Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays

1.   Introduction

Human immunodeficiency virus (HIV) mainly infects CD4$^+$ T-cells in human body and destroys the immune system gradually, which leads to the occurrence of various opportunistic infections. HIV can persist in a latent form in resting CD4$^+$ T-cells, and it can give rise to infectious virus upon stimulation in vitro, which becomes an obstacle to the eradication of virus ^[1,2]. The mathematical model of HIV infection has attracted more and more scholars' attention, and the classical virus dynamics model is composed of three populations: healthy T-cells, infected T-cells and virus ^[3,4]. Recently, the four-dimensional mathematical models incorporating latent infection have been developed to describe the mechanism of the latency ^[5,6].

When HIV invades the human body, the human body makes two modes of immune responses: one is the humoral immune response mediated by B-cell, and the other is the cellular immune response mediated by cytotoxic T lymphocyte (CTL). It is not clear which immune response mode is more effective. Previous studies have established that the humoral immune response is more effective than the cellular immune reponse in malaria infection ^[7]. Therefore, we focus on the humoral immune response mediated by B-cell here.

In the process of virus infection, we are more concerned about the number of new infectious virions in unit time, that is, the incidence rate. The incidence rate function of virus infection model is mostly bilinear $\beta TV$, and then gradually develops to saturation function $\frac{\beta TV}{1+aT}$ ^[8], Holling type II function $\frac{\beta TV}{1+bV}$ ^[9] and Beddington-DeAngelis function $\frac{\beta TV}{1+aT+bV}$ ^[10,11,12] ($\beta$ is the infection rate, and $a, b \geq 0$ are the inhibition constants). The Beddington-DeAngelis function is firstly proposed by Beddington ^[14] and DeAngelis et al. ^[15], and it is a generalized infection function, which includes the cases of bilinear, saturation and Holling type II functions. Hence, we will consider the Beddington-DeAngelis incidence rate.

In the process of HIV infection, it takes time for the virus to infect healthy T-cells and then release infectious virus particles. Herz et al. first introduced an intracellular delay to reflect this time period and showed that the model with time delay could shorten the half-life of free virus ^[16]. Since then, many researchers have analyzed the effect of intracellular delay on the stability of the equilibrium of the virus infection models. Their results showed that the intracellular delay could prolong the progress of the virus, but did not affect the stability of the equilibrium essentially ^{[9,11,12,13,17,18,19,20,21,22,23,24,25,26,27,28]}. In addition, it also takes time for antigenic stimulation to generate an immune response ^[29]. It shows that the immune delay makes the positive equilibrium unstable and leads to Hopf bifurcation. So, both the intracellular delay and the immune delay will be included in our model, which can reflect the progress of virus more practically.

Motivated by the works of ^[17,18,19], we propose an HIV virus infection model incorporating the latent infection, the Beddington-DeAngelis function infection rate, the humoral immunity and mutiple time delays, which can be described as

Here, $T(t)$, $L(t)$, $I(t)$, $V(t)$ and $B(t)$ denote the number of healthy T-cells, latently infected T-cells, actively infected T-cells, virions and B-cells at time $t$, respectively. The healthy T-cells are assumed to be input with a constant rate $\lambda$, and it can be infected by the virus with the Beddington-DeAngelis functional response $\frac{\beta T(t)V(t)}{1+aT(t)+bV(t)}$. The constant $\eta$ is the fraction of the infected T-cells leading to latency. Latently infected T-cells can be activated and then becomes the productively infected T-cells with a constant $\alpha$. Parameter $k$ is the production rate of each infected T-cell, and $d_i$ $(i = 1,\, 2,\, 3,\, 4,\, 5)$ is the death rate of each population. $p$ and $q$ are the B-cell effectiveness and responsiveness, respectively. Here, we consider four time delays: (ⅰ) $\tau_1$ represents a time delay between the initial virus entering into the cell and the subsequent viral latency; (ⅱ) $\tau_2$ represents a time delay between the cell infection and the subsequent viral generation; (ⅲ) $\tau_3$ is the time it takes from the newly produced virus to be mature and then infectious; (ⅳ) $\tau_4$ is the time for the B-cell immune system to be activated. Factors $e^{-m\tau_1}$ and $e^{-m\tau_2}$ denote the probability that an infected T-cell survives the interval $\tau_1$ and $\tau_2$, respectively.

