1 Introduction

Nowadays, virus infection is related to the global health problems. Many diseases which are caused by viruses, such as hepatitis B virus (HBV), human immunodeficiency virus (HIV), hepatitis C virus (HCV), have drawn the attention of researchers. Based on the virus infection models proposed in [1–18], several different mathematical models which are valuable to obtaining comprehensive views to the virus dynamics have been investigated, for example, in a form of ordinary differential equations (ODEs) [12, 19–21], delayed differential equations (DDEs) [1–5, 7, 9, 11, 22], partial differential equations (PDEs) [8, 22–28] and fractional-order differential equations (FODEs) [15–18]. Nowak and Bangham [19] pointed out the basic virus infection model which plays a critical role in understanding the virus replication dynamics in vivo. They considered the following basic mathematical model for uninfected host cells u, infected host cells w, free virus v and the magnitude of the CTL response z:

$$\begin{aligned}& {\frac{du(t)}{dt}}=\lambda-du(t)-\beta u(t)v(t), \\& {\frac{dw(t)}{dt}}=\beta u(t)v(t)-h w(t)-pw(t)z(t), \\ & \\ & {\frac{dv(t)}{dt}}=kw(t)-\mu v(t), \\& {\frac{dz(t)}{dt}}=cw(t)z(t)-qz(t), \end{aligned}$$
(1)

where the uninfected host cells are produced at rate λ, die at rate d and become infected at rate β. Infected host cells die at rate h and are killed by the CTL response at rate p. Free virus is produced from infected cells at rate k and is removed at rate μ. The magnitude of the CTL response, which expands in response to viral antigen derived from infected cells at rate c, decays in the absence of antigenic stimulation at rate q.

It should be mentioned here that, in model (1), the rate of infection is assumed to be bilinear, that is, \(\beta uv\). However, this assumption is not biologically sensible all the time. Recently, many researchers have performed the virus dynamics models with Crowley-Martin infection rate (see [20, 21]). The Crowley-Martin type of functional response, that is, \(\frac{\beta uv}{(1+au)(1+bv)}\), was introduced by Crowley and Martin in [29], where a, b are constants. Particularly, when \(a=0\), \(b=0\), the Crowley-Martin infection rate becomes bilinear infection rate. Thus, it is necessary to study virus infection model with Crowley-Martin infection rate. In addition, based on the epidemiological background, time delays play a critical role in the virus infection model. To incorporate the intracellular phase of the virus life-cycle, we assume that virus production occurs after the virus entry by the intracellular delay \(\tau_{1}\). The recruitment of virus producing cells at time t is given by the number of uninfected cells that were newly infected at time \(t-\tau_{1}\) and are still alive at time t (see [2, 5, 9, 11]). The constant m is assumed to be the death rate for newly infected cells during time period \([t-\tau_{1},t]\). \(e^{-m \tau_{1}}\) denotes the surviving rate of infected cells during the delay period. Virus replication delay \(\tau_{2}\) represents the time necessary for the newly produced viruses to become mature and then infectious (see [10, 14, 22]). The constant n is assumed to be the death rate of a new virus during time period \([t-\tau_{2},t]\). \(e^{-n \tau_{2}}\) denotes the surviving rate of a virus during the delay period.

Until now, there has been a large number of works about virus infection models which considered delays, but in many biological systems, the species under consideration may disperse spatially as well as evolve in time (see [30]). As a matter of fact, many models ignored the spatial mobility of cells and viruses. Recently, many authors argued that the virus moves freely in body and follows the Fickian diffusion (see [31]) and investigated the global stability properties of virus infection models with diffusion in [8, 22–28, 31]. But the research is relatively small, a lot of work needs to be further done to provide theoretical evidence for controlling disease.

In this paper, motivated by the work of [6, 22, 23], we further neglect the mobility of susceptible cells, infected cells and immune cells and consider a delayed virus infection model with Crowley-Martin infection rate and spatial diffusion:

$$ \begin{aligned} &{\frac{\partial u}{\partial t}}=\lambda -du(x,t)-{ \frac{\beta u(x,t)v(x,t)}{ (1+au(x,t) ) (1+bv(x,t) )}}, \\ &{\frac{\partial w}{\partial t}}=e^{-m\tau_{1}}{\frac{\beta u(x,t- \tau_{1})v(x,t-\tau_{1})}{ (1+au(x,t- \tau_{1}) ) (1+bv(x,t-\tau_{1}) )}}-hw(x,t)-pw(x,t)z(x,t), \\ &{\frac{\partial v}{\partial t}}= D\Delta v(x,t)+ke ^{-n\tau_{2}}w(x,t- \tau_{2})-\mu v(x,t), \\ &{\frac{\partial z}{\partial t}}=cw(x,t)z(x,t)-qz(x,t) \end{aligned} $$
(2)

for \(t>0\), \(x\in\Omega\subset\mathbb{R}^{n}\) with the initial conditions

$$ \begin{aligned} &u(x,\theta)=\phi_{1}(x, \theta)\geq 0,\qquad w(x,\theta)=\phi_{2}(x, \theta)\geq0, \\ &v(x,\theta)=\phi_{3}(x,\theta)\geq0, \qquad z(x,\theta)= \phi_{4}(x, \theta)\geq0,\quad x\in\bar{\Omega}, \theta\in [-\tau ,0], \end{aligned} $$
(3)

and the homogeneous Neumann boundary conditions

$$ \frac{\partial v}{\partial\vec{n}}=0, \quad t>0, x\in\partial\Omega, $$
(4)

where \(u(x, t)\), \(w(x, t)\), \(v(x, t)\) and \(z(x, t)\) represent the densities of uninfected cells, infected cells, free virus and immune cells at location x and time t, respectively. The Laplacian operator and the diffusion coefficient are denoted by Δ and D, respectively. \(\tau=\max\{\tau_{1}, \tau_{2}\}\), Ω is a connected, bounded domain in \(\mathbb{R}^{n}\) with smooth boundary ∂Ω. \(\frac{\partial}{\partial\vec{n}}\) denotes the outward normal derivative on ∂Ω. \(\phi_{i}(x,\theta)\) (\(i=1,2,3,4\)) are nonnegative and Hölder continuous in \(\bar{\Omega}\times[-\tau,0]\). The boundary conditions in (4) imply that the virus particles do not move across the boundary ∂Ω.

