GLOBAL STABILITY OF A DELAYED HIV-1 DYNAMICS MODEL WITH SATURATION RESPONSE WITH CURE RATE, ABSORPTION EFFECT AND TWO TIME DELAYS

,


INTRODUCTION
HIV is called human immunodeficiency infection, taints and annihilates the cells of the human resistant framework, challenging it to battle off other illnesses. When HIV severely weakens the human immune system, it causes acquired immunodeficiency syndrome (AIDS) [1].
HIV contamination happens through the exchange of blood, bosom milk, vaginal liquid, semen, or pre-ejaculation. In these body fluids, HIV may exist in the form as free-virus and virus within the infected cells. HIV attacks to the immune system of the body, which consists of a variety of biological structures to prevent or control a variety of infections and diseases.
CD4 + T cells are focal middle mediators of the immune system in people, critically organizing cellular and humoral immune reactions against infections. The use of CD4 + T lymphocytes in the fight against infections is reduced because HIV only infects CD4 + T lymphocytes, which make up a large portion of white blood cells [2]. Therefore, if the infected body is not given external treatment, the amount of CD4 + T in the human body can be drastically reduced and it may increasing trend to receive the associated diseases. An opportunistic infection occurs when a certain number of CD4 + T cells are destroyed, resulting in weakened immune systems.
Therefore, if not treated with HIV drugs, HIV can gradually destroy the immune system and develop into the acquired immunodeficiency syndrome (AIDS), which is the advanced stage of HIV infection (see [3,4]).
Current statistics published by the World Health Organization indicate that HIV remains as a major global public health problem. Certain antiretroviral (ARV) drugs are currently available to help the immune system fight off HIV infection. Protease Inhibitors (PIs), is one of chemotherapy drugs that inhibit virus production from actively infected CD4 + T cells. Reverse transcriptase inhibitors (RTIs) are another chemotherapy that opposes the conversion of the virus's RNA into DNA (reverse transcription), so that the viral population will be low and the CD4 + T count will high and the host can survive (see [5,6]). Even though HIV treatment and prevention have made significant progress, appropriate or relevant drug therapy must still be promoted. The dynamics of HIV infection, its progression, and the immune system's cooperation with HIV can all be studied with the help of mathematical modeling.
Bonhoeffer and Nowak proposed the fundamental virus dynamics model (1.1) in section [7], and Korobeinikov performed the model's entire theoretical calculation in [8]. In model (1.1), the rate of disease infection per host and per virus is assumed to be a bilinear incidence rate, and virus particles produce virus immediately upon binding to a target cell without a time delay. The tests detailed in [9] unequivocally recommended that the contamination pace of microparasitic diseases is a rising capability of the parasite portion, and is normally sigmoidal in shape (see e.g [10]). In [11], it has been demonstrated biologically that there is a delay known as the intracellular time delay between the time a virus enters a target cell and the time it takes that infected cell to produce another virus. Therefore, researchers are motivated to propose and develop the mathematical dynamics models by introducing time delays and the different functional responses instead of the bilinear functional response (see e.g. [12,13,14]).
Li-Ming et al. in [13] Have examined the HIV-1 infection model using the Beddington-DeAngelis functional response and intracellular time delay, and they demonstrated that the system is permanent and infection equilibrium is globally asymptotically stable if the reproduction number is greater than one. In fact, when a pathogen enters into uninfected cells, the number of pathogens in the blood volume decreases, which is known as the retention impact or absorption effect (see e.g [14]). In [15], Pradeep and Ma have looked at a HIV-1 virus dynamics model with an absorption effect, a Beddington-DeAngelis functional response, and an intracellular time delay. The authors have determined the local and global stability of the infection-free equilibrium by using Lyapunov method and determine only local stability of the chronic infection equilibrium and show that the stability properties are totally dependent on the reproduction number of the model. A HIV-1 model with double delays (intracellular deferral and immune delay) and infection depreciation term is proposed in [16]. The authors have done a total analytical calculation and numerical simulation. Further suggested that intracellular delay is more important than the immune delay.
In [17], Hattaf and Noura have considered the mathematical model with the general form of infection process by incorporating the cure rate and absorption effect without considering the delay terms. In this paper the authors have determined the stability of the disease free equilibrium by direct Lyapunov method and geometrical approach have used for chronic infection.
Geo and Ma [18], have considered the Mathematical dynamics model of HIV with apoptosis by incorporation intracellular time delay and considering the general nonlinear infection processes. The authors have determined that if the basic reproduction number of the model is less than unit, the infection-free equilibrium is globally asymptotically stable and global attractors if equal to unity. They further show that the system is permanent if the reproduction number greater than unity.
In [19], Zhuang and Zhu have suggested a model for the dynamics of HIV with cure rate.
The authors have studied the existence of Hopf bifurcation by analysing the transcendental characteristic equation and used the Hopf bifurcation theory for global existence of bifurcating periodic solutions. Culshaw and Ruan have presented a mathematical model with intercellular time delay in [20] and the authors have discussed the effect of the time delay on the stability of the endemically infected equilibrium and they have given criteria to ensure the chronic infection is asymptotically stable for all delay.
A HIV-1 dynamics model by incorporating with both saturation infection rate and intracellular time delay has been proposed by Xu in [21]. The author has determined the local and global stability of the both infection-free equilibrium and a chronic-infection equilibrium of the model. In addition, Xu conducted a comprehensive mathematical analysis of the HIV-1 infection model in [22], which included an intracellular delay and absorption effect.
The time required for the virus to ripen after the infected cells have replicated the new virus is known as the maturation time. A newly produced virus cannot attach directly to uninfected target cells without maturing biologically. A mathematical model in [23] has been presented by incorporating both intercellular time and maturation time and a thorough theoretical analysis has been performed to show the global stability of the model.
In [24], the authors present a mathematical model that incorporates an intracellular time delay, a maturation time delay, and an absorption effect. They also demonstrate that the model's equilibrium points are stable both locally and globally. Further researcher have shown that the system is permanent.
Inspired by the above researches, we can consider the following dynamic model of HIV infection together with the saturation infection rate by incorporating absorption effects, intercellular time delay, maturation time delay, and cure rate. (1.2) where, τ ≥ 0 represent the intercellular time delays, the term e −pτ is the probability of surviving from t − τ to t, 1/α is half saturation constant, when the number of susceptible cells increases the inhibition effect from the behavioral change is measured by 1/(1 + αv) and the infectivity of the virus is measured by β v, σ ≥ 0 present the maturity time delay, the term e −uσ present the probability of surviving from time t − σ to t, γ is the cure rate of infected cells. In addition, it is presumptively assumed that all parameters are positive, and the remaining variables have the same biological significance as the models discussed earlier.
The remainder of the content of the paper is arranged as follows. Section 2 present the solution of system (1.2)'s positive, bounded, and limiting behavior. The existence of the unique equilibrium of the system (1.2), as well as the local and global stability of each equilibrium point, are all discussed in Section 3. In Section 4, the permanency of system (1.2) is discussed.
In Section 5, the numerical simulations of system (1.2) are discussed. Section 6 provides a discussion and conclusion.

