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• ISSN 1 746-7233, England, UKWorld Journal of Modelling and Simulation

Vol. 9 (2013) No. 2, pp. 139-149

Numerical Behavior of a Fractional Order HIV/AIDS Epidemic Model

Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran

(Received April 18 2012, Revised September 30 2012, Accepted April 16 2013)

Abstract. In this paper, a fractional order HIV/AIDS (FOHA) epidemic model with treatment is investigated.The first step in the proposed procedure is represent the FOHA system as an equivalent system of ordinarydifferential equations. In the second step, we solved the system obtained in the first step by using the wellknown fourth order Runge-Kutta method. Numerical simulations are also presented to verify the obtainedresults. We solved the FOHA system for 0.8 < 1.

Keywords: HIV/AIDS epidemic model, fractional order, numerical method

1 Introduction

South Africa is currently experiencing a serious HIV/AIDS epidemic in history. Nearly, one in every fiveSouth Africans is infected with HIV, corresponding to approximately 5.7 million individuals. Mathematicalmodels have been used to model the dynamics of HIV/AIDS. Many researchers discussed on this models.Nyabadza , proposed a simple deterministic HIV/AIDS model incorporating sexual partner acquisition,behavior change and treatment as HIV/AIDS control strategies is formulated using a system of ordinary differ-ential equations with the object of applying it to the current South African situation. Samanta , considereda nonautonomous stage-structured HIV/AIDS epidemic model having two stages of the period of infectionaccording to the developing progress of infection before AIDS defined in, with varying total population sizeand distributed time delay to become infectious. The authors of , proposed a SIR mathematical modelof HIV transmission dynamics to explain the epidemiology of infectious diseases, and to assess the poten-tial benefits of proposed control strategies. A general SIR model is considered where there is no interventionand that resulted to an endemic situation. Also the same authors , examined and extended the HIV/AIDSmodel incorporating complacency by Flugentius et al for the adult population. Complacency is assumed afunction of the number of AIDS cases in a population with an inverse relation. Zunyou and coauthors ,reviews the epidemic of HIV infection and AIDS, the Chinese national policy development in response to theepidemic, and disparities between policies and the need for AIDS prevention in China. Dynamic models andcomputer simulations are experimental tools for comparing regions or risk groups, testing theories, assessingquantitative conjectures, and answering questions . Also see [2, 5, 7, 28, 33].

The authors of , reviews various mathematical models already proposed in the context of HIV trans-mission and the AIDS epidemic. Emphasis is placed on the various forms of HIV transmission models andthe assumptions under which the models were based. We also trace how transmission models evolved fromthe simplest population of homosexuals with homogeneity with respect to susceptibility, infectiousness andsexual mixing. Some HIV transmission dynamics models are stochastic, with probabilities of moving to thenext stage at each time step. Stochastic models assume that the response variables are a family of randomvariables indexed by time so that the HIV epidemic is a stochastic process .

Corresponding author. E-mail address: mo javidi@yahoo.com.

• 140 M. Javidi & N. Nyamorady: Numerical Behavior of a Fractional Order HIV/AIDS

AIDS has developed into a global pandemic since the first patients were identified in 1981. Since 1999,the year in which it is thought that the epidemic peaked, globally, the number of new infections has fallenby 19%. Of the estimated 15 million people living with HIV in middle-income countries who need treatmenttoday, 5.2 million have accesstranslating into fewer AIDS-related deaths. For the estimated 33.3 million peopleliving with HIV after nearly 30 years into a very complex epidemic, the gains are real but still fragile.

In 2009 alone, 1.2 million people received HIV antiretroviral therapy for the first time an increase inthe number of people receiving treatment of 30% in a single year. Overall, the number of people receivingtherapy has grown 13-fold, more than five million people in middle-income countries, since 2004. Expandingaccess to treatment has contributed to a 19% decline in deaths among people living with HIV between 2004and 2009. This is just the beginning: 10 million people living with HIV who are eligible for treatment underthe new WHO guidelines are still in need. Also, it is reported that 33.3 million people currently live withHIV-1 infection, 2.6 million people have been newly infected and 1.8 million AIDS deaths occurred in 2009(http://www.unaids.org/epi/2005/index.asp). It is well known that HIV mainly targets a hosts CD4+ T -cells,the main driver of the immune response. Chronic HIV infection causes gradual depletion of the CD4+ T -cellpool, and thus progressively compromises the hosts immune response, leading to humoral and cellular im-mune function loss (the marker of the on set of AIDS), making the host susceptible to opportunistic infections.The fact that HIV replicates rapidly, producing on average 1010 viral particles per day, led to the realizationthat HIV evolves so rapidly that treatment with a single drug is bound to fail . In a normal healthy in-dividuals peripheral blood, the level of CD4+ T -cells is between 800mm3 and 1200mm3 and once thisnumber reaches 200 or below in an HIV infected patient, the person is classified as having AIDS. Without drugtreatment, HIV-1 infection is nearly uniformly fatal within 5-10 years. With drug therapies, such as HAART(highly active antiretroviral therapy), treated individuals can live longer free of HIV-related symptoms .