The rest of this paper is arranged as follows. In section 2, we analyze the well-posedness of the model solution, and obtain the basic reproduction number, the immune response reproduction number and three equilibria. In section 3, by analyzing the characteristic equation at each equilibrium and employing appropriate Lyapunov functions, the global asymptotic stability of uninfected, B-cell-inactivated and B-cell-activated equilibria is obtained, respectively. Furthermore, Hopf bifurcation occurs at the B-cell-activated equilibrium if the system including the immune response delay. In section 4, numerical simulations are given to further investigate the delays and their effects on the stability of the B-cell-activated equilibrium. Finally, we conclude our work in section 5.

2.   Preliminary results

2.1.   Well-posedness

Assuming that system (1.1) satisfies the following initial value:

$\tau = \max\{\tau_1, \tau_2, \tau_3, \tau_4\}$. Here, $\psi_2$ is a given non-negative constant, $\psi_1(\theta)$, $\psi_3(\theta)$, $\psi_4(\theta)$, $\psi_5(\theta)$ $\in C([-\tau, 0]$, $\mathbb{R}_{+})$ with $\mathbb{R}_{+} = [0, +\infty)$, and $\psi = (\psi_1, \psi_2, \psi_3, \psi_4, \psi_5)\in C\times \mathbb{R}_{+}\times C\times C\times C$.

Theorem 2.1. For system (1.1), there exists a unique non-negative solution with initial value (2.1), and the solution is ultimately bounded for all $t\geq 0$.

Proof. By the basic theory of the functional differential equations ^[30], there exists a unique solution satisfying the initial condition (2.1). According to the proof process in Theorem 3.1 of reference ^[20], we can obtain the nonnegativity of the solution and we omit the proof process here. In the following, we prove the boundedness of the solution.

By the first equation of system (1.1), we calculate that $\underset{t\rightarrow +\infty}{\lim}\sup \,T(t)\leq T_0$, and see Eq (2.2) for the expression of $T_0$.

Define

Calculating the derivative of $P(t)$ along the solution of system (1.1), we obtain

where $d = \mathrm{min}\{d_1, d_2, \frac{d_3}{2}, d_4, d_5\}$. Therefore,

which indicates that $T(t)$, $L(t)$, $I(t)$, $V(t)$ and $B(t)$ are all ultimately bounded. This completes the proof.

Suppose that there exists a positive constant $M>0$ such that $T\leq T_0,\, L,\, I,\, V,\, B\leq M$ for large $t$. We will analyze the dynamics of system (1.1) in the following bounded feasible region

2.2.   Reproduction numbers and equilibria

System (1.1) always has one uninfected equilibrium $E_0$ $(T_0,\, 0,\, 0,\, 0,\, 0)$, where

For convenience, we define

Following the derivation method ^[31,32], we obtain the basic reproduction number of model (1.1),

We also define the B-cell immune response reproduction number,

and see Eq (2.6) for the expressions of $T_2$ and $V_2$.

System (1.1) has a B-cell-inactivated equilibrium $E_1$ $(T_1,\, L_1,\, I_1,\, V_1,\, 0)$ if $\mathcal{R}_0>1$, where,

and

Let

Since $T\leq T_0$ in the bounded region $\Gamma$, from the expressions of $\mathcal{R}_0$ and $\mathcal{R}^*$, we can derive that

System (1.1) also has a B-cell-activated equilibrium $E_2$ $(T_2,\, L_2,\, I_2,\, V_2,\, B_2)$ if $\mathcal{R}_1>1$, where,

and

3.   Stability analysis

To study the stability at equilibrium $\bar{E}$ $(\bar{T},\, \bar{L},\, \bar{I},\, \bar{V},\, \bar{B})$, we let $Y_1(t) = T(t)-\bar{T}$, $Y_2(t) = L(t)-\bar{L}$, $Y_3(t) = I(t)-\bar{I}$, $Y_4(t) = V(t)-\bar{V}$, $Y_5(t) = B(t)-\bar{B}$, and $Y(t) = (Y_1(t),\, Y_2(t),\, Y_3(t),\, Y_4(t),\, Y_5(t))$. The linearization of system (1.1) at $\bar{E}$ $(\bar{T},\, \bar{L},\, \bar{I},\, \bar{V},\, \bar{B})$ becomes

where,

The characteristic equation of linearized system (3.1) at the equilibrium $\bar{E}$ can be written as

3.1.   Global asymptotic stability of the uninfected equilibrium $E_0$

Theorem 3.1. If $\mathcal{R}_0<1$, the uninfected equilibrium $E_0$ is locally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3,\, 4$, and it is unstable if $\mathcal{R}_0>1$.

