MATHEMATICAL ANALYSIS OF HIV INFECTION OF CD4+ T-CELLS WITH DISCRETE DELAYS

In this study, we introduce a discrete time to the model to describe the time delays between infection of a CD4+ T-cells, and the emission of viral particles on a cellular level. We begin by determining the existence and stability of the equilibrium. Further We investigate the global stability of the infection-free equilibrium and give sufficient condition for the local stability of the infected steady state is asymptotically stable for all delays. Finally, the numerical simulations are presented to illustrate the analytical results.


INTRODUCTION
In recent years, there has been a lot of interest in mathematical modelling of HIV/AIDS infection, in order to predict the evolution of this modern plague. Since the discovery of the human immunodeficiency virus type 1 (HIV-1) in the early 1980s, the disease has spread in successive waves to most regions around the globe. It is reported that HIV has infected more than 60 million people, and over a third of them subsequently died [1]. Considerable scientific effort has been devoted to the understanding of viral pathogenesis, host/virus interactions, immune response to infection, and antiretroviral therapy [2]. HIV primarily attacks a host's CD4 + T − cells (the main driver of the immune response). The amount of viruses in the blood is a good predictor of the stage of the disease. The amount of CD4 + T − cells in a typical healthy person's peripheral blood ranges between 800/mm3 and 1200/mm3. When this value falls below 200, an HIV-positive patient is diagnosed with Acquired Immune Deficiency Syndrome (AIDS). HIV differs from most viruses in that it is a retrovirus: Viruses do not have the ability to reproduce independently, and they must be rely on a host to aid reproduction. Most viruses carry copies uninfected healthy CD4 + T − cells, latently infected CD4 + T − cells, actively infected CD4 + T − cells and free virus [4]. This model has been important in the field of mathematical modelling of HIV infection and many other models have been proposed which take the model of Perelson, Krischner and De Boer [3].
In [5] Liming Cai have studied stability properties for delay differential equations and applied the results obtained   to analyze the stability of the equilibria for the model of HIV-1 infection. To our knowledge, no works are contributed to the analysis for HIV infection of CD4 + T − cells with two independent delays or two proportional delay terms. Motivated by this situation, we introduce a HIV infection model with independent time delays proposed by Culshaw and Ruan [6]. Here τ 1 and τ 2 are two time delays were included in our model. The first delay "τ 1 " is the time between viral entry latent infection. The second delay "τ 2 " is the time between cell infection and viral production.
So, we assume that CD4+ T cells (healthy and infected) are governed by a full Logistic growth term. Therefore, we shall establish a mathematical model as follows where τ 1 and τ 2 are positive.
where T(t) represents the concentration of healthy CD4 + T − cells at time t, I(t) represents the concentration of infected CD4 + T − cells and V(t) represents the concentration of free HIV at time t. To explain the parameter, we note that s is the source of CD4 + T − cells from precursors, r is their growth rate of T-cells(thus , r > µ 1 in general) and T max is the maximum level of CD4 + Tcells concentration in the body. The parameter k represents the rate of infection of T-cells with free virus and so is given as a loss term for both healthy cells and virus, since they are both lost by binding to one another, and is the source term for infected cells. µ i (i = 1, 2, 3) are the nature death rates of the uninfected T-cells, infected T-cells and the virus particles, respectively. It is reasonable to assume that µ 1 ≤ µ 2 , i.e., the infected T cells have a relatively shorter life than the uninfected T cells due to an HIV viral burden. N is number of virus produced by infected CD4 + T-cells during its lifetime. It is clear that according to the viral life cycle. We assume that all parameters are non-negative constant.
The organization of this paper is as follows. In the next section, we verify the boundedness of the solutions and existence of feasible equilibria of the system (1). In Section 3 , the local asymptotic stability of feasible equilibria is established. In Section 4 , we investigate the global asymptotic stability of feasible equilibria. We also performed numerical simulation to illustrate the main analytical result in the section 5. The paper ends with a conclusion.

BOUNDEDNESS OF SOLUTIONS AND FEASIBLE EQUILIBRIA
In the following, we first show all solution of the system (1) with (2) are positive and ultimately bounded. Proof: First, let us prove the positivity by contradiction.
By the definition of t 1 , this is a contradiction.Therefore, I(t) > 0 for all t > 0. Similarly, we easily show that V(t) is always positive.Thus, we can conclude that all solutions of system (1) with initial conditions (2) remain positive for all t > 0.
Next, we shall discuss the boundedness of solutions of the system (1). In the absence of HIV infection, the dynamics of healthy CD4 + T-cells are governed bẏ It can be shown that, the CD4 + T-cells concentration stabilizes at a level T 0 , which is given by and T 0 satisfy the following equation, By the first equation of system (1),we havė Let M 1 = k s + rT max 2 and solving equation (5), we obtain According to inequality (6), we get W (t) < 2M 1 µ 1 for sufficiently large t. Recall that T (t) > 0 and I(t) > 0 combining with inequality (4), T(t) and I(t) have ultimately above bound M 2 > 0.
Similiary, the third equation of system (1), we have solving inequality (7), we have It follows from inequality (8), that V(t) has an ultimately above bound M 3 > 0,for sufficiently large t. Hence we proved that all solutions of system (1) are ultimately bounded. Then the proof of Theorem 3.1 is completed. Let then Ω is the positive invariant set of system (1).
Next,we shall investigate the existence of equilibrium of system (1). The equilibrium of system (1) satisfy the following equation Clearly, the system (1) has always the infection free equilibrium E 0 (T 0 , 0, 0). From the third equation of (9), we have Substituting this expression into the second equation of (9) and solving for T results in, Rewriting the first equation of (9) as substituting (10) and (11) into (12), we obtain The critical number N crit is defined by, It is easy to verify that the equation s = (A + BV )(C + DV ) has a unique positive root if and if only if N > N crit . Thus, we also obtain