In this paper, the purpose is to investigate the dynamical properties of model (2), expressly the stability of equilibria. Our paper is organized as follows. In the next section, we discuss the positivity and boundedness of solutions, the threshold values and the existence of equilibria of model (2). In Section 3, by constructing Lyapunov functionals, we establish global stability of all equilibria of model (2). In Section 4, we further illustrate the dynamical behavior by numerical simulations. In the last section, we give brief conclusions.

2 Positivity, boundedness and equilibrium

In this section, our main purpose is to prove the existence, positivity and boundedness of solutions of model (2).

Theorem 2.1

For any given initial data satisfying condition (3), there exists a unique solution of model (2) defined on \([0,+\infty)\), and this solution remains nonnegative and bounded for all \(t\geq0\).

Proof

By standard existence theory [32–34], it is easy to establish the local existence of the unique solution \((u(x,t), w(x,t), v(x,t),z(x,t))\) of model (2) for \(x\in\overline{\Omega}\) and \(t\in[0,T_{\max}]\), where \(T_{\max}\) is the maximal existence time for solution of the model (2).

It is not hard to see that \(\textbf{0}=(0,0,0,0)\) and \(\textbf{M}=(M _{1}, M_{2}, M_{3}, M_{4})\) are a pair of coupled lower-upper solutions to model (2), where \(M_{1}\), \(M_{2}\), \(M_{3}\) and \(M_{4}\) satisfy

$$ \begin{aligned} &M_{1}=\max \biggl\{ {\frac{\lambda}{d}}, \sup_{-\tau\leq s\leq0} \bigl\Vert \phi_{1}(\cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \\ &M_{2}=\max \biggl\{ {\frac{\beta M_{1}}{lb}}e^{-m\tau_{1}}, \sup _{-\tau\leq s\leq0} \bigl\Vert \phi_{1}(\cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \\ &M_{3}=\max \biggl\{ {\frac{k\beta M_{1}}{\mu lb}}e^{-(m\tau_{1}+n\tau _{2})}, \sup _{-\tau\leq s\leq0} \bigl\Vert \phi_{1}( \cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \\ &M_{4}=\max \biggl\{ {\frac{c\beta M_{1}}{plb}}e^{-m\tau_{1}}, \sup _{-\tau\leq s \leq0} \bigl\Vert \phi_{1}(\cdot, s) \bigr\Vert _{C(\overline{\Omega},R)} \biggr\} , \end{aligned} $$
(5)

and \(l=\min\{q,h\}\). From the comparison principle [35], we have \(0\leq u(x,t)\leq M_{1}\), \(0\leq w(x,t)\leq M_{2}\), \(0\leq v(x,t) \leq M_{3}\), \(0\leq z(x,t)\leq M_{4}\) for \(x\in\overline{\Omega}\) and \(t\in[0, T_{\max}]\). Then \(u(x,t)\), \(w(x,t)\), \(v(x,t)\) and \(z(x,t)\) are bounded on \(\overline{\Omega}\times[0, T_{\max})\). By the standard theory for semilinear parabolic systems [36, 37], we can deduce that \(T_{\max}=+\infty\). This completes the proof.

In the following, we will discuss the existence of equilibria of model (2). Model (2) always has an infection-free equilibrium \(E_{0}=(u_{0}, 0, 0, 0)\), where \(u_{0}={\frac{\lambda}{d}}\). The basic reproduction number is given by

$$ R_{0}=\frac{\beta k \lambda}{\mu h(d+a\lambda) e^{m\tau_{1}+n\tau _{2}}}. $$
(6)

We can rewrite \(R_{0}\) in the following form:

$$ R_{0}=k\cdot\frac{1}{\mu}\cdot e^{-n\tau_{2}}\cdot \frac{\beta \cdot\frac{\lambda}{d}}{1+a\cdot\frac{\lambda}{d}}\cdot e^{-m\tau _{1}}\cdot\frac{1}{h}. $$

Here, k is the rate of new virus particles produced by infected cells, \(\frac{1}{\mu}\) is the surviving period of virus, \(e^{-n\tau_{2}}\) is the surviving rate of a new virus in time period \([t-\tau_{2}, t]\), \(\frac{\beta\cdot\frac{\lambda}{d}}{1+a\cdot\frac{\lambda}{d}}\) denotes the newly infected cells which are infected by the first virus, \(e^{-m\tau_{1}}\) is the surviving rate of newly infected cells in time period \([t-\tau_{1}, t]\), and \(\frac{1}{h}\) is the surviving period of infected cells. Therefore, we easily see that \(R_{0}\) denotes the average number of the free viruses released by the infected cells which are infected by the first virus.

