Stability Analysis of an In-Host Viral Model with Cure of Infected Cells and Humoral Immunity

An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completely determined by the basic reproductive number 𝑅 0 and show that the infection-free equilibrium 𝐸 0 is globally asymptotically stable if and only if 𝑅 0 ≤ 1 . Moreover, if 𝑅 0 > 1 , the infection equilibrium is locally asymptotically stable when the time delay 𝜏 is small and it loses stability as the length of the time delay increases past a critical value 𝜏 0 . Finally, we confirm our analysis by providing several numerical examples.


Introduction
The humoral immunity is a kind of immunologic mechanism which uses B lymphocytes to produce antigen to prevent virus and there are evidences to prove that the humoral immunity is more effective than the cell-mediated immune in some infections such as malaria infection [1][2][3].Many authors present and develop mathematical systems for the humoral immunity [4][5][6][7].And the cure of virus is also important especially in HBV models [8,9].In the present paper, we analyze an in-host viral model with humoral immunity and intracellular delay, and we incorporate a "cure" of infected cells into it.We propose the following system:   () = Λ −  ()  () −  () +  () ,   () =  ()  () − ( + )  () ,   () =  () −  () −  ()  () ,   () =  ( − ) −  () , (1) where , , , and  represent the uninfected cells, the infected cells, the virus, and the B cells, respectively.Λ and  are assumed as the birth rate and death rate of uninfected cells. is the infection rate and () represents the number of free virus which is produced during the average infected cell life span. is the death rate of infected cells and  represents the death rate of virus. and  represent the birth rate and death rate of B cells.The B cells neutralization rate is represented by .
The following form is taken as the initial conditions: where ), the space of continuous functions mapping the interval (−∞, 0] into R 4 +0 , and The organization of this paper is as follows.In the next section, we will find threshold parameters  0 of system (1) and it determines the existence of the equilibriums.In Section 3, by structuring suitable Lyapunov functionals and using LaSalle's invariance principle we attain the global stability of the uninfected equilibrium if  0 ≤ 1.In Section 4, we consider the stability of the infected equilibrium and the occurrence of local Hopf bifurcation.In Section 5, we present the numerical simulations to illustrate our results.Finally, we offer concluding remarks in the last section.

Global Attraction of Infection-Free Equilibrium
Theorem 1.The infection-free equilibrium  0 of system (1) is globally attractive if  0 ≤ 1.

Permanence of the System for 𝑅
is locally asymptotically stable when  = 0.
Let  = V and then substitute it into (8).Separating the real and imaginary parts, we will gain Squaring and adding (11) yield Denote We have that Hence if () = 0 has a positive root  = V 2 , then  = ±V are a couple of purely imaginary characteristic roots.Moreover, for a real practice model with fully known coefficients, the exactly four roots of () = 0 can be numerically calculated with the help of computational software such as Matlab.Let   (1 ≤  ≤ , 1 ≤  ≤ 4) be the positive roots of () = 0 and V  = √  .Solving (11) with respect to , we can obtain where 1 ≤  ≤ ; 1 ≤  ≤ 4;  = 0, 1, 2, . ... Let This means that  0 is the least value of  which can be used to make the characteristic equation have purely imaginary roots.
have the same sign.
Proof.Let the characteristic equation be in the following form: We can obviously know from (8) that Calculating the derivative of ( 17) with respect to , we can obtain that And after some calculations, we can get This completes the proof.
Applying Theorems 2 and 3 and the Hopf bifurcation theorem for functional differential equation [10], we derive the existence of a Hopf bifurcation as stated in Theorem 4.
Theorem 4. Suppose that () = 0 has at least one simple positive root and  0 is defined in (16).Then there is a Hopf bifurcation for the system (1) as s passes upwards through  0 leading to a periodic solution that bifurcates from  * , where (1) When  = 0.75,  * is locally asymptotically stable and the stability is illustrated by Figure 2. (2) When  = 0.9, the delay can destabilize  * and lead to Hopf bifurcation.We will show that in Figure 3.

Discussions
In this paper, we consider an in-host viral model humoral immunity.In addition, a "cure" of infected cells is incorporated.Firstly, it has been shown that if the basic reproductive number is less than unity, the infection-free equilibrium  0 is globally attractive, and the time delay has no effect on the dynamics of the system.Then, a detailed analysis on the local asymptotic stability of the infection equilibrium  * of the model is carried out.If  * is feasible, the basic reproduction number of the virus is greater than unity;  * is globally attractive for any time delay under some parameter conditions.By taking the discrete time delay as a bifurcation parameter, it is shown that this system undergoes a sequence of Hopf bifurcations and stability switches are observed by using simulations.

Figure 2 :Figure 3 :
Figure 2: Equilibrium  * is locally asymptotically stable when  is very small.