LOCAL STABILITY ANALYSIS
In this section, we study the local stability of the infection -free equilibrium and the infected equilibrium points.
Theorem 3.2: If N ≤ N crit , then system (1) has only the uninfected equilibrium E 0 (T 0 , 0, 0); if N > N crit , the system (1) has the two equilibria; the infected free equilibria E 0 and the chronic infection equilibrium E * (T * , I * ,V * ).

Proof:
Let E * (T * , I * ,V * ) be an arbitrary equilibrium. Thus, linerarizing the system (1) at the equilibrium E * (T * , I * ,V * ), we obtain the charateristic equation about E * as follows where, Thus, for the uninfected equilibrium E 0 (T 0 , 0, 0), the charateristic equation has been reduces to Clearly, the equation (13) has a characteristic root , and the rest charateristic roots of equation (13)satisfy the following equation, Criterion, E 0 is locally asymptotically stable. If N = N crit , one eigenvalue is zero, and it is For the time delays τ 1 , τ 2 > 0, we can show that equation (13) has no root with positive real part as N < N crit . In fact, assume λ = (u 1 ± iv 1 ), where V 1 > 0 and i = √ −1. Substituting (13) and seperating the real and imaginary parts, we obtain Squaring and adding both equations (15), we have Since The left side of equation (16) is larger than zero, while the right side of equation (16) is less than zero (since u 1 ≥ 0, and if N < N crit , then B 0 > C 0 ). This results in contradiction. Therefore, u 1 < 0, and E 0 is locally asymptotically that the equation (14) has at least one positive real root. Hence, the characteristic equation (13) has at least one positive real root. Hence, E 0 is unstable. This complete the proof.
Therefore, E * is locally asymptotically stable.

GLOBAL STABILITY ANALYSIS
In this section, we construct a suitable Lyapunov function to study the global dynamics of the infection-free equilibrium and chronic infection equilibrium for system (1).
Using the equation; Using the equation, Rewritten dW dt interms of the critical number, we get Thus, the function g has a global minimum at 1 and satisfies g(1) = 0.

Proof:
We define a Liapunov function as follows: At infected steady state, we have The derivative of U 1 (t) with respect to 't' along the solution of (30), we get  In order to illustrate the system dynamics we have used the default parameter values in Table 1.
The average CD4 + T -cells count in healthy human body is 1000 cells mm 3 [53] which is taken as its initial value. Since there is no infected CD4 + T -cells immediately after first effective contact between a healthy CD4 + T -cells and a human immunodeficiency virus, so that the initial value of infected CD4 + T -cells is taken to be zero. The initial viral load is considered as 1 × 10 −3 mm 3 . We therefore fix the initial value for each iteration as (1000, 0, 1 × 10 −3 ). Thus, the parameter values and initial condition of the system relate to real world scenarios.
First we have simulated the non -delayed system (1) [3]. Thus the infected steady state E is asymptotically stable for all τ 1 , τ 2 > 0. Take N = 60, τ 1 = τ 2 = 0.1 and other parameter values given in Table 1. Numerical solution show that the infected steady state E is asymptotically stable (Figure 1).  Table 1 In Figure 3, we plot the time series solution of the delayed system (2) for τ 1 = 0.1 & τ 2 = 3, with N = 60 ( Figure 3). Following the analytical results, we observe that the larger delay τ 2 can not produce any oscillations and the system populations remain stable for all values of τ 1 < τ 2 .
The only difference between the two cases is that the viral blip occurs earlier with high peak when τ 2 is smaller and it occurs later with low peak if the delay is higher (τ 2 = 3). Further under the condition of τ 1 = 1 when τ 2 = 0.1, E is asymptotically stable (see Figure 2). while at τ 1 = 1.21 and τ 2 = 0.1, E loses stability and the Hopf bifurcation occurs. From Figure 3.5, the periodic solution bifurcated from the infected steady state E , when τ 1 .5, τ 3 > 0. Take N = 60, and other parameter values given in table 1. Numerical solution show that the infected steady state E is asymptotically unstable.

CONCLUSIONS
In this work, we have proposed a delayed model to describe the dynamics of HIV infection of CD4 + T-cells by taking two independent delays, we first proved that proposed model is mathematically and virologically well-posed. In addition, we have proved that the disease-free equilibrium E 0 is globally asymptotically stable. If the critical number N crit ≤ N, which means that the HIV particals are eradicated. When N crit > N, E 0 become unstable and there occurs the HIV infection equilibrium E * which is globally asymptotically stable.
Finally we investigate the delay induced oscillations could occur via instability. Numerical simulations shows that the bifurcation is super critical and the bifurcating periodic solution is absolutely asymptotically stable. Sufficient conditions are established for the local asymptotic stability of the uninfected steady state and the infected steady state. The influence of the time delays in the stability of equilibrium states is discussed. We shared that the local stability of the uninfected steady state is discrete of the size of the delay; on the other hand, we proved that increasing the delay can destabilize the infected steady state leading to a Hopf bifurcation periodic solutions.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.