If \(R_{0}>1\), there exists a unique immune-free equilibrium \(E_{1}=(u_{1},w_{1},v_{1},0)\), where \(u_{1}\) is a positive root of the following equation:

$$ abkdu^{2}+ \bigl(\beta k+bkd-abk\lambda-a\mu h e^{m\tau_{1}+n\tau_{2}} \bigr)u- \bigl(bk \lambda+\mu h e^{m\tau_{1}+n\tau_{2}} \bigr)=0, $$

and \(w_{1}={\frac{\mu e^{n\tau_{2}}}{kb}}(R^{*}-1)\), \(v_{1}={\frac{1}{b}}(R ^{*}-1)\) if and only if

$$ R^{*}={\frac{\beta ku_{1}}{h\mu(1+au_{1})e^{m\tau_{1}+n\tau_{2}}}}>1. $$

Also, an immune response reproduction number is

$$ R_{1}={\frac{c}{q}}w_{1}={\frac{c\mu e^{n\tau_{2}}}{qkb}} \bigl(R^{*}-1 \bigr). $$

Note that when \(R_{0}>1\) model (2) has a unique immune-free equilibrium \(E_{1}=(u_{1},w_{1},v_{1},0)\). This shows that virus infection is successful and the numbers of free viruses and infected cells at equilibrium \(E_{1}\) are \(v_{1}\) and \(w_{1}\), respectively. Furthermore, we have that \(\frac{1}{q}\) is the average life-span of CTL cells and c is the rate at which the CTL response is produced. Hence, \(R_{1}\) denotes the average number of the CTL immune cells activated by infected cells when virus infection is successful.

If \(R_{1}>1\), then model (2) has a unique interior equilibrium \(E_{2}=(u_{2}, w_{2}, v_{2}, z_{2})\), where \(u_{2}\) is a positive root of the following equation:

$$\begin{aligned}& \bigl(a\mu cd+abkqde^{-n\tau_{2}} \bigr)u^{2}+ \bigl(\beta kqe^{-n\tau_{2}}+d\mu c+bkqde ^{-n\tau_{2}}-a\mu c\lambda -abkq\lambda e^{-n\tau_{2}} \bigr)u \\& \quad{}- \bigl(\lambda\mu c+bkq\lambda e^{-n\tau_{2}} \bigr)=0, \end{aligned}$$

and

$$ w_{2}={\frac{q}{c}}, \qquad v_{2}={ \frac{ke^{-n\tau_{2}}}{\mu}}w_{2}, \qquad z_{2}={\frac{\lambda-du_{2}-hw_{2}e^{m\tau_{1}}}{pw_{2}e^{m\tau_{1}}}}. $$

 □

3 Stability analysis

In this section, we investigate the global stability of equilibria for model (2), namely, infection-free equilibrium \(E_{0}\), immune-free equilibrium \(E_{1}\) and interior equilibrium \(E_{2}\) of model (2), respectively. For the sake of convenience, we let \(y=y(x,t)\) and \(y_{\tau_{i}}=y(x,t-\tau_{i})\) for any \(y\in\{u, w, v, z\}\) and \(i\in{1,2}\).

3.1 Stability of equilibrium \(E_{0}\)

Theorem 3.1

If \(R_{0} \leq1\), then the infection-free equilibrium \(E_{0}\) of model (2) is globally asymptotically stable.

Proof

Construct a Lyapunov functional

$$ W(t)= \int_{\Omega} \bigl(W_{1}(x,t)+W_{2}(x,t) \bigr)\,\mathrm{d}x, $$

where

$$\begin{aligned}& W_{1}(x,t)={\frac{u_{0}}{1+au_{0}}}\biggl({\frac{u}{u_{0}}}-1-\ln{ \frac {u}{u_{0}}}\biggr)+e^{m\tau_{1}}w+ {\frac{he^{m\tau_{1}+n\tau_{2}}}{k}}v+{ \frac{pe^{m\tau_{1}}}{c}}z, \\& W_{2}(x,t)= \int_{0}^{\tau_{1}} {\frac{\beta u_{\theta}v_{\theta }}{(1+au_{\theta}) (1+bv_{\theta})}} \,\mathrm{d} \theta+ he^{m\tau_{2}} \int_{0}^{\tau_{2}} w_{\theta}\, \mathrm{d}\theta. \end{aligned}$$

We have

$$\begin{aligned}& {\frac{\partial W_{1}(x,t)}{\partial t}}+{\frac{\partial W_{2}(x,t)}{\partial t}} \\& \quad= {\frac{1}{1+au_{0}}}\biggl(1-{\frac{u_{0}}{u}}\biggr){ \frac{\partial u}{\partial t}}+e^{m\tau _{1}}{\frac{\partial w}{\partial t}}+{\frac{he^{m\tau_{1}+n\tau _{2}}}{k}} { \frac{\partial v}{\partial t}}+{\frac{pe^{m\tau _{1}}}{c}} {\frac{\partial z}{\partial t}} \\& \quad\quad{}+{\frac{\beta uv}{(1+au) (1+bv)}}-{\frac{\beta u_{\tau _{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} +he^{m\tau _{2}}w-he^{m\tau_{2}}w_{\tau_{2}} \\& \quad= {\frac{1}{1+au_{0}}}\biggl(1-{\frac{u_{0}}{u}}\biggr) \biggl(\lambda -du-{\frac{\beta uv}{(1+au) (1+bv)}} \biggr)+{\frac{he^{m\tau_{1}+n\tau _{2}}}{k}}D \Delta v \\& \quad\quad{}-{\frac{h\mu e^{m\tau_{1}+n\tau_{2}}}{k}}v-{\frac {pqe^{m\tau_{1}}}{c}}z+{\frac{\beta uv}{(1+au) (1+bv)}} \\& \quad= -{\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}- {\frac{1}{1+au_{0}}} {\frac{\beta uv}{(1+au) (1+bv)}}+{ \frac {1}{1+au_{0}}} {\frac{\beta u_{0}v}{(1+au) (1+bv)}} \\& \quad\quad{}+{\frac{\beta uv}{(1+au) (1+bv)}}-{\frac{h\mu e^{m\tau _{1}+n\tau_{2}}}{k}}v- {\frac{pq}{c}}e ^{m\tau_{1}}z+{\frac{he^{m\tau_{1}+n\tau _{2}}}{k}}D\Delta v \\& \quad= -{\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}+ {\frac{1+au}{1+au_{0}}} {\frac{\beta u_{0}v}{(1+au) (1+bv)}}-{ \frac{h \mu e^{m\tau_{1}+n\tau_{2}}}{k}}v \\& \quad\quad{} -{\frac{pq}{c}}e^{m\tau_{1}}z+{\frac{he^{m\tau _{1}+n\tau_{2}}}{k}}D\Delta v. \end{aligned}$$