Fractional differential equations have gained considerable importance due to their application in varioussciences, such as physics, mechanics, chemistry, engineering [6, 8, 17, 2024, 26]. In the recent years, the dynamicbehaviors of fractional-order differential systems have received increasing attention.

The existence of solutions of initial value problems for fractional order differential equations have beenstudied in the literature [14, 19, 31, 32] and the references therein.

In this paper, we first introduce a fractional order HIV/AIDS epidemic model with treatment. We ex-plained a numerical method to converting the system of fractional differential equations to system of ordinarydifferential equations. Finally, numerical simulations are presented to illustrate the obtained results.

2 Preliminaries

Definition 1. The Riemann-Liouville fractional integral operator of order > 0, of function f L1(R+) isdefined as

It0f(t) =1

()

tt0

(t s)1f(s)ds,

where () is the Euler gamma function.

Definition 2. The Riemann-Liouville fractional derivative of order > 0, n 1 < < n, n N is definedas

Dt0f(t) =1

(n )

( ddt

)n tt0

(t s)n1f(s)ds,

where the function f(t) have absolutely continuous derivatives up to order (n 1).

The initial value problem related to Definition 2 is{Dx(t) = f(t, x(t)),x(t)|t=0+ = x0,

(1)

WJMS email for contribution: submit@wjms.org.uk

• World Journal of Modelling and Simulation, Vol. 9 (2013) No. 2, pp. 139-149 141

where 0 < < 1 and D = D0 .In  the following results about the existence and uniqueness of solutions for Eq. (1) are further pre-

sented.

Theorem 1. Assume that < : [0, T ] [x0, x0 +] with some T > 0 and some > 0, and let the functionf : < R be continuous. Furthermore, define T := min

{T ,

( (+1)||f ||

) 1

}, then there exists a function

x : [0, T ] R solving the initial value problem (1). Notice that ||f || is the norm of function f .

Theorem 2. Assume that < : [0, T ] [x0, x0 +] with some T > 0 and some > 0, and let the functionf : < R be bounded on < and fulfill a Lipschitz condition with respect to the second variable, i.e.

|f(t, x) f(t, y)| L|x y|

with some constant number L > 0 independent of t, x, y. Then denoting T as Theorem 1, there exists at mostone function x : [0, T ] R solving the initial value problem Eq. (1).

Furthermore, the above definition in one dimension can naturally be generalized to the case of multipledimensions. That is, let X(t) = (x1(t), x2(t), , xn(t))T Rn and = (1, 2, , n)T Rn, 0 2 is satisfiedfor all eigenvalues of matrix A. Also, this system is stable if and only if | arg()| 2 is satisfied for alleigenvalues of matrixAwith those critical eigenvalues satisfying | arg()| = 2 having geometric multiplicityof one. The geometric multiplicity of an eigenvalue of the matrixA is the dimension of the subspace of vectorsv for which Av = v.

Theorem 4. (). Consider the following commensurate fractional-order system:

dx

dt= f(x), x(0) = x0, (6)

with 0 < 1 and x Rn. The equilibrium points of system Eq. (6) are calculated by solving the followingequation: f(x) = 0. These points are locally asymptotically stable if all eigenvalues i of the Jacobian matrixJ = fx evaluated at the equilibrium points satisfy: | arg(i)| >

2 .

According to [, Theorem 2], the basic reproduction number of system Eq.(3) is

R0 =cK(+ k2 + + bk1)(+ k1)(+ k2) +

. (7)

3.1 Equilibrium points and stability

To evaluate the equilibrium points of Eq. (3), let

dSdt

= 0,dIdt

= 0,dJdt