We give a consistent discretization of a continuous model of HIV infection, with distributed time delays to express the lag between the times when the virus enters a cell and when the cell becomes infected. The global stability of the steady states of the model is determined and numerical simulations are presented to illustrate our theoretical results.

1. Introduction

Nowadays, human immunodeficiency virus (HIV) that causes acquired immunodeficiency syndrome (AIDS) is a major health problem worldwide. From the World Health Organization (WHO), more than 35 million people are living with HIV/AIDS, and 1.6 million people died of this disease in 2012 [1]. Recent studies have been developed to know the dynamics of HIV infection, such as [2–9]. All these studies are based on continuous mathematical models. In reality, the statistical data are collected in discrete time, and the numerical simulations of continuous-time models are obtained by discretizing the models.

In this paper, we consider the model presented in [9] and we ignore the effect of the adaptive immune response. This model becomes as follows:
(1)x˙(t)=λ-dx(t)-βx(t)v(t),y˙(t)=β∫0θf(τ)e-mτx(t-τ)v(t-τ)dτ-ay(t),v˙(t)=ky(t)-μv(t),
where x(t), y(t), and v(t) denote the concentration of uninfected cells, infected cells, and free virus particles at time t, respectively. The uninfected cells are produced at a constant λ, die at a rate dx, and become infected by free virus at a rate βxv. Infected cells are lost at a rate ay. Free viruses are produced by infected cells at a rate ky and cleared at a rate μv. The authors [9] assumed that the uninfected cells are contacted by the virus particles at time t-τ and become infected cells at time t, where τ is a random variable with a probability distribution f(τ) over the interval [0,θ] and θ is limit superior of this delay. This probability distribution is assumed, for simplicity, to be a positive and integrable function on [0,θ], satisfying ∫0θf(τ)dτ=1. The term e-mτ is the probability of surviving from time t-τ to time t, where m is the death rate of infected but not yet virus-producing cells.

Using the result presented in [9], it is not hard to see that the basic reproduction number of system (1) is given by R0*=(λkβ/adμ)∫0θf(τ)e-mτdτ.

We recall that the number R0* is defined as the average number of secondary infections produced by one infected cell over its average life time, when all cells are uninfected.

In addition, the system (1) always has a disease-free equilibrium E0(λ/d,0,0) which is globally asymptotically stable if R0*≤1 and a unique endemic equilibrium E*((λ/d)(1/R0*),(μd/kβ)(R0*-1),(d/β)(R0*-1)) is globally asymptotically stable when R0*>1.

Motivated by the works [10–15] and that the statistical data are collected in discrete time, we propose the following discrete model obtained from (1) by using the rectangle method to approximate the integral and by applying the backward Euler discretization:
(2)xn+1-xnh=λ-dxn+1-βxn+1vn+1,yn+1-ynh=β∑i=0Nf(i)e-mix(n-i)v(n-i)-ayn+1,vn+1-vnh=kyn+1-μvn+1,
where N=θ-1 if θ is an integer, and if not, N is the integer part of θ (N=[θ]). The sequences xn, yn, and vn denote the concentration of uninfected cells, infected cells, and free virus particles at time n, respectively. The parameters in the system (2) are the same as those in (1). For simplicity, we may assume that ∑i=0Nf(i)=1. Similar to the continuous system (1), system (2) always has a disease-free equilibrium E0(λ/d,0,0) and an endemic equilibrium point E*((λ/d)(1/R0),(μd/kβ)(R0-1),(d/β)(R0-1)), where R0 is the basic reproduction number of (2) which is defined by
(3)R0=λkβadμ∑i=0Nf(i)e-mi.

The aim of this work is to show that the discretization scheme used in system (2) preserves the positivity and boundedness of solutions and the global stability of both equilibria for the continuous model (1). Therefore, this discretization is dynamically consistent which means that all of the critical, qualitative properties of the solutions to the system of differential equations should also be satisfied by the solutions of the discrete scheme.

The paper is organized as follows. Section 2 deals with positivity and boundedness of solutions. In Section 3, we discuss the global stability of the equilibria. The numerical simulations are presented in Section 4 and the paper ends with a conclusion in Section 5.

2. Positivity and Boundedness of Solutions

Model (2) describes the evolution of a cell population that the cell number is nonnegative and bounded. For these biological reasons, we assume that the initial data for system (2) satisfy
(4)x(s)≥0,y(s)≥0,v(s)≥0y(s)≥1,∀s=-N,-(N-1),…,0.

Proposition 1.

All solution of system (2) subject to condition (4) remains nonnegative and bounded.

Proof.

From (2), we have
(5)xn+1=λh+xn1+dh+βhvn+1yn+1=yn+hβ∑i=0Nf(i)e-mix(n-i)v(n-i)1+ahvn+1=vn+hkyn+11+hμ.
Hence, by recurrence and (4), we have yn nonnegative and, thereafter, vn and xn are nonnegative.