Calculating the derivative of \(W(t)\) along the positive solution of model (2), we get

$$\begin{aligned} {\frac{dW(t)}{dt}}&=- \int_{\Omega} {\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}\,\mathrm{d}x+ \int_{\Omega}{\frac{\beta u_{0}v}{(1+au_{0}) (1+bv)}}\,\mathrm{d}x \\ &\quad{}- \int_{\Omega}{\frac{h\mu e^{m\tau_{1}+n\tau_{2}}}{k}}v\,\mathrm{d}x- \int_{\Omega} {\frac{pqe^{m\tau_{1}}}{c}}z\,\mathrm{d}x \\ &\quad{}+ \int_{\Omega} {\frac{he^{m\tau_{1}+n\tau_{2}}}{k}}D\Delta v\,\mathrm{d}x \\ &=- \int_{\Omega} {\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}\,\mathrm{d}x + \int_{ \Omega} {\frac{h\mu e^{m\tau_{1}+n\tau_{2}}v}{k(1+bv)}}(R_{0}-1) \, \mathrm{d}x \\ &\quad{}- \int_{\Omega} {\frac{h\mu be^{m\tau_{1}+n\tau_{2}}}{k(1+bv)}}v^{2} \, \mathrm{d}x - \int_{\Omega} {\frac{pqe^{m\tau_{1}}}{c}}z\,\mathrm{d}x \\ &\quad{}+ \int_{\Omega} {\frac{he^{m\tau_{1}+n\tau_{2}}}{k}}D\Delta v\,\mathrm{d}x. \end{aligned}$$

Owing to the divergence theorem and the homogeneous Neumann boundary conditions (4), we obtain

$$ \int_{\Omega} \Delta v\,\mathrm{d}x= \int_{\partial\Omega} \frac{ \partial v}{\partial\vec{n}}\,\mathrm{d}x=0. $$

Thus, we further have

$$\begin{aligned} {\frac{dW(t)}{dt}}&=- \int_{\Omega} {\frac{d(u-u_{0})^{2}}{u(1+au_{0})}}\,\mathrm{d}x + \int_{ \Omega} {\frac{h\mu e^{m\tau_{1}+n\tau_{2}}v}{k(1+bv)}}(R_{0}-1) \, \mathrm{d}x \\ &\quad{}- \int_{\Omega} {\frac{h\mu be^{m\tau_{1}+n\tau_{2}}}{k(1+bv)}}v^{2} \, \mathrm{d}x - \int_{\Omega} {\frac{pqe^{m\tau_{1}}}{c}}z\,\mathrm{d}x. \end{aligned}$$

Therefore, \(\frac{dW(t)}{dt}\leq0\) if \(R_{0} \leq1\). We have \(\frac{dW(t)}{dt}= 0\) if and only if \(u=u_{0}\), \(w=0\), \(v=0\) and \(z=0\). It follows that the largest invariant set \(\{(u,w,v,z)\in R ^{4}_{+}: \frac{dW(t)}{dt}=0\}\) is the singleton \(E_{0}\). By using LaSalle’s invariance principle [30], we see that the equilibrium \(E_{0}\) of model (2) is globally asymptotically stable when \(R_{0}\leq1\). □

3.2 Stability of equilibrium \(E_{1}\)

Theorem 3.2

If \(R_{1}\leq1< R_{0}\), then the immune-free equilibrium \(E_{1}\) of model (2) is globally asymptotically stable.

Proof

Let \(G(x,t)=g(x,t)-1-\ln g(x,t)\). We have that \(G(x,t)\geq0\) for all \(g(x,t)>0\) and \(G(x,t)=0\) if and only if \(g(x,t)=1\). Define a Lyapunov functional

$$ V(t)= \int_{\Omega} \bigl(V_{1}(x,t)+hw_{1}V_{2}(x,t) \bigr)\,\mathrm{d}x, $$

where

$$\begin{aligned}& V_{1}(x,t)= e^{-m\tau_{1}} \biggl(u-u_{1}- \int_{u_{1}}^{u}{\frac{1+a\theta}{1+au_{1}}} { \frac{u_{1}}{\theta}} \,\mathrm{d}\theta \biggr) +w_{1}G\biggl( { \frac{w}{w_{1}}}\biggr) +{\frac{he^{n\tau_{2}}}{k}}v_{1}G\biggl({ \frac{v}{v_{1}}}\biggr) +{\frac{p}{c}}z, \\& V_{2}(x,t)= \int_{0}^{\tau_{1}}G \biggl({\frac{e^{-m\tau_{1}}\beta u_{\theta }v_{\theta}}{hw_{1}(1+au_{\theta}) (1+bv_{\theta})}} \biggr) \,\mathrm {d}\theta+ \int_{0}^{\tau_{1}}G\biggl({\frac{w_{\theta}}{w_{1}}}\biggr)\, \mathrm{d}\theta. \end{aligned}$$

We have

$$\begin{aligned}& {\frac{\partial V_{1}(x,t)}{\partial t}} = e^{-m\tau_{1}}\biggl(1-{\frac {1+au}{1+au_{1}}} { \frac{u_{1}}{u}}\biggr){\frac{\partial u}{\partial t}} +\biggl(1- {\frac{w_{1}}{w}} \biggr){\frac{\partial w}{\partial t}} \\& \hphantom{{\frac{\partial V_{1}(x,t)}{\partial t}} =}+{\frac{he^{n\tau_{2}}}{k}}\biggl(1-{\frac{v_{1}}{v}}\biggr){\frac{\partial v}{\partial t}} +{ \frac{p}{c}} {\frac{\partial z}{\partial t}}, \\& {\frac{\partial V_{2}(x,t)}{\partial t}}= {\frac{e^{-m\tau_{1}}\beta uv}{hw_{1}(1+au) (1+bv)}}-{\frac{e^{-m\tau_{1}}\beta u_{\tau_{1}}v_{\tau _{1}}}{hw_{1}(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \hphantom{{\frac{\partial V_{2}(x,t)}{\partial t}}=}+\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau _{1}})}} {\frac{(1+au) (1+bv)}{uv}} \\& \hphantom{{\frac{\partial V_{2}(x,t)}{\partial t}}=}+{\frac{w-w_{\tau_{1}}}{w_{1}}}+\ln{\frac{w_{\tau_{1}}}{w}}. \end{aligned}$$