Proof. The characteristic equation of the linearized system (3.1) at the uninfected equilibrium $E_0$ is

where,

It is clear that $\xi = -d_1<0$ and $\xi = -d_5<0$. The remaining roots of Eq (3.2) can be determined by the following equation

It is equivalent to the following equation

It is assumed that $\xi = x+iy$ ($x\geq0$) is a solution of Eq (3.4), and we take the modulus on both sides,

If $\mathcal{R}_0<1$, the above inequality does not hold. Therefore, Eq (3.3) has only negative real part roots. Hence, $E_0$ is locally asymptotically stable if $\mathcal{R}_0<1$ for all $\tau_i \geq 0$, $i = 1,\, 2,\, 3,\, 4$.

If $\mathcal{R}_0>1$, $g_1(0) = d_3d_4(\alpha+d_2)-kN_0\rho(\tau_1, \tau_2)(\alpha+d_2) = d_3d_4(\alpha+d_2)(1-\mathcal{R}_0)<0$, and $\lim\limits_{\xi\rightarrow+\infty}g_1(\xi) = +\infty$. So, Eq (3.3) has at least one positive real root. Therefore, the uninfected equilibrium $E_0$ is unstable for all time delays.

Theorem 3.2. If $\mathcal{R}_0<1$, the uninfected equilibrium $E_0$ is globally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3,\, 4$.

Proof. We employ a particular function

and it was first proposed in literature ^[33]. The function $G(x)>0$ for $x>0$, and $G(x) = 0$ when and only when $x = 1$. We define a Lyapunov function $W$: $C\times \mathbb{R}_{+}\times C\times C\times C\rightarrow \mathbb{R}$

The derivative of $W_1$ and $W_2$ along the solution of system (1.1) is

Using the equality $\lambda = d_1T_0$ at the uninfected equilibrium and Eq (2.3), we calculate that

If $\mathcal{R}_0<1$, then $\dot{W}\leq 0$. Clearly, the singleton $E_0$ is the largest invariant set in $\{X\in \Gamma|\dot{W} = 0\}$. By the LaSalle's invariance principle ^[34], we obtain the global attractivity of $E_0$. Combining with Theorem 3.1, we derive the global asymptotic stability of $E_0$.

3.2.   Global asymptotic stability of B-cell-inactivated equilibrium $E_1$

Define a critical condition

Theorem 3.3. If $\mathcal{R}_2<1<\mathcal {R}_0$, then the B-cell-inactivated equilibrium $E_1$ is locally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3,\, 4$, and it is unstable if $\mathcal{R}_2>1$.

Proof. The characteristic equation of the linearized system (3.1) at the B-cell-inactivated equilibrium $E_1$ is

where,

(Ⅰ) Transcendental equation

If $\tau_4 = 0$, then $\xi = qV_1-d_5$. Under the condition $\mathcal{R}_2<1$, that is $qV_1-d_5<0$. If $\tau_4>0$, suppose $\xi = i\omega$ ($\omega>0$), then Eq (3.6) can be written as

That is to say,

Under the condition $\mathcal{R}_2<1$, there is no positive $\omega$ such that the above equation holds. Therefore, Eq (3.6) only has negative real roots. If $\mathcal{R}_2>1$, then Eq (3.6) has positive real roots.

(Ⅱ) Equation

and it is equivalent to the following equation

Assume $\xi = x+iy$ ($x\geq0$) is a solution of Eq (3.8), and we take the modulus on both sides. Clearly, the modulus of the left hand side of Eq (3.8) is greater than one. Suppose that the modulus of the right hand side of Eq (3.8) is $\Lambda_1$, use Eq (3.5) and equation

at the B-cell-inactivated equilibrium $E_1$, we have

If $\mathcal{R}^*>1$, that is $\mathcal{R}_0>1$ (see Eq (2.5)), then $\Lambda_1<1$, which leads to a contradiction. Therefore, Eq (3.7) has only negative real part roots.

