Global Stability and Hopf Bifurcation for a Virus Dynamics Model with General Incidence Rate and Delayed CTL Immune Response

In this work, we investigate an HIV-1 infection model with a general incidence rate and delayed CTL immune response. The model admits three possible equilibria, an infection-free equilibrium *0 E , CTL-inactivated infection equilibrium *1 E and CTL-activated infection equilibrium *2 E . We prove that in the absence of CTL immune delay, the model has exactly the basic behaviour model, for all positive intracellular delays, the global dynamics are determined by two threshold parameters 0 R and 1 R , if 0 1 R ≤ , *0 E is globally asymptotically stable, if 1 0 1 R R ≤ < , *1 E is globally asymptotically stable and if 1 1 R > , *2 E is globally asymptotically stable. But if the CTL immune response delay is different from zero, then the behaviour of the model at *2 E changes completely, although 1 1 R > , a Hopf bifurcation at *2 E is established. In the end, we present some numerical simulations.


Introduction
Immunity can be broadly categorized into adaptive immunity and innate immunity. Adaptive immunity is mediated by clonally B-cells and T-cells are also called lymphocytes, humoral and cellular immunity is characterized by specificity and memory. Cytotoxic T Lymphocytes (CTLs) play an important role in antiviral defense by attacking infected cells. Many mathematical models have been How to cite this paper: Ndongo, A.S. (2021) Global Stability and Hopf Bifurcation for a Virus Dynamics Model with General Incidence Rate and Delayed CTL Immune Response. Applied Mathematics, 12, developed to describe HIV-1 (human immunodeficiency virus type 1) (see for example [1]- [15]). First of all, we introduced the standard viral infection model with CTL immune response considered by Nowak and Bangham [16] as follows: , d x t s dx t x t v t t y t x t v t y t py t z t t v t ky t uv t t z t cy t z t bz t t This incidence rate considered in this paper generalized many forms of commonly used incidence rate, including simple mass action, saturation incidence rate, Beddington-DeAngelis functional response form and the Crowly-Martin functional response form introduced by Crowly-Martin (see [20]). The global dynamic of a virus dynamics model with Crowly-Martin functional response was discussed in [21]. In this paper, we propose a virus dynamics model with three delays and general incidence rate as follows: where the parameters have the same meanings as in system (1.

Preliminary Results
In this section we established the positivity and the boundedness of solutions of (1.2) and we define the basic reproduction numbers 0 R and 1 R and the existence of three possible equilibrium points is studied. The following theorem establishes the non-negativity and boundedness of solutions of (1.2). : Due to lemma 2 in [23], any solution of (1.2) with Next we show that the solutions are also bounded.   By substituting this into (2.3) 2 , we obtain: If Since, and: ( ) Hence, we obtain the CTL-inactivated infection equilibrium: x is the unique zero in ( ) x is the unique zero of L in ( ) 0, M and z is given by (2.9). This completes the proof.
So, as K is increasing in the interval 0, s d

Global Stability of the Infection-Free Equilibrium
In this section, we study the global stability of the infection-free equilibrium is globally asymptotically stable for all 1 2 3 , , 0 τ τ τ ≥ .
Proof. Define a Lyapunov function 0 U as follows: . Therefore we have two cases: By the above discussion, we deduce that It follow from LaSalle invariance principle [24] that the infection-free equilibrium * 0 E is globally asymptotically stable.

Global Stability of the Infected Equilibria
In this section, we study the global stability of the CTL-inactivated infection equilibrium We set: It is clear that for any 0 , e e x m x m m U is nonnegative defined with respect to the endemic equilibrium * E , which is a global minimum.
We now prove that the time derivative of 2 U is non-positive. Calculating the time derivative of 1 V along the positive solutions of (1.2), we obtain: , ,  .
From (H 2 ), we have: and from (H 4 ) we have: ,

The CTL-Activated Equilibrium and Hopf Bifurcation
In this section we will take where: , , , , , , We have already seen in Theorem 4.1 for 1 0 τ ≥ and 2 0 τ ≥ that the equilibrium * 2 E is globally asymptotically stable in the case 3 0 τ τ = = and in particular is locally asymptotically stable in this case. Now, let see to what value of 3 0 τ > , this stability persists. If is a solution of (5.1), separating real and imaginary parts, it follows that: where θ is one of cube roots of the complex number 0 2 Q D − + and According to [25] we have the following Lemma. Lemma 1. [25] For the polynomial Equation (5.6), the following states are true.   Differentiating the two sides of Equation (5.1) with respect to τ , we obtained  3  2  2  3  2  1  3  2  1  4  3  2  3  2  3  2  1  0  3  2  1  0   4  3  2  3   A. S. Ndongo

Numerical Simulations
In this section, we give some numerical simulations supporting the theoretical analysis given in Sections 3, 4 and 5. We assume ( ) , 1 Parameters in Figure 1 are

Examples
In this section, we give some particular examples.

Conclusion
In this paper, we have considered a virus dynamics model with a general incidence rate and three delays 1 2 , τ τ and 3 τ . This general incidence represents a variety of possible incidence functions that could be used in virus dynamics models. We establish that the global dynamics are determined by two threshold parameters the basic reproduction ratios for viral infection and CTL immune