Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington–DeAngelis incidence and three delays

ABSTRACT In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington–DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.


Introduction
In recent years, the virus infection models provide comprehensive views for our understanding of diseases, such as HIV, influenza, HBV, Ebola, HTLV and HCV (see 23,24]). Theoretical analysis for these mathematical models are important to obtain complete insights for the viral dynamics in vivo. In particular, the stability and the bifurcation will provide specific information for our understanding about disease control. The adaptive immune system reacts against virus and infected cells during virus infections. The antibody and cytotoxic T-lymphocyte (CTL) responses play the significant role in preventing infections. Hence, effective strategies to prevent virus infection need both antibody and CTL responses (see [1,14,17,19]). In [17], Wodarz proposed a basic model to describe the interactions of antibody and CTL immune responses with bilinear incidence which includes uninfected target cells x(t), productively infected cells y(t), free virus v(t), CTL response cells z(t) and antibody response cells w(t). Then, Yousfi et al. [20] gave the global analysis for this model. Yan and Wang [19] incorporated an intracellular delay into the infected cell equation in the model and studied the effect of the delay on the global dynamics. However, Wang and Xu [15] suggested that the incidence rate is probably not linear over the large number of virus and susceptible cells. Balasubramaniam et al. [1] developed a HIV model with Beddington-DeAngelis incidence and investigated the global stability and the existence of Hopf bifurcation.
However, only single immune response delay is considered in [1]. We know that there are usually three delays in a virus infection disease with the interactions of antibody and CTL immune responses: the intracellular delay, virus replication delay and immune response delay in the transmission process of virus infection. An important and interesting problem is how dynamical properties in virus infection disease will befallen when three delays exist simultaneously. Particularly, how stability properties will occur at possible equilibrium stations.
Therefore, in this paper we consider a five-dimensional virus infection model with three time delays which describes the interactions of antibody, CTL immune responses and Beddington-DeAngelis incidence rate , where , k, c and g are the birth rate of the uninfected cells, the virus, the CTL responses and the antibody responses, respectively; β is the infection rate; d,r,u,h and α represent the death rate of uninfected target cells, productively infected cells, virus, CTL responses and antibody responses, respectively; p represents the killing rate of infected cells by CTL response cells; q is the B cells neutralize rate; τ 1 denotes the intracellular delay, and e −mτ 1 denotes the surviving rate of infected cells during delay period [t − τ 1 , t] (see [4,[7][8][9][10]12,19,24]); τ 2 is virus replication delay, and e −nτ 2 denotes the surviving rate of virus during delay period [t − τ 2 , t] (see [5,18]) and τ 3 denotes immune response delay which is suggested in [1,8,11,23]. In this paper, our purpose is to investigate the dynamical properties of model (1), expressly the stability of equilibria and the existence of Hopf bifurcation. The organization of our paper is as follows. In Section 2, the basic properties of model (1) for the non-negativity and boundedness of solutions, the threshold values and the existence of five equilibria are discussed. In Section 3, the threshold conditions on the global stability and instability for the infection-free equilibrium, infection equilibrium without immune response and infection equilibrium with only antibody response are stated. When τ 3 = 0, the threshold conditions on the global stability and instability for the infection equilibrium with only CTL response and infection equilibrium with both CTL and antibody responses are proved. In Section 4, when τ 3 > 0, the sufficient conditions on the existence of Hopf bifurcation for the infection equilibrium with only CTL response and infection equilibrium with both CTL and antibody responses are established. In Section 5, the numerical simulations are presented to further illustrate the dynamical behaviour of the model. Finally, we will give a conclusion.

Boundedness and equilibrium
Let τ = max{τ 1 , τ 2 , τ 3 } and R 5 . The initial conditions for model (1) are given as follows where . By the fundamental theory of functional differential equation [6], It is easy to see that model (1) admits a unique solution (x(t), y(t), v(t), z(t), w(t)) satisfying initial conditions (2). We have the following basic result of model (1).
be the solution of model (1) satisfying initial conditions (2), then x(t), y(t), v(t), z(t) and w(t) are positive and ultimately bounded.