Combined with the above two steps, we conclude that, under the condition $\mathcal{R}_2<1<\mathcal{R}_0$, $E_1$ is locally asymptotically stable for all time delays, and it is unstable if $\mathcal{R}_2>1$.

Theorem 3.4. If $\mathcal{R}_2<1<\mathcal{R}_0$, then the B-cell-inactivated equilibrium $E_1$ is globally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3,\, 4$.

Proof. Define a Lyapunov function $U: C\times \mathbb{R}_{+}\times C\times C\times C\rightarrow \mathbb{R}$

where, the expression of $F(T_1, V_1)$ can be found in Eq (2.4). Computing the derivative of $U_1$ and $U_2$ along the solution of system (1.1), we obtain

where,

Thus,

where,

It follows that,

Thereafter,

where,

Note that

Then, we have

where

Therefore,

If $\mathcal{R}_2<1$, we obtain $\dot{U}\leq 0$. When $\dot{U} = 0$ if and only if

That is, $T(t) = T_1$, $L(t) = L_1$, $I(t) = I_1$, $V(t) = V_1$ and $B(t) = 0$. It is obvious that the singleton $E_1$ is the largest invariant set in $\{X\in \Gamma|\dot{U} = 0\}$. By the LaSalle's invariance principle ^[34], we obtain the global attractivity of $E_1$. Combining with Theorem 3.3, if $\mathcal{R}_2<1<\mathcal{R}_0$, we conclude the global asymptotic stability of $E_1$ for $\tau_i \geq 0$, $i = 1,\, 2,\, 3,\, 4$.

3.3.   Global asymptotic stability of the B-cell-activated equilibrium $E_2$ if $\tau_4=0$

Theorem 3.5. If $\mathcal{R}_1>1$, the B-cell-activated equilibrium $E_2$ is locally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3$ and $\tau_4 = 0$.

Proof. The characteristic equation of the linearized system (3.1) at the infected state $E_2$ is

where,

If $\tau_4 = 0$, then

Divide both sides by $\xi+\alpha+d_2$, and the equation simplifies to

Let $\xi = x+iy$ ($x\geq 0$) is a solution of Eq (3.10), then

Take the modulus on both sides, and it is easy to see that

Assume that $\Lambda_2$ and $\Lambda_3$ are the modulus of the left-hand side and the right-hand side for above equation, respectively. Then,

At the B-cell-activated equilibrium $E_2$, we have

So,

Obviously, this leads to a contradiction. Therefore, under the condition $\mathcal{R}_1>1$, $E_2$ is locally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3$ and $\tau_4 = 0$.

Theorem 3.6. If $\mathcal{R}_1>1$, the B-cell-activated equilibrium $E_2$ is globally asymptotically stable for $\tau_i \geq 0$, $i = 1,\, 2,\, 3$ and $\tau_4 = 0$.

Proof. Define a Lyapunov function $Q: C\times \mathbb{R}_{+}\times C\times C\times C\rightarrow \mathbb{R}$

Take the derivative of $Q_1$ along the solution of system (1.1), and we obtain

Use the equalities at the B-cell-activated equilibrium $E_2$,

and we have

where,

Thus,

where,

Take the derivative of $Q_2$ along the solution of system (1.1), and we obtain

where,

Then, we have

where,

Then,

If $\mathcal{R}_1>1$, we obtain $\dot{Q}\leq 0$. When $\dot{Q} = 0$ if and only if

That is, $T(t) = T_2$, $L(t) = L_2$, $I(t) = I_2$, $V(t) = V_2$ and $B(t) = B_2$. Clearly, the singleton $E_2$ is the largest invariant set in $\{X\in \Gamma|\dot{Q} = 0\}$. By the LaSalle's invariance principle ^[34], we obtain the global attractivity of $E_2$. Combining with Theorem 3.5, under the condition $\mathcal{R}_1>1$, we derive the global asymptotic stability of $E_2$ for $\tau_i \geq 0$, $i = 1,\, 2,\, 3$ and $\tau_4 = 0$.

3.4.   Stability of the B-cell-activated equilibrium $E_2$ if $\tau_4>0$

For the characteristic Eq (3.9) at the B-cell-activated equilibrium $E_2$, let $\tau_1 = \tau_2 = \tau_3 = 0$, then it yields

For convenience, let $h_1 = d_1+A_2$, $h_2 = \alpha+d_2$, $h_3 = d_3$, $h_4 = d_4+pB_2$ and $h_5 = d_5$, the above equation can be rewritten as the following,

where,

From Theorem 3.5, we know that the B-cell-activated equilibrium $E_2$ is locally asymptotically stable when $\tau_i = 0$, $i = 1,\, 2,\, 3,\, 4$.