For the boundedness, we put Tn=emxn+yn+hβ∑i=0Nf(i)∑j=n-ine-m(n-j)x(j)v(j).

We have
(6)Tn+1-Tn=hem[λ-dxn+1-βxn+1vn+1]+h[β∑i=0Nf(i)e-mix(n-i)v(n-i)-ayn+1]+hβ∑i=0Nf(i)∑j=n+1-in+1e-m(n+1-j)x(j)v(j)-hβ∑i=0Nf(i)∑j=n-ine-m(n-j)x(j)v(j)=h[emλ-emdxn+1-ayn+1]+h[1-em]β∑i=0Nf(i)∑j=n+1-in+1e-m(n+1-j)x(j)v(j)≤h[λem-δTn+1],
with δ=inf{d,a,(em-1)/h}. Then,
(7)Tn+1≤(11+hδ)(n+1)T0+λemδ(1-(11+hδ)(n+1))≤T0+λemδ.
Then, xn and yn are bounded.

By the third equation of (2), we have vn+1=(1/(1+hμ))vn+(hk/(1+hμ))yn+1. Since yn is bounded, then there is M such that yn≤M, for all n∈ℕ. Then, vn+1≤(1/(1+hμ))vn+(hk/(1+hμ))M; hence, vn+1≤(1/(1+hμ))n+1v0+(kM/μ)[1-(1/(1+hμ))n+1]≤v0+(kM/μ), and then vn is bounded.

3. Global Stability

In this section, we will give the following main result that characterizes the global behavior of our model.

Theorem 2.

If R0≤1, then E0 is globally asymptotically stable.

R0>1; then E* is globally asymptotically stable.

Proof.

For (i), we consider the following sequence {Un}n=0+∞ defined by
(8)Un=Ax*g(xnx*)+yn+hβ∑i=0Nf(i)e-mi[∑j=0ix(n-j)v(n-j)]+akvn,
with A=∑i=0Nf(i)e-mi, x*=λ/d, and g(s)=s-1-ln(s).

It is clear that g(s)≥0 for any s>0 and g has the global minimum g(1)=0.

Consider
(9)Un+1-Un=Ax*[xn+1x*-xnx*+ln(xnxn+1)]+yn+1-yn+hβ∑i=0Nf(i)e-mi[x(n+1)v(n+1)-x(n-i)v(n-i)]+ak[vn+1-vn]≤A[(xn+1-xn)(1-x*xn+1)]+yn+1-yn+hβ∑i=0Nf(i)e-mi[x(n+1)v(n+1)-x(n-i)v(n-i)]+ak[vn+1-vn]≤hA(1-x*xn+1)[λ-dxn+1-βxn+1vn+1]+hβ∑i=0Nf(i)e-mix(n-i)v(n-i)-hayn+1+hβ∑i=0Nf(i)e-mi[x(n+1)v(n+1)-x(n-i)v(n-i)]+hak[kyn+1-μvn+1].
Since x*=λ/d, we have that
(10)Un+1-Un≤h[∑i=0Nf(i)e-mi[x(n-i)v(n-i)]-Axn+1(xn+1-x*)2-Aβxn+1vn+1+Aβx*vn+1+β∑i=0Nf(i)e-mix(n-i)v(n-i)-aμkvn+1+(β∑i=0Nf(i)e-mi)x(n+1)v(n+1)-β∑i=0Nf(i)e-mi[x(n-i)v(n-i)]]≤h[-Axn+1(xn+1-x*)2-(1-R0)aμkvn+1]≤0.
We consider the set S={(xn,yn,vn)∈ℝ+3/Un+1-Un=0}.

We have (xn,yn,vn)∈S⇒xn=x* and, by (2), we have yn=vn=0. By LaSalle’s invariance principle (see [16, Theorem 4.24]), we have E0 that is globally asymptotically stable.

For (ii), we consider the following sequence {wn}n=0+∞ defined by
(11)wn=x*g(xnx*)+1Ay*g(yny*)+βx*v*A×h∑i=0Nf(i)e-mi∑j=0ig(x(n-j)v(n-j)x*v*)+akAv*g(vnv*),
with x*=λ/dR0, y*=μd/kβ(R0-1), and v*=d/β(R0-1).