PROPERTIES OF SOLUTION
Since the model depicts the growth of a cell population, we present the positivity and boundingness of model (1.2) solutions in this section. As a result, the cell numbers ought to remain fixed and not negative. The solutions' global presence is inferred by these properties.

Positivity and boundedness.
Let the initial condition of (1.2) is , v(t)) be the solution of the model (1.2) satisfy the initial condition (2.1), then the x(t), y(t) and v(t) are positive and ultimately bounded for all t ≥ 0. Further we 1 + αv(s) ds, and δ = min{d, p}.
Proof. To start with, we will confirm the solution of system (1.2) is positive. Using the constant variable formula, we get the following solution for the first and second equations of system (1.2).
Then using the initial condition (2.1) and equation (2.2) and (2.3), we deduce that x(t) ≥ 0, y(t) ≥ 0 for t ∈ [0, ζ ]. This method can repeat on [ζ , 2ζ ] to show that x(t) and y(t) are non-negative, and then successive intervals Next, we show that v(t) is non-negative for t > 0. Assuming contrary, and considering t 1 > 0 be the first time such that v(t 1 ) = 0. Then by the third equation of the model (1.2) and the similar argument above we havev(t 1 ) = ke −nσ y(t 1 − σ ) > 0, and hence we have v(t) < 0 for some t ∈ (t 1 −ε,t 1 ) for sufficiently small ε > 0. This contradicts v(t) > 0 for t ∈ [0,t 1 ]. Therefore it follows that v(t) > 0 for t > 0. This confirms the positivity of the solution.
Next, we have to show that the model (1.2) has bounded solutions.
From the first and second equations of (1.2), we have Equation (2.4) can be deduced as .
The third equation of (1.2) can be simplify as Then, we have .
This confirms that the solutions of the model (1.2) are bounded.

Limiting behavior of solution.
It is important to focus our attention on the existence of solutions to the finite behavior of the dynamical model.
, v(t)) be the solution of the model (1.2) subject to the initial condition (2.1), then the limit of X(t) exists as t → +∞. Additionally, we have Then adding the first and second equation of (1.2) and the derivative of z(t), we have . (2.13) If p ≥ d, using similar methods to get equation (2.12), we have and, using Lemma 3.3 in [25] and from equation (2.14), we get (2.15) By comparing system (2.13) and (2.15), for the parameters p > 0, and d > 0, we have (2.16) Then, from system (2.16), we have Thus, easily we can prove that Then, from (2.16), we have Since z(t) ≥ 0 and y(t) ≥ 0 for all t ≥ 0, from equation (2.17), we have (2.8).