Now, assume that $\tau_4>0$ and let $\xi = i\omega$, $\omega>0$ of Eq (3.11). Separating real and imaginary parts yields

Squaring and adding these equation, we obtain

where,

and

Let $z = \omega^2>0$, then Eq (3.13) becomes

and the derivative of $H(z)$ with respect to $z$ is

From the expressions of $m_0$ in Eq (3.14), we know that we could not determine the sign of $m_0$, and it depends on the size of $\mathcal{R}_1$. According to the relationship between root and coefficient, from the expressions of $m_4$ in Eq (3.14), we derive that Eq (3.15) has at most four positive real roots.

Theorem 3.7. Assume $\mathcal{R}_1>1$ and $\tau_1 = \tau_2 = \tau_3 = 0$, then the B-cell-activated equilibrium $E_2$ is locally asymptotically stable when $\tau_4 \in [0, \tau_{40})$ and unstable when $\tau_4>\tau_{40}$. Furthermore, if $H'(\omega_0^2)\neq 0$, system (1.1) undergoes a Hopf bifurcation to periodic solutions at $E_2$ when $\tau_4 = \tau_{40}$.

Proof. Without loss of generality, suppose that Eq (3.15) has four positive real roots $z_1, z_2$, $z_3$ and $z_4$. Then Eq (3.13) has positive roots $\omega_i = \sqrt{z_i}$ $(i = 1,\, 2,\, 3,\, 4)$, and thus $\pm i\omega_i$ is a pair of purely imaginary roots of Eq (3.11). From Eq (3.12), the corresponding $\tau_{4i}$ are

It is clear to see that

Define

Now, differentiating Eq (3.11) with respect to $\tau_4$, we obtain

Then,

Using Eqs (3.12) and (3.16), we get

where,

Thus,

Thereafter,

Since $(q_3\omega_0^2-q_1)^2\omega_0^2+(q_4\omega_0^4-q_2\omega_0^2+q_0)^2>0$, then

Therefore, a Hopf bifurcation occurs at $\tau_4 = \tau_{40}$ under the condition $H'(\omega_0^2)\neq 0$. Furthermore, a family of periodic solutions occurs for system (1.1) when $\tau_4>\tau_{40}$ and $\tau_i = 0$, $i = 1,\, 2,\, 3$.

4.   Numerical simulations

In this section, we perform numerical simulations to show the dynamical behaviour at the B-cell-activated equilibrium $E_2$ of system (1.1), for different values of the cell infection delay $\tau_2$ and the immune system delay $\tau_4$. We choose parameter values from the modelling literature (see Table 1), and let $a = b = 0.015$ in the following numerical simulations.

In order to study the effect of cell infection delay on the dynamical behaviour of system (1.1), we fix $\tau_1 = \tau_3 = 1$, $\tau_4 = 0$ and choose $\tau_2 = 0$, $\tau_2 = 2$, $\tau_2 = 6$ and $\tau_2 = 12$, respectively. Using Data values in Table 1, we calculate that $\mathcal{R}_0>1$ and $\mathcal{R}_1>1$ when $\tau_2 = 0, \tau_2 = 2$ and $\tau_2 = 6$, while $\mathcal{R}_0<1$ when $\tau_2 = 12$. From Figure 1, we can see that, the B-cell activated equilibrium $E_2$ is globally asymptotically stable for $\tau_2$ equals 0, 2 and 6, respectively, which is consistent with Theorem 3.6; the uninfected equilibrium $E_0$ is globally asymptotically stable for $\tau_2 = 12$, which is consistent with Theorem 3.2. Moreover, as the cell infection delay $\tau_2$ increases, both the number of actively infected T-cells ($I$) and B-cells ($B$) decrease, while the number of healthy T-cells ($T$) and virus ($V$) remain unchanged. It is also observed that the cell infection delay $\tau_2$ can cause the T-cell, virus and B-cell populations to oscillate in the early stage of infection, and prolong the time for system to reach the B-cell-activated equilibrium. In fact, for the immune delay $\tau_4 = 0$, when system (1.1) has only the latency delay $\tau_1$ or the virus maturation delay $\tau_3$ and the remaining delay values are fixed, we can obtain graphs similar to Figure 1 (not shown here).