By using the equilibrium equation, we get \(\lambda=du_{1}+e^{m\tau _{1}}hw_{1}\), \({\frac{h\mu e^{n\tau_{2}}}{k}}={\frac{hw_{1}}{v_{1}}}\) and \({\frac{\beta u_{1}v_{1}}{(1+au_{1})(1+bv_{1})}}=hw _{1}e^{m\tau_{1}}\). Thus, we have

$$\begin{aligned}& {\frac{\partial V_{1}(x,t)}{\partial t}}+hw_{1}{\frac{\partial V_{2}(x,t)}{\partial t}} \\& \quad=e^{-m\tau_{1}}{\frac{\partial u}{\partial t}}+{\frac{\partial w}{\partial t}}+ { \frac{he^{n\tau_{2}}}{k}} {\frac{\partial v}{\partial t}}+{\frac {p}{c}} { \frac{\partial z}{\partial t}}+hw _{1}{\frac{\partial V_{2}(x,t)}{\partial t}} \\& \quad\quad{}-e^{-m\tau_{1}}{\frac{1+au}{1+au_{1}}} {\frac{u_{1}}{u}} { \frac{\partial u}{\partial t}} -{\frac{w_{1}}{w}} {\frac{\partial w}{\partial t}} -{\frac{he^{n\tau_{2}}}{k}} {\frac{v_{1}}{v}} {\frac{\partial v}{\partial t}} \\& \quad= e^{-m\tau_{1}}(du_{1}-du)+hw_{1}-{ \frac{h\mu e^{n\tau _{2}}}{k}}v-{\frac{pq}{c}}z \\& \quad\quad{} +hw_{1}\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau _{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} \\& \quad\quad{} +hw_{1}\ln{\frac{w_{\tau_{1}}}{w}}+ {\frac{he^{n\tau_{2}}}{k}}D \Delta v -e^{-m \tau_{1}}{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}}du_{1} \\& \quad\quad{} +e^{-m\tau_{1}} {\frac{1+au}{1+au_{1}}}du_{1} -{ \frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}}hw _{1}+ {\frac{v}{v_{1}}} { \frac{1+bv_{1}}{1+bv}}hw_{1} \\& \quad\quad{}+hw_{1} \biggl(1-{\frac{w_{1}}{w}} { \frac{u_{\tau_{1}}}{u_{1}}} {\frac{v_{\tau_{1}}}{v_{1}}} {\frac{(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \\& \quad\quad{} +pw_{1}z +hw_{1}\biggl(1-{\frac{w_{\tau_{2}}}{w_{1}}} {\frac {v_{1}}{v}}\biggr)-{\frac{he^{n\tau_{2}}}{k}} {\frac{v_{1}}{v}} D \Delta v \\& \quad= -{\frac{de^{-m\tau _{1}}(u-u_{1})^{2}}{u(1+au_{1})}}+hw_{1}\biggl(-1-{\frac{v}{v_{1}}}+ { \frac {v}{v_{1}}} {\frac{1+bv_{1}}{1+bv}}+{\frac{1+bv}{1+bv_{1}}}\biggr) \\& \quad\quad{} +hw_{1} \biggl(4-{\frac{w_{1}}{w}} { \frac{u_{\tau_{1}}}{u_{1}}} {\frac{v_{\tau_{1}}}{v_{1}}} {\frac{(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \quad\quad{}-{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}} - {\frac{w_{\tau_{2}}}{w_{1}}} { \frac{v_{1}}{v}}-{\frac{1+bv}{1+bv_{1}}} \biggr)+pz\biggl(w_{1}-{ \frac{q}{c}}\biggr) \\& \quad\quad{} +hw_{1}\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau _{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} { \frac{w_{\tau_{1}}}{w}} \\& \quad\quad{}+{\frac{he^{n\tau_{2}}}{k}}D\Delta v-{\frac{he^{n\tau_{2}}}{k}} { \frac{v_{1}}{v}}D \Delta v \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{1})}}(u-u_{1})^{2}- { \frac{bhw_{1}}{v_{1}(1+bv) (1+bv_{1})}} (v-v_{1})^{2} \\& \quad\quad{}+hw_{1} \biggl(4-{\frac{w_{1}}{w}} { \frac{u_{\tau_{1}}}{u_{1}}} {\frac{v_{\tau_{1}}}{v_{1}}} {\frac{(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \quad\quad{}-{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}} - {\frac{w_{\tau_{2}}}{w_{1}}} { \frac{v_{1}}{v}}-{\frac{1+bv}{1+bv_{1}}} \biggr)+pz\biggl(w_{1}-{ \frac{q}{c}}\biggr) \\& \quad\quad{} +hw_{1}\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau _{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} { \frac{w_{\tau_{1}}}{w}} \\& \quad\quad{}+{\frac{he^{n\tau_{2}}}{k}}D\Delta v-{\frac{he^{n\tau_{2}}}{k}} { \frac{v_{1}}{v}}D \Delta v \end{aligned}$$

due to

$$\begin{aligned}& \ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} {\frac{w_{\tau_{1}}}{w}} \\& \quad= \ln{\frac{u_{1}}{u}} {\frac{1+au}{1+au_{1}}}+\ln {\frac{w_{\tau_{1}}}{w_{1}}} { \frac{v_{1}}{v}} +\ln{\frac{1+bv}{1+bv_{1}}} \\& \quad\quad{}+\ln{\frac{w_{1}u_{\tau_{1}}v_{\tau_{1}}}{wu_{1}v_{1}}} {\frac {(1+au_{1}) (1+bv_{1})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}}. \end{aligned}$$