Consider
(12)wn+1-wn=x*[xn+1-xnx*+ln(xnxn+1)]+y*A[yn+1-yny*+ln(ynyn+1)]+βx*v*Ah∑i=0Nf(i)e-mi×∑j=0i(x(n+1-j)v(n+1-j)-x(n-j)v(n-j)x*v*)+βx*v*Ah×∑i=0Nf(i)e-mi∑j=0i(ln(x(n-j)v(n-j)x(n+1-j)v(n+1-j)))+av*kA[vn+1-vnv*+ln(vnvn+1)]≤[(xn+1-xn)(1-x*xn+1)]+1A[(yn+1-yn)(1-y*yn+1)]+βx*v*Ah×∑i=0Nf(i)e-mi(x(n+1)v(n+1)-x(n-i)v(n-i)x*v*)+βx*v*Ah×∑i=0Nf(i)e-mi∑j=0i(ln(x(n-j)v(n-j)x(n+1-j)v(n+1-j)))+akA[(vn+1-vn)(1-v*vn+1)]≤h[(β∑i=0Nf(i)e-mix(n-i)v(n-i)-ayn+1)[(λ-dxn+1-βxn+1vn+1)(1-x*xn+1)]+1A[(β∑i=0Nf(i)e-mix(n-i)v(n-i)-ayn+1)×(1-y*yn+1)(β∑i=0Nf(i)e-mix(n-i)v(n-i)-ayn+1)]+βxn+1vn+1-βx*v*A×∑i=0Nf(i)e-mi(x(n-i)v(n-i)x*v*)+βx*v*A∑i=0Nf(i)e-mi(ln(x(n-i)v(n-i)x(n+1)v(n+1)))+akA[(kyn+1-μvn+1)(1-v*vn+1)](β∑i=0Nf(i)e-mix(n-i)v(n-i)-ayn+1)].
Using the fact that λ=dx*+βx*v*, we have
(13)wn+1-wn≤h[akA[-kv*yn+1vn+1+μv*]-dxn+1(xn+1-x*)2+βx*v*-βx*v*x*xn+1-βy*Ayn+1∑i=0Nf(i)e-mix(n-i)v(n-i)+aAy*+βx*v*A∑i=0Nf(i)e-mi(ln(x(n-i)v(n-i)x(n+1)v(n+1)))+akA[-kv*yn+1vn+1+μv*]].

Using the relations ln(x(n-i)v(n-i)/x(n+1)v(n+1))=ln(x(n-i)v(n-i)y*/x*v*y(n+1))+ln(x*/x(n+1))+ln(v*y(n+1)/v(n+1)y*), βAx*v*=ay*, and ky*=μv*, we obtain
(14)wn+1-wn≤h[[v*y*yn+1vn+1-1-ln(v*y*yn+1vn+1)]-dxn+1(xn+1-x*)2-βx*v*×[x*xn+1-1-ln(x*xn+1)]-βx*v*A×∑i=0Nf(i)e-mi[x(n-i)v(n-i)y*x*v*yn+1-1-ln(x(n-i)v(n-i)y*x*v*yn+1)]-βx*v*[v*y*yn+1vn+1-1-ln(v*y*yn+1vn+1)]]≤h[∑i=0N-dvn+1(xn+1-x*)2-βx*v*g(x*xn+1)-βx*v*A∑i=0Nf(i)e-mig(x(n-i)v(n-i)y*x*v*yn+1)-βx*v*g(v*y*yn+1vn+1)∑i=0N]
since g(s)≥0 for any s>0; then wn+1-wn≤0.

We consider the set T={(xn,yn,vn)∈ℝ+3/wn+1-wn=0}.

We have (xn,yn,vn)∈T⇒xn=x* and, by (2), we have yn=y* and vn=v*. From LaSalle’s invariance principle, we deduce that E* is globally asymptotically stable.

4. Numerical Simulations

In this section, we present the numerical simulations to illustrate our theoretical results. In this section, we choose f(x)=2x/N(N+1). First, we use the following data set: λ=10, d=0.02, β=0.000024, m=0.5, a=0.5, k=600, μ=3, N=5, and h=0.1. In this case, the basic infection reproduction number R0 is 0.9483. By using Theorem 2 (i), we deduce that E0 is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 1).

Demonstration of the stability of E0.

In Figure 2, we choose β=0.00024 and do not change the other parameter values. By calculation, we have R0=9.4830 which satisfies the condition (ii) of Theorem 2. Hence, E* is globally asymptotically stable. Numerical simulation illustrates our result (see Figure 2).

Demonstration of the stability of E*.

In Figure 3, the parameter values are the same as those in Figure 1. Figure 3 gives R0 in function of N and shows that the growth of N decreases the value of R0 below 1, making the disease-free equilibrium globally asymptotically stable.

Plot of the basic reproduction number R0 as a function of the delay N. Here, λ=10, d=0.02, β=0.000024, m=0.5, a=0.5, k=600, μ=3, and h=0.1.

5. Conclusion

In this work, we have proposed a discrete mathematical model of HIV infection by applying the backward Euler discretization, with distributed time delay. We have proved that, when R0≤1, the disease-free equilibrium E0 is globally asymptotically stable. When R0>1, the endemic equilibrium E* is globally asymptotically stable. More precisely, it is proved that this discretization guarantees the correct dynamic behavior regardless of the size of the time step.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions that have led to improvement of the quality of this work.

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