Proof:
It is now easy to show that all solutions of model (1) with initial conditions (2) are defined on R 5 + and remain positive for all t ≥ 0. Denote Calculating the derivative of N(t) along solutions of model (1), we havė where s = min{d, r/2, u, h, α}. This implies that N(t) is ultimately bounded for large t. So, and w(t) are also ultimately bounded. Next, we discuss the existence of equilibria of model (1). Firstly, we directly obtain that model (1) always has a unique infection-free equilibrium E 0 = (x 0 , 0, 0, 0, 0) with x 0 = /d. The basic reproductive number of viral infection for model (1) is Here, k is the rate of the newly virus produced by infected cells, 1/u is the surviving period of virus, e −nτ 2 is the surviving rate of newly virus in [t − τ 2 , t], β( /d)/(1 + a( /d)) denotes the newly infected cells which are infected by the first virus, e −mτ 1 is the the surviving rate of newly infected cells in [t − τ 1 , t], and 1/r is the surviving period of infected cells. Therefore, we easily see that R 0 denotes the average number of the free virus released by the infected cells which are infected by the first virus. Obviously, R 0 > 1 implies that kβ e −mτ 1 −nτ 2 − ur(d + a ) > 0 and k(β + bd) − aure mτ 1 +nτ 2 > 0.
When R 0 > 1, model (1) has a unique immune-free equilibrium E 1 = (x 1 , y 1 , v 1 , 0, 0), where The antibody immune reproductive number for model (1) is Note that when R 0 > 1 model (1) has a unique immune-free equilibrium E 1 = (x 1 , y 1 , v 1 , 0, 0). This shows that virus infection is successful and the number of free virus at equi- Here, R 2 denotes the average number of the CTL immune cells activated by infected cells when virus infection is successful and antibody immune responses have not been established. Note that the number of infected cells at equilibrium E 1 is kβ e −mτ 1 − ure nτ 2 (d + a )/r[k(β + bd) − aure mτ 1 +nτ 2 ], 1/h is the average life-span of CTL cells and c is the rate at which the CTL responses are produced. We see that R 1 > 1 is equivalent to α − gv 1 < 0, and R 2 > 1 is equivalent to h − cy 1 < 0.
When R 1 > 1, model (1) has a unique infection equilibrium with only antibody response E 2 = (x 2 , y 2 , v 2 , 0, w 2 ), where and x 2 is the unique positive zero point of the following function In fact, from k e −mτ 1 −nτ 2 /ur > v 1 , by R 1 > 1 we obtain k g − αure mτ 1 +nτ 2 > 0. From the expression of w 2 it follows that the existence of equilibrium E 2 is equivalent to x 2 ∈ (0, kg − αure mτ 1 +nτ 2 /kgd). Noticing that L(x) is a quadratic function and L(0) < 0, we know that the existence and uniqueness of equilibrium E 2 is equivalent to from R 1 > 1, we have Therefore, when R 1 > 1, we get When R 2 > 1, model (1) has a unique infection equilibrium with only CTL response and x 3 is the unique positive zero point of the following function In fact, since e −mτ 1 /r > y 1 , by R 2 > 1 we obtain c − rhe mτ 1 > 0. From the expression of z 3 it follows that the existence of CTL-present infection equilibrium E 3 is equivalent to x 3 ∈ (0, c − rhe mτ 1 /cd). Noticing that L(x) is a quadratic function and L(0) < 0, we know that the existence and uniqueness of CTL-present equilibrium E 3 is equivalent to Therefore, when R 2 > 1, we get The CTL immune competitive reproductive number for model (1) is In fact, when R 1 > 1, model (1) has a unique infection equilibrium with only antibody response E 2 = (x 2 , y 2 , v 2 , 0, w 2 ). This predicates that CTL immune responses have been established, and the number of infected cells at equilibrium E 2 is ( − dx 2 )e −mτ 1 /r. Hence, R 3 denotes the average number of the CTL immune cells activated by infected cells under the condition that antibody immune responses have been established. The antibody immune competitive reproductive number for model (1) is In fact, when R 2 > 1, model (1) has a unique infection equilibrium with only CTL response In fact, from the above discussion on the existence of equilibrium E 2 , we directly have v 4 = v 2 and This shows that equilibrium E 4 uniquely exists.

