethelm K. A fractional calculus based model for the simulation of an outbreak of denguefever. Nonlinear Dynamics. 2013;71(4):613-619.

[50] Sardar T, Rana S, Bhattacharya S, Al-Khaled K, Chattopadhyay J. A generic model for a singlestrain mosquito-transmitted disease with memory on the host and the vector. MathematicalBiosciences. 2015;263:18-36.

11


[51] Odibat ZM, Shawagfeh NT. Generalized Taylor’s formula. Applied Mathematics andComputation. 2007;186(1):286-293.

[52] Lin W. Global existence theory and chaos control of fractional differential equations. Journalof Mathematical Analysis and Applications. 2007;332(1):709-726.

[53] Matignon D. Stability results for fractional differential equations with applications to controlprocessing. Computational Engineering in Systems Applications. 1996;2:963-968.

[54] Diethelm K, Ford NJ, Freed AD. A predictor-corrector approach for the numerical solution offractional differential equations. Nonlinear Dynamics. 2002;29(1):3-22.

[55] Diethelm K, Ford NJ, Freed AD. Detailed error analysis for a fractional Adams method.Nonlinear Dynamics. 2004;36(1):31-52.

[56] Al-Sulami H, El-Shahed M, Nieto JJ, Shammakh W. On fractional order dengue epidemicmodel. Mathematical Problems in Engineering. 2014;vol. 2014, Article ID 456537, 6 pages.doi:10.1155/2014/456537.

—————————————————————————————————————————————-c⃝2016 Okyere et al.; This is an Open Access article distributed under the terms of the Creative CommonsAttribution License http://creativecommons.org/licenses/by/4.0, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Peer-review history:The peer review history for this paper can be accessed here (Please copy paste the total link in your browseraddress bar)http://sciencedomain.org/review-history/13115

12

http://creativecommons.org/licenses/by/4.0

fractional order sir model with constant population · authors fto, ska, ikd supervised the study...

Documents