Then we have

$$\begin{aligned}& {\frac{\partial V_{1}(x,t)}{\partial t}}+hw_{1}{\frac{\partial V_{2}(x,t)}{\partial t}} \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{1})}}(u-u_{1})^{2}- { \frac{bhw_{1}}{v_{1}(1+bv) (1+bv_{1})}} (v-v_{1})^{2}-hw_{1} \biggl(G \biggl( {\frac{u_{1}(1+au)}{u(1+au_{1})}} \biggr) \\& \quad\quad{}+G\biggl({\frac{w_{\tau_{2}}v_{1}}{w_{1}v}}\biggr)+G\biggl( {\frac{1+bv}{1+bv_{1}}} \biggr) +G \biggl({\frac{w_{1}u_{\tau_{2}}v_{\tau _{1}}(1+au_{1}) (1+bv_{1})}{wu_{1}v_{1}(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \biggr) \\& \quad\quad{}+pz\biggl(w_{1}-{\frac{q}{c}}\biggr)+{ \frac{he^{n\tau_{2}}}{k}}D\biggl(1- {\frac{v_{1}}{v}}\biggr)\Delta v. \end{aligned}$$

We know that

$$ \int_{\Omega} \triangle v\,\mathrm{d}x=0,\qquad \int_{\Omega} {\frac{\triangle v}{v}} \,\mathrm{d}x= \int_{\Omega} {\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}}\, \mathrm{d}x. $$

Thus,

$$\begin{aligned} {\frac{dV(t)}{dt}} &= \int_{\Omega} \biggl( {\frac{\partial V_{1}(x,t)}{\partial t}}+hw_{1}{ \frac{\partial V_{2}(x,t)}{\partial t}} \biggr)\, \mathrm{d}x \\ &=- \int_{\Omega} {\frac{de^{-m\tau_{1}}(u-u_{1})^{2}}{u(1+au_{1})}}\,\mathrm{d}x- \int_{\Omega} {\frac{bhw_{1}(v-v_{1})^{2}}{v_{1}(1+bv) (1+bv_{1})}}\,\mathrm{d}x \\ &\quad{}-hw_{1} \int_{\Omega} \biggl(G \biggl({\frac{u_{1}(1+au)}{u(1+au_{1})}} \biggr)+G \biggl({\frac{w_{\tau_{2}}v_{1}}{w_{1}v}}\biggr) +G \biggl({\frac{w_{1}u_{\tau _{2}}v_{\tau_{1}}(1+au_{1}) (1+bv_{1})}{wu_{1}v_{1}(1 +au_{\tau_{1}}) (1+bv_{\tau _{1}})}} \biggr) \\ &\quad{}+G\biggl({\frac{1+bv}{1+bv_{1}}}\biggr) \biggr)\,\mathrm{d}x+ \int_{\Omega}pz\biggl(w_{1} -{\frac{q}{c}}\biggr) \,\mathrm{d}x+ {\frac{hDe^{n\tau_{2}}v_{1}}{k}} \int_{\Omega}{\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}} \,\mathrm{d}x. \end{aligned}$$

Also, \(R_{1}\leq1\) implies that \(w_{1}\leq{\frac{q}{c}}\). Therefore, if \(R_{1}\leq1< R_{0}\), we can obtain \({\frac{dV(t)}{dt}}\leq0\). Obviously, \({\frac{dV(t)}{dt}}= 0\) if and only if \(u=u_{1}\), \(w=w_{1}\), \(v=v_{1}\) and \(z=0\). It follows that the largest invariant set \(\{(u,w,v,z)\in R ^{4}_{+}: \frac{dV(t)}{dt}=0\}\) is the singleton \(E_{1}\). By LaSalle’s invariance principle [30], we finally conclude that the equilibrium \(E_{1}\) is globally asymptotically stable when \(R_{1}\leq1< R_{0}\). □

3.3 Stability of equilibrium \(E_{2}\)

Theorem 3.3

If \(R_{1}>1\), then the interior equilibrium \(E_{2}\) of model (2) is globally asymptotically stable.

Proof

Consider a Lyapunov functional \(L(t)\) as follows:

$$ L(t)= \int_{\Omega} \bigl(L_{1}(x,t)+(hw_{2}+pw_{2}z_{2})L_{2}(x,t) \bigr)\,\mathrm{d}x, $$

where

$$\begin{aligned}& L_{1}(x,t)= e^{-m\tau_{1}} \biggl(u-u_{2}- \int_{u_{2}}^{u} {\frac{1+a\theta}{1+au_{2}}} { \frac{u_{2}}{\theta}} \,\mathrm{d}\theta \biggr) \\& \hphantom{L_{1}(x,t)= } +w_{2}G\biggl({\frac{w}{w_{2}}}\biggr)+ { \frac{e^{n\tau_{2}}v_{2} (h+pz_{2})}{k}}G\biggl({\frac{v}{v_{2}}}\biggr) +{\frac {pz_{2}}{c}}G \biggl({\frac{z}{z_{2}}}\biggr), \\& L_{2}(x,t)= \int_{0}^{\tau_{1}}G \biggl({\frac{e^{-m\tau_{1}}\beta u_{\theta }v_{\theta}}{(hw_{2}+pw_{2}z_{2}) (1+au_{\theta}) (1+bv_{\theta})}} \biggr) \,\mathrm{d}\theta+ \int_{0}^{\tau_{2}}G\biggl({\frac{w_{\theta}}{w_{2}}}\biggr)\, \mathrm{d}\theta. \end{aligned}$$

