Analysis of a Viral Infection Model with Delayed Nonlytic Immune Response

We investigate the dynamical behavior of a virus infection model with delayed nonlytic immune response. By analyzing corresponding characteristic equations, the local stabilities of two boundary equilibria are established. By using suitable Lyapunov functional and LaSalle’s invariance principle, we establish the global stability of the infection-free equilibrium. We find that the infection free equilibrium E 0 is globally asymptotically stable when R 0 ⩽ 1, and the infected equilibrium without immunity E 1 is local asymptotically stable when 1 < R 0 ⩽ 1+ bβ/cd. Under the condition R 0 > 1+ bβ/cd we obtain the sufficient conditions to the local stability of the infected equilibrium with immunity E 2 . We show that the time delay can change the stability of E 2 and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions are studied and numerical simulations to our theorems are provided.


Introduction
Mathematical models have been proven valuable in understanding the population dynamics of viral load in vivo.A proper model may play significant role in a better understanding of the disease and the various drug therapy strategies.Viral infection models have received great attention in recent years [1][2][3][4][5][6][7].In most viral infections, cytotoxic T lymphocytes (CTLs) play a critical role in antiviral defense by providing a cell-mediated response to specific foreign antigens associated with cells.Therefore, the immune response after a viral infection is universal and necessary to eliminate or control the disease.
Recently, there have been a lot of papers on virus dynamics within host; some include the immune response directly [8][9][10][11][12][13].During viral infections, the host immune system reacts with innate and antigen-specific immune response.Both types of response can be subdivided broadly into lytic and nonlytic components.Lytic components kill infected cells, whereas the nonlytic components inhibit viral replication through soluble mediators.As a part of the innate response, cytotoxic T lymphocytes (CTLs) kill infected cells, while antibodies neutralize free virus particles and thus inhibit the infection of susceptible cells.In addition, CD4+ and CD8+ T cells can secrete cytokines that inhibit viral replication [12].In order to investigate the role of direct lytic and nonlytic inhibition of viral replication by immune cells in viral infections, a mathematical model was constructed to describe the basic dynamics of the interaction among susceptible host cells, a virus population, and immune response, which is described by the following differential equations [14,15]: where (), (), and () represent the densities of uninfected target cells, virus, and CTL cells at time , respectively.Uninfected cells are produced at rate , die at rate , and become infected by virus at rate  without the immune 2 Discrete Dynamics in Nature and Society response; to model nonlytic antiviral, viral replication is inhibited by the immune response at rate 1+; infected cells die at rate  and are removed at rate  by the CTL immune response.The virus-specific CTL cells proliferate at rate  by contact with infected cells and die at rate .The parameter  expresses the strength of the lytic component; the parameter  expresses the efficacy of the nonlytic component.Time delays cannot be ignored in models for immune response.Antigenic stimulation generating CTLs may need a period of time  [16][17][18]; the CTL response at time  may depend on the population of antigen at a previous time  − .Under the assumption of retarded immune response, Wang et al. [18] studied the effects of the time delay for immune response and assumed the time evolution of the population of CTL cells  is governed by the delayed nonlinear differential equation   () = ( − ) − .Li and Shu [19] and Xie et al. [20] investigated the effects of a time delay on a threedimensional system with   () = ( − )( − ).However, both studies do not consider the nonlytic component.In this paper we propose a more general model, with the initial conditions where C + = C([−, 0], R + ) which is the Banach space of continuous functions mapping the interval [−, 0] into R + with the topology of uniform convergence.Clearly, the models studied in [18][19][20] correspond to the case  = 0 in our general model (2).In this work we study (2) in the more general case  ⩾ 0. This paper is organized as follows.In Section 2, we study the local asymptotic stability of the infection-free equilibrium and the immune-exhausted equilibrium of system (2), and the global asymptotic stability of the infection-free equilibrium also is investigated.In Section 3, we analyze the local stability of the positive equilibrium and the existence of Hopf bifurcations.In Section 4, the direction and stability of Hopf bifurcation are analyzed by the normal form theory and center manifold approach.In Section 5, we present numerical simulations to illustrate our results.Finally, in Section 6 we provide concluding remarks.

The Stability Analysis of Equilibrium
Considering the existence of the three equilibria, then we have the following conclusions.Proposition 1.Let  0 = /.Then, consider the following.
(3) If  0 > 1 + /, in addition to the infectionfree equilibrium  0 , system (2) has another infected equilibrium  2 = ( 2 ,  2 ,  2 ) that corresponds to the survival of free virus and CTL, where By the similar method in [20], we have the following result.