Stability analysis
Theorem 3.1: Proof: Consider conclusion (a). Define a Lyapunov functional Calculating the derivative of V 1 (t) along any positive solution of model (1) and noting that From the LaSalle's invariance principle [6], we have that E 0 is globally asymptotically stable when R 0 ≤ 1. Next, we consider conclusion (b). The characteristic equation of the linearized system of model (1) at the equilibrium E 0 is Hence, there is at least a positive s * such that f (s * ) = 0. Therefore, when R 0 > 1, E 0 is unstable. This completes the proof.

Remark 3.2:
Theorem 3.1 shows that if only infection-free equilibrium E 0 exists, then it is globally asymptotically stable, and delays τ 1 , τ 2 and τ 3 do not impact the stability of E 0 . Biologically, we see that in this case the virus is cleared up.
Calculating the derivative of V 2 (t) along the solution of model (1) gives Notice that .
Next, we consider conclusion (b). The characteristic equation of the linearized system of model (1) at the equilibrium E 1 is This completes the proof.
Remark 3.4: Theorem 3.3 shows that delays τ 1 , τ 2 and τ 3 do not impact the stability of E 1 . Biologically, we see that when R 0 > 1, R 1 ≤ 1 and R 2 ≤ 1 then the establishments of both CTLs and antibody immune responses are unsuccessful.
(a) If R 3 ≤ 1, then the infection equilibrium E 2 with only antibody response is globally asymptotically stable; (b) If R 3 > 1, then the equilibrium E 2 is unstable.

Proof: Consider conclusion (a). Define a Lyapunov functional
Calculating the derivative of V 3 (t) along the solution of model (1), it follows that Notice that .
Next, we consider conclusion (b). The characteristic equation of the linearized system of model (1) at the equilibrium E 2 is If R 3 > 1, then we have f 1 (0) = h − cy 2 < 0 and lim s→+∞ f 1 (s) = +∞. Hence, there is at least a positive root s * such that f 1 (s * ) = 0. Therefore, when R 3 > 1, E 2 is unstable. This completes the proof. Remark 3.6: From Theorem 3.5 we see that delays τ 1 , τ 2 and τ 3 do not impact the stability of E 2 . Biologically, Theorem 3.5 implies that when R 0 > 1, R 1 > 1 and R 3 ≤ 1, the antibody immune response can be established, but the infected cells are too weak so that it can not stimulate CTL immune response. Theorem 3.7: Let R 0 > 1 and R 2 > 1.
(a) If R 4 ≤ 1 and τ 3 = 0, then the infection equilibrium E 3 with only CTL response is globally asymptotically stable; (b) If R 4 > 1, then the equilibrium E 3 is unstable.
Next, we consider conclusion (b). The characteristic equation of the linearization system of model (1) at the equilibrium E 3 is If R 4 > 1, then we have a positive root s * = gv 3 − α. Therefore, when R 4 > 1, E 3 is unstable for any τ 1 ≥ 0, τ 2 ≥ 0 and τ 3 = 0. This completes the proof.
Remark 3.8: Theorem 3.7 shows that delays τ 1 and τ 2 do not impact the stability of E 3 . Biologically, we see that, when τ 3 = 0, for any τ 1 ≥ 0 and τ 2 ≥ 0 as long as R 0 > 1, R 2 > 1, R 4 ≤ 1 then the CTL immune response can be determined, but the virus loads are so small that it can not activate the antibody immune responses.

then the infection equilibrium E 4 with CTL and antibody responses is globally asymptotically stable.
Proof: Define a Lyapunov functional x 4 1 + aθ + bv 4 1 + ax 4 + bv 4 Calculating the derivative of V 5 (t) along the solution of model (1), it follows that Notice that bv(t)) .

Remark 4.4:
We here only establish the criteria on the existence of Hopf bifurcations at equilibria E 3 and E 4 for model (1) in the case of delays τ 1 = τ 2 = 0 and τ 3 > 0. However, when τ 1 > 0 or τ 2 > 0 whether we also can obtain similar results still is a very interesting and estimable problem. In the following section, we will give a discussion by means of the numerical simulations.