By calculating the time derivative of \(L_{1}(x,t)\) and \(L_{2}(x,t)\), we have

$$\begin{aligned} {\frac{\partial L_{1}(x,t)}{\partial t}}&=e^{-m\tau_{1}}\biggl(1-{\frac {1+au}{1+au_{2}}} { \frac{u_{2}}{u}}\biggr){\frac{\partial u}{\partial t}} +\biggl(1- {\frac{w_{2}}{w}} \biggr){\frac{\partial w}{\partial t}}+{\frac {(h+pz_{2})e^{n\tau_{2}}}{k}}\biggl(1-{\frac{v_{2}}{v}} \biggr){\frac{\partial v}{\partial t}} \\ &\quad{}+{\frac{p}{c}}\biggl(1-{\frac{z_{2}}{z}}\biggr){ \frac{\partial z}{\partial t}}, \end{aligned}$$

and

$$\begin{aligned} {\frac{\partial L_{2}(x,t)}{\partial t}} &={\frac{e^{-m\tau_{1}}\beta uv}{(hw_{2}+pw_{2}z_{2}) (1+au) (1+bv)}} + {\frac{w-w_{\tau_{2}}}{w_{2}}}+\ln{ \frac{w_{\tau_{2}}}{w}} \\ &\quad{}-{\frac{e^{-m\tau_{1}}\beta u_{\tau_{1}}v_{\tau _{1}}}{(hw_{2}+pw_{2}z_{2}) (1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\ &\quad{}+\ln{\frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}}. \end{aligned}$$

Note that the interior equilibrium \(E_{2}(u_{2}, w_{2}, v_{2}, z_{2})\) satisfies the following equations:

$$ \begin{aligned} & \lambda-du_{2}-{\frac{\beta u_{2}v_{2}}{(1+au_{2}) (1+bv_{2})}}=0, \\ &e^{-m\tau_{1}}{\frac{\beta u_{2}v_{2}}{(1+au_{2}) (1+bv_{2})}}-hw_{2}-pw_{2}z_{2}=0, \\ &ke ^{-n\tau_{2}}w_{2}-\mu v_{2}=0, \\ & cw_{2}z_{2}-qz_{2}=0. \end{aligned} $$
(7)

We obtain that \(\lambda=du_{2}+e^{m\tau_{1}}(hw_{2}+pw_{2}z_{2})\), \({\frac{pw_{2}}{v_{2}}}={\frac{p\mu}{k}}\), \(w_{2}={\frac{q}{c}}\) and \(e^{m\tau_{1}}(hw _{2}+pw_{2}z_{2})={\frac{\beta u_{2}v_{2}}{(1+au_{2})(1+bv_{2})}}\). Thus, we have

$$\begin{aligned}& {\frac{\partial L_{1}(x,t)}{\partial t}}+(hw_{2}+pw_{2}z_{2}) { \frac{\partial L_{2}(x,t)}{\partial t}} \\& \quad= e^{-m\tau_{1}}{\frac{\partial u}{\partial t}}+{\frac {\partial w}{\partial t}} +{ \frac{(h+pz_{2})e^{n\tau_{2}}}{k}} {\frac{\partial v}{\partial t}}+{\frac{p}{c}} { \frac{\partial z}{\partial t}} \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2}){ \frac{\partial L_{2}(x,t)}{\partial t}} -e^{-m\tau _{1}}{\frac{1+au}{1+au_{2}}} {\frac{u_{2}}{u}} { \frac{\partial u}{\partial t}} \\& \quad\quad{}-{\frac{w_{2}}{w}} {\frac{\partial w}{\partial t}} - {\frac{(h+pz_{2})e^{n\tau_{2}}}{k}} { \frac{v_{2}}{v}} {\frac{\partial v}{\partial t}} -{\frac{p}{c}} { \frac{z_{2}}{z}} {\frac{\partial z}{\partial t}} \\& \quad= e^{-m\tau_{1}}(du_{2}-du)+hw_{2}+pw_{2}z_{2}+pz_{2}w_{\tau_{2}} -(hw_{2}+pw_{2}z_{2}){\frac{v}{v_{2}}} \\& \quad\quad{} +(hw_{2}+pw_{2}z_{2})\ln {\frac{u_{\tau_{1}} v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} \\& \quad\quad{} +(hw_{2}+pw_{2}z_{2}) {\frac{w_{\tau_{2}}}{w}}+{\frac{e^{n\tau_{2}}(h+pz_{2})}{k}}D \Delta v \\& \quad\quad{}-{\frac{u_{2}}{u}} {\frac {1+au}{1+au_{2}}}(hw_{2}+pw_{2}z_{2})+(hw_{2}+pw_{2}z _{2}){\frac{v}{v_{2}}} {\frac{1+bv_{2}}{1+bv}} \\& \quad\quad{} -e^{-m\tau_{1}}{\frac{u_{2}}{u}} {\frac {1+au}{1+au_{2}}}du_{2}+e^{-m\tau_{1}}{ \frac{1+au}{1+au_{2}}}du _{2} \\& \quad\quad{} -(hw_{2}+pw_{2}z_{2}) { \frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} {\frac{v_{\tau_{1}}}{v_{2}}} { \frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \\& \quad\quad{}+hw_{2}-(hw_{2}+pw_{2}z_{2}) {\frac{w_{\tau_{2}}}{w_{2}}} {\frac{v_{2}}{v}}+hw_{2}+pw _{2}z_{2} \\& \quad\quad{}-{\frac{e^{n\tau_{2}}(h+pz_{2})}{k}} {\frac{v_{2}}{v}}D\Delta v -pwz_{2}+pw_{2}z _{2} \\& \quad= -{\frac{de^{-m\tau_{1}}(u-u_{2})^{2}}{u(1+au_{2})}}+(hw_{2}+pw_{2}z_{2}) \biggl(-1-{\frac{v}{v_{2}}}+ {\frac{v}{v_{2}}} {\frac{1+bv_{2}}{1+bv}} \\& \quad\quad{} +{\frac{1+bv}{1+bv_{2}}}\biggr) +(hw_{2}+pw_{2}z_{2}) \biggl(4-{\frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}} - {\frac{w_{\tau_{2}}}{w_{2}}} { \frac{v_{2}}{v}} \\& \quad\quad{}-{\frac{1+bv}{1+bv_{2}}} -{\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} {\frac{v_{\tau_{1}}}{v_{2}}} {\frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2})\ln { \frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} {\frac{w_{\tau_{2}}}{w}} \\& \quad\quad{}+{\frac{(h+pz_{2})e^{n\tau_{2}}}{k}}\biggl(1- {\frac{v_{2}}{v}}\biggr)D \Delta v \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{2})}}(u-u_{2})^{2}- { \frac{b(hw_{2}+pw_{2}z_{2})}{v_{2}(1+bv) (1+bv_{2})}} (v-v_{2})^{2} \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2}) \biggl(4- { \frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}} -{\frac{w_{\tau _{2}}}{w_{2}}} { \frac{v_{2}}{v}}-{\frac{1+bv}{1+bv_{2}}} \\& \quad\quad{}-{\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} {\frac{v_{\tau_{1}}}{v_{2}}} { \frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \\& \quad\quad{}+(hw_{2}+pw_{2}z_{2})\ln { \frac{u_{\tau_{1}}v_{\tau_{1}}}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} {\frac{(1+au) (1+bv)}{uv}} {\frac{w_{\tau_{2}}}{w}} \\& \quad\quad{} +{\frac{e^{n\tau_{2}}(h+pz_{2})}{k}}\biggl(1- {\frac{v_{2}}{v}}\biggr)D\Delta v \\& \quad= -{\frac{de^{-m\tau_{1}}}{u(1+au_{2})}}(u-u_{2})^{2}- { \frac{b(hw_{2}+pw_{2}z_{2})}{v_{2}(1+bv) (1+bv_{2})}} (v-v_{2})^{2} \\& \quad\quad{}-(hw_{2}+pw_{2}z_{2}) \biggl(G \biggl( {\frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}}\biggr)+G\biggl({ \frac{w_{\tau _{2}}}{w_{2}}} {\frac{v_{2}}{v}}\biggr)+G\biggl({\frac{1+bv}{1+bv_{2}}} \biggr) \\& \quad\quad{}+G \biggl({\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} { \frac {v_{\tau_{1}}}{v_{2}}} {\frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \biggr)+ {\frac{e^{n\tau_{2}}(h+pz_{2})}{k}} \biggl(1-{\frac{v_{2}}{v}}\biggr)D\Delta v. \end{aligned}$$