MODEL ANALYSIS
In this section, we study the existence of the infection-free equilibrium point E f (x 0 , y 0 , v 0 ) and the chronic infection equilibrium point E ch (x 1 , y 1 , v 1 ) of model (1.2) and their stabilities.

Basic reproduction number. The average number of infected cells produced by one
infected cell during the time of admixture when all cells are uninfected is referred to as the basic reproductive number. The similar approach that was provided in [26] can be used to determine the model reproduction number and which is denoted as R 0 .
where 1 p+γ represents the average length of time an infected cell can be expected to live that is less than 1 p because a portion of infected cells recover by removing all DNA from their nuclei at a rate γ [17]. The dynamics of the system are significantly influenced by R 0 .
(i) If R 0 ≤ 1, then model (1.2) has a unique infection free equilibrium of the form Proof. For any equilibrium points, the following system of equations holds.
Hence, there is no equilibrium point for the system if x > λ d . By direct calculation from the equations (3.2)-(3.4), it is easy to show that, if R 0 > 1, system . Let E(x, y, v) be an arbitrary equilibrium point. Then the characteristic equation of the system Theorem 3.2. For any τ ≥ and σ ≥ 0, Proof. At the point E f , the characteristic equation (3.6) can be reduced to As d = −s demonstrates, the first root of equation (3.7) has a negative root, and equation (3.8) must be used to determine the other roots.
Therefore, if R 0 < 1, all the roots of equation (3.7) are negative real parts. Hence, if R 0 < 1, the point E f is locally asymptotically stable.
When R 0 = 1, from equation (3.9) we have (3.16) It is obvious that s = 0 is a solution of (3.16).
Further, any other root of (3.16) can be proved to be negative real.
Assume that s = α 1 + iα 2 is a root of equation (3.16) for some α 1 , α 2 ≥ and τ, σ ≥ 0. Then, by substituting s into (3.16) and separating the imaginary and real parts, we have the following equations. (3.17) By squaring equations (3.17) and (3.18) and adding, we have the inequality, (3.19) Clearly inequality (3.19) cannot be fulfilled, which prompts a logical inconsistency. This implies every roots of the equation (3.16) are negative real except s = 0.

Proof. Let prove Theorem 3.3 under the two cases.
Consider (x(t), y(t), v(t)) be any solution of model (1.2) with initial condition (2.1).
Case II: When ke −uσ −pτ − (γ + p) ≤ 0, Let us define a Lyapunov functional as For t ≥ 0, taking the derivative of V 2 (t) along the solutions of (1.2), equation (3.24) can be derive as follows.
Clearly,V 2 ≤ 0 for all y, v > 0 when ke −uσ −pτ − (p + γ) ≤ 0. Then, at this point, by a similar argument as above, it is not difficult to show that E f is globally asymptotically stable. This prove Theorem 3.3.

Stability of the chronic infection equilibrium.
In this subsection we study the local stability of the chronic infection equilibrium point E ch (x 1 , y 1 , v 1 ) by using the characteristic equation of model (1.2) considering three special cases that τ = 0, σ = 0, τ ≥ 0, σ = 0, and τ = 0, σ ≥ 0 and the global stability of the chronic equilibrium points by using the Barbalat's lemma in [30].
Proof. Let Then, from (3.6) the characteristic equation at E ch (x 1 , y 1 , v 1 ) can be derived as where, When σ = τ = 0, equation (3.27) reduced to the form and by direct computation, we have and Therefore, according to the Routh-Hurwitz criterion, any root of equation (3.28) has negative real part for τ = 0 and σ = 0. Hence, for τ = σ = 0, the chronic infection equilibrium E ch of system (1.2) is locally asymmetrically stable.
As a second case, consider σ = 0 and τ ≥ 0. Then equation (3.27) can be written as Let s = iω (ω > 0) is the purely imaginary root of equation (3.29). Then, by substituting S into (3.29) and separating the imaginary and real parts, we obtain following two equations By squaring equation (3.30) and (3.31), and adding together we have

Then by direct calculation, it can be show that
Hence, based on the Routh-Hurwitz criterion, when τ ≥ 0, (3.32) has no positive real roots for ω 2 . Consequently, this implies that all roots of equation (3.29) have negative real parts. That is, if R 0 > 1, the chronic disease equilibrium E ch in model (1.2) is locally asymptotically stable [27].
As the third case, when σ ≥ 0 and τ = 0, equation (3.27) can be written as Let s = iω(ω > 0) is the purely imaginary root of equation (3.33). Then by substituting S into (3.33) and separating imaginary and real parts of the equation, we obtain following two equations.
Then by direct calculation, it can be show that Again, based on the Routh-Hurwitz criterion, (3.32) has no positive real roots for ω 2 when σ ≥ 0. Consequently, this implies that all roots of equation (3.29) have negative real parts. That is, if R 0 > 1, the chronic disease equilibrium E ch in model (1.2) is locally asymptotically stable [27].
(i) If R 0 ≤ 1, then the infection free equilibrium E f is globally asymptotically stable.
(ii) If R 0 > 1, then the chronic infection equilibrium E ch is globally asymptotically stable.