Numerical simulations
In the above sections, we establish the global asymptotic stability of equilibria E 3 and E 4 when τ 1 ≥ 0, τ 2 ≥ 0 and τ 3 = 0, and by using the theory of bifurcation, we obtain the existence of the Hopf bifurcation and stability switches at equilibria E 3 and E 4 when τ 1 = 0, τ 2 = 0 and τ 3 ≥ 0. However, aim at the case: τ 1 ≥ 0, τ 2 ≥ 0 and τ 3 ≥ 0, the theoretical analysis is very complicated. In this section, by using the numerical simulation, it is shown that the Hopf bifurcation and stability switches occur at these equilibria as τ 3 increases. In model (1), we choose a, α, β, τ 1 , τ 2 and τ 3 as free parameters and fix all other parameters as displayed in Table 1.

Discussion
In this paper, we have investigated a virus infection model (1) with intracellular delay τ 1 , virus replication delay τ 2 and immune response delay τ 3 . We assume that the production of CTL immune response depends on the infected cells and CTL cells based above important biological meaning. We see that similar assumption also is given in [1,9,11,12,16,18,24]. Similarly, the production of antibody response depends on the virus and antibody (see [1,13,14,16]). Dynamical analysis shows that τ 1 , τ 2 and τ 3 play different roles in the stability of the model.
By the analysis, model (1) has five possible equilibria, an infection-free equilibrium E 0 , immune-free equilibrium E 1 , infection equilibrium E 2 with only antibody response, infection equilibrium E 3 with only CTL response and infection equilibrium E 4 with both CTL and antibody responses. A combination of basic reproductive ratio of viral infection R 0 , for antibody response R 1 , for CTL immune response R 2 , for CTL immune response competitive R 3 and for antibody response competitive R 4 determines the existence of these equilibria. Furthermore, they also determine the global properties of the model. We have shown that when R 0 ≤ 1, E 0 is globally asymptotically stable, which means that the viruses are cleared and immune is not active. When R 0 > 1, R 1 ≤ 1 and R 2 ≤ 1, E 1 is globally asymptotically stable, which means that the infection becomes chronic but with no persistent CTL immune responses and antibody responses. When R 1 > 1 and R 3 ≤ 1, E 2 is globally asymptotically stable, which means that the infection becomes chronic with persistent antibody responses, but the infected cells can not stimulate and activate CTL immune responses. As respect to the analysis of E 3 , we consider special case τ 1 ≥ 0, τ 2 ≥ 0 and τ 3 = 0, when R 2 > 1 and R 4 ≤ 1, E 3 is globally asymptotically stable, which means that the infection becomes chronic with persistent CTL immune responses, but the virus loads can not activate the antibody responses. About the stability of E 4 , we have obtained that for special case τ 1 ≥ 0, τ 2 ≥ 0 and τ 3 = 0, when R 3 > 1 and R 4 > 1, E 4 is globally asymptotically stable, that is, susceptible cells, infected cells, free virus, antibody responses and CTL responses coexist in vivo. We see that τ 1 and τ 2 do not affect the stability of the equilibria.
When τ 3 > 0, by using the bifurcation theory, we obtain the sufficient conditions on the existence of Hopf bifurcation at E 3 and E 4 . Meanwhile, by means of numerical simulations, it is shown that the Hopf bifurcation and stability switches occur at E 3 and E 4 as τ 3 increases. Figures 1-4 indicate that E 3 remains stable as τ 3 > 0 is small, and along with the increase of τ 3 , E 3 becomes unstable and periodic oscillations appear. It shows that stability switches occur as τ 3 increases. Similarly, from Figures 5-8, we see that along with the increase of τ 3 the dynamical behaviours of model (1) at E 4 appear very large diversification. Particularly, when τ 3 is small enough, E 4 is asymptotically stable and when τ 3 is increasing, the stability switches occur at E 4 , and when E 4 is unstable, Hopf bifurcation occurs. Finally, when τ 3 is large enough, E 4 is always unstable. Summarizing these discussions, we point out that τ 3 affects markedly the stability of E 3 and E 4 . This illustrates the fact that τ 3 plays a negative part in the disease prevalence and control. Motivated by the above discussion, one can realize that the emergence in the broadly neutralizing antibodies reacted to change in infected cells that could kill viruses. On the whole, this paper somehow provides better understanding about neutralizing antibodies and might help to design a powerful vaccine, which prevents at least uninfected peoples from ever becoming infected with virus. However, by considering some other factors such as the antibody delay, diffusion and a time-varying drug concentration, whether we also can obtain that the global asymptotic stability of equilibria will also be a very estimable and significative subject. This is left to the future research.