The Stability of Infected Equilibrium and Hopf Bifurcation
Let  2 /(1 +  2 ) = , apply  2 /(1 +  2 ) =  +  2 to the characteristic equation of system (2) at the positive equilibrium  2 , and we have that is, where If  = 0, (16) becomes where Then, By the Routh-Hurwitz criterion, we know that all roots of (18) have negative real parts.From the above analysis, the following theorem holds.
From Theorem 6, when  = 0, all roots of ( 16) lie to the left of imaginary axis.But, as  is increased from zero, some of the roots may cross the imaginary axis to the right.Then the equilibrium  2 becomes unstable.Now we suppose (16) has a pure imaginary root  =  ( > 0).Obviously,  =  ( > 0) is a root of ( 16) if and only if  satisfies Separating the real and imaginary parts, we have From ( 21), we also obtain So we have where Denote Therefore, if ( 16) has a pure imaginary root , then equation has a positive real root  2 .Suppose that (27) has , 1 ⩽  ⩽ 3, positive real roots, which are   , 1 ⩽  ⩽ , respectively.From (22), we have cos sin Let where 1 ⩽  ⩽  and  = 0, 1, 2, . ... Then ±√  is a pair of pure imaginary roots of ( 16).

Direction and Stability of Hopf Bifurcation
In this section, we will study the direction of the Hopf bifurcations and stability of bifurcating periodic solutions by applying the normal theory and the center manifold theorem introduced in [22].We always assume that system (2) undergoes Hopf bifurcation at the positive equilibrium  2 for  =  0 , and then ± are pure imaginary roots of the characteristic equation at the positive equilibrium  2 .
then system (33) is equivalent to where ), the adjoint operator of (0) is defined by and define a bilinear inner product where () = (, 0).By the discussion in Section 3 we know that ± 0 are eigenvalues of (0).Hence, they are also eigenvalues of  * .We define () and  * () to be the eigenvectors of  and  * corresponding to the eigenvalues  0 and − 0 , respectively.Assume that () = (1,  1 ,  2 ) ⊤   0  is the eigenvector of (0) corresponding to  0 ; then (0)() =  0 ().It follows from the definition of (0) that we have Then we obtain ( In order to assure ⟨ * (), ()⟩ = 1, we need to determine the value of .From (40) we have Hence Using the method in [22] we obtain the following coefficients: where ). (47) Then we get which determine the quantities of bifurcating periodic solutions reduced on the central manifold at the critical value  0 .More precisely, ] 2 determines the direction of Hopf bifurcation and  2 determines the stability of bifurcating periodic solution.Summarizing the above discussion, we have the following main result.exist for  >  0 ( <  0 ).Moreover,  2 determines the stability of the bifurcating periodic solutions.The bifurcating periodic solutions are stable (unstable) if  2 < 0 ( 2 > 0).

Conclusion
In this paper, we have studied a virus infection model with delayed nonlytic immune response.We obtain the sufficient and necessary conditions for the existence of the equilibria.Global stability of the infection-free equilibrium has been given by the Lyapunov-LaSalle type theorem.We find that the infection-free equilibrium  0 is globally asymptotically stable if  0 ⩽ 1.In this case, the virus is finally cleared up.By analyzing corresponding characteristic equation, we obtain that the immune-exhausted equilibrium  1 is locally asymptotically stable if 1 <  0 ⩽ 1 + /.In this case,   the infection becomes chronic but without CTL immune response.By choosing time delay as a bifurcation parameter, we have shown that a limit cycle emerges via Hopf bifurcation when the delay passes through the critical value  0 .By applying normal form theory and center manifold theorem, the direction of Hopf bifurcation and stability of bifurcating periodic solutions have been investigated.Some numerical simulations are performed to verify the correctness of theoretical analysis.Clearly, compared to the earlier studies, our analysis shows that the new time delay we introduce can change the stability of the equilibrium  2 .We would like to mention that whether the equilibrium  2 is globally asymptotically stable when  0 > 1 + /.We leave this for our future work.

Theorem 10 .Figure 1 :Figure 2 :
Figure 1: The phase diagram of system (2) with different initial values and different .

Figure 7 :
Figure 7: A bifurcation diagram in which  is bifurcating parameter.

The characteristic equation about 𝐸 0 is given by
has negative roots  1 = − and  2 = −.