PERMANENCE
In this subsection, we will investigate the uniform persistence of model (1.2) with given initial values (2.1). Biologically, the uniform persistence or permanence suggests that the virus v(t) and infected cells y(t) not possible totally removed from the body and will ultimately continue. To continue, we present the terminology and notation as follows. Let the Banach space of continuous functions which composed with sup-norm For t ≥ 0., let P(t) be the set of solutions operators for model (1.2). Defined ω− limit set as ω(x) = {y ∈ X|there is a sequencet n → ∞ as n → ∞ with P(t n )x → y as n → ∞}. Then, we refer Theorem 4.2 in [31].
Lemma 4.1. Assume that we have (i) X 0 ∈ X is a open set with X 0 X 0 = / 0 and X 0 X 0 = X, (ii) P(t) satisfy P(t) : X 0 → X 0 , P(t) : X 0 → X 0 , and point dissipative in X, (iii) P(t) is asymptotically smooth, (iv) If Y 2 is the global attractor of P(t) limited to X 0 and N = k i=1 N i , then is isolate and has acyclic covering N, (v) for each N i ∈ N,W s (N i ) X 0 = / 0, where W s is stable set.
Then we have a uniform repeller P(t) with respect to X 0 . That is, there exists an η > 0 such that for arbitrary x ∈ X 0 , lim inf  First we say the set M ∂ = {( λ d , 0, 0)}. Assume that P(t) ∈ M ∂ , for all t ≥ 0. Then it is enough to show that y(t) = 0 and v(t) = 0 for all t ≥ 0. We then use contradiction to prove this claim.
Let Ω 2 = x∈Y 2 ω(x), where Y 2 is the global attractor of P(t) restricted to the set ∂ X. Now we need to show that Ω 2 = {E f }. Indeed, it follows from Ω 2 ⊆ M ∂ and using the first equation Next, we need to prove that W s (E f ) X 0 = / 0. To prove this result, we assume the contradiction. Suppose (x(t), y(t), v(t)) ∈ X 0 is a positive solution for system (1.2) such that Then, for any small constant ε > 0, we have positive constant t 0 > 0, such that Thus, for the selected constant ε, from model (1.2) that for t ≥ t 0 + τ and t ≥ t 0 + σ , we havė System (4.2) follows the quasi-monotone structure since the right hand sides of the first and second equations of system (4.2) is increasing with the functions v(t − τ) and y(t − σ ) (see e.g. [13]).
Consider the following system of differential equations to apply the comparison principle.
It implies that, This completes the proof of Theorem (4.1).

NUMERICAL SIMULATIONS
The purpose of this section is to show that the theoretical results that we obtained in Sections (3) and (4) are valid, by using the parameter values that have been reported in the literature and are shown in Table 1.  R 0 ≤ 1. Moreover, Figure 8 shows that sufficiently large τ and σ reduce the reproductive rate.
Hence, it is clear that both τ and σ contribute greatly to the removal of the virus from the system.  Table 1, and infection-free equilibrium point  Table 1, and infection-free equilibrium point   Table 1, and infection-free equilibrium point with parameter values Case V in Table 1, and infection equilibrium point  Table 1, and infection equilibrium point   and Theorem 3.4, respectively. Further, if R 0 ≤ 1, in Theorem 3.3, it has been shown that the E f is globally stable for any time delay σ , τ ≥ 0, by using the appropriate Lyapunov function and Lassalle's invariance principle and it further has been verified by using the Barbalat's lemma in Theorem 3.5. In this case, the virus will be removed from the system by activating the immune system of the body or giving external medical treatment, the illness will be cured after some time.
If R 0 of the model is greater than unity, then E f become unstable (Theorem 3.3) and E ch is locally asymptotically stable (Theorem 3.4). Further it has been shown in Theorem 3.5 chronic infection equilibrium E ch is globally asymptotically stable. In this condition, the infection is chronic and persistent, it biologically means that the host cannot control the infection with drugs or the immune system. In Section (4), it has been confirmed that model (1.2) is uniformly persistence for any σ , τ ≥ 0, if R 0 > 1. Numerical simulations obviously propose that the chronic infection equilibrium of model(1.2) is globally asymptotically steady if R 0 > 1. Figure   8 shows that we can choose sufficiently large enough σ and τ that to satisfy the condition R 0 ≤ 1 if all other parameters remain constant, which causes the virus to be cleaned out.