Next, we investigate the system behaviour at the B-cell-activated equilibrium $E_2$ when the immune delay $\tau_4>0$ and $\tau_1 = \tau_2 = \tau_3 = 0$. Choose $\beta = 0.0032$ and all the other parameter values can be seen in Table 1. By calculating, we obtain that $\mathcal{R}_1 = 1.2781>1$, Eq (3.13) has only one positive real root $\omega\approx0.0925$, and $\tau_{40} \approx 4.9066$. In this situation, Theorem 3.7 is satisfied. Figure 2 shows that the B-cell-activated equilibrium is locally asymptotically stable when the immune time delay $\tau_4 = 4<\tau_{40}$, while Figure 3 demonstrates that Hopf bifurcation and period solutions occur at the B-cell-activated equilibrium when $\tau_4 = 7>\tau_{40}$.

Now, we examine the dynamics of system (1.1) when all time delays $\tau_i>0$, $i = 1,\, 2 ,\, 3,\, 4$. The two delays $\tau_1 = 1$ and $\tau_3 = 1$ are fixed, and all the other parameter values are from Table 1 in the following numerical simulations. Let $\tau_4 = 6$ be fixed, and $\tau_2\in(0, 12]$ varies. With the increase of the cell infection delay $\tau_2$, from the left panel of Figure 4, we observe that, (ⅰ) the system firstly undergoes the Hopf bifurcation at the B-cell-activated equilibrium $E_2$, and the amplitude of the periodic solution decreases gradually; (ⅱ) then the system experiences the local asymptotic stability of the B-cell-inactivated equilibrium $E_1$, and finally reaches the local asymptotic stability of the uninfected equilibrium $E_0$. Similar figures can be obtained when $\tau_1$ or $\tau_3$ varies and the other delays are fixed (not shown here). Moreover, let $\tau_2 = 2$ be fixed, and $\tau_4\in(0, 20]$ varies. With the increase of the immune delay $\tau_4$, from the right panel of Figure 4, we find that, the system firstly reaches the local asymptotic stability of the B-cell-activated equilibrium $E_2$, then undergoes the Hopf bifurcation at the B-cell-activated equilibrium. In fact, the periodic solutions still exist when $\tau_4$ is greater than 20 (not shown here).

5.   Conclusions

In this paper, we have formulated an HIV infection model including the latently infected cell, the Beddington-DeAngelis infection function, the B-cell immune response and multiple delays. The latency delay, the cell infection delay, the virus maturation delay and the immune delay of B-cell activation are considered in our model. Two basic reproduction numbers and three equilibria are obtained. Theoretically, by constructing the appropriate Lyapunov function, we obtain that both the uninfected and B-cell-inactivated equilibria are globally asymptotically stable for the four time delays. The B-cell-activated equilibrium is globally asymptotically stable for the first three delays, while the immune delay may destabilize the stability of the B-cell-activated equilibrium and lead to Hopf bifurcation.

It is a challenging problem to analyze system (1.1) for the joint effect of four time delays theoretically. So, numerical simulations are performed to investigate the dynamical behaviour at the B-cell-activated equilibrium $E_2$ for $\tau_i>0$ $i = 1,\, 2,\, 3,\, 4$. It was discovered that, the latency delay $\tau_1$, the cell infection delay $\tau_2$, and the virus maturation delay $\tau_3$ can stabilize the B-cell-activated equilibrium, and the bifurcation disappears gradually; while the immune response delay $\tau_4$ can lead to its instability, and bifurcation occurs.

As far as we know, HIV infection model considering the latent infection, the Beddington-DeAngelis incidence function and the B-cell immune response is rare, and the theoretical study on the analysis of multiple time delay models is also rare. In this paper, we analyze the global stability of the equilibrium through constructing suitable Lyapunov functions, which is a generalization of the existing models.

Acknowledgments

The authors would like to thank the referees for their valuable suggestions. This work is supported by National Natural Science Foundation of China (No. 11801566, No. 11871473, No. 11401589), the Fundamental Research Funds for the Central Universities (No. 18CX02049A), and Shandong Provincial Natural Science Foundation.

Conflict of interest

The authors declare that they have no conflict of interest.