Since

$$ \int_{\Omega}\triangle v\,\mathrm{d}x=0,\qquad \int_{\Omega} {\frac{\triangle v}{v}} \,\mathrm{d}x= \int_{\Omega} {\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}}\, \mathrm{d}x, $$

we further have

$$\begin{aligned} {\frac{dL(t)}{dt}}&= \int_{\Omega} \biggl( {\frac{\partial L_{1}(x,t)}{\partial t}}+(hw_{2}+pw_{2}z _{2}){\frac{\partial L_{2}(x,t)}{\partial t}} \biggr)\,\mathrm{d}x \\ &=- \int_{\Omega} {\frac{de^{-m\tau_{1}}(u-u_{2})^{2}}{u(1+au_{2})}}\,\mathrm{d}x- \int_{\Omega} {\frac{b(hw_{2}+pw_{2}z_{2}) (v-v_{2})^{2}}{v_{2}(1+bv) (1+bv_{2})}} \, \mathrm{d}x \\ &\quad{}-(hw_{2}+pw_{2}z_{2}) \biggl(G\biggl( { \frac{u_{2}}{u}} {\frac{1+au}{1+au_{2}}}\biggr)+G\biggl({\frac{w_{\tau _{2}}}{w_{2}}} { \frac{v_{2}}{v}}\biggr)+G\biggl({\frac{1+bv}{1+bv_{2}}}\biggr) \\ &\quad{}+G \biggl({\frac{w_{2}}{w}} {\frac{u_{\tau_{1}}}{u_{2}}} { \frac{v_{\tau_{1}}}{v_{2}}} {\frac{(1+au_{2}) (1+bv_{2})}{(1+au_{\tau_{1}}) (1+bv_{\tau_{1}})}} \biggr) \biggr) \\ &\quad{}-{\frac{(h+pz_{2})De^{n\tau_{2}}v_{2}}{k}} \int_{\Omega} {\frac{\Vert \nabla v \Vert ^{2}}{v^{2}}}\,\mathrm{d}x. \end{aligned}$$

Therefore, we have \({\frac{dL(t)}{dt}}\leq0\) when \(R_{1}>1\). \({\frac{dL(t)}{dt}}=0\) if and only if \(u=u_{2}\), \(w=w_{2}\), \(v=v_{2}\) and \(z=z_{2}\). It follows that the largest invariant set \(\{(u,w,v,z) \in R^{4}_{+}: \frac{dL(t)}{dt}=0\}\) is the singleton \(E_{2}\). Based on LaSalle’s invariance principle [30], we conclude that the equilibrium \(E_{2}\) is globally asymptotically stable when \(R_{1}>1\). □

4 Numerical simulations

In the previous section, we analyzed the global stability of a model of virus infection with diffusion and time delay. The object of this section is to further illustrate the obtained theoretical results by some numerical simulations. The Runge-Kutta scheme of fourth order is applied for the reaction part and an explicit Euler scheme for the diffusion part of the PDE. Next, we consider model (2) with the homogeneous Neumann boundary conditions

$$ \frac{\partial v}{\partial\vec{n}}=0, \quad t>0, x=0, 1, $$
(8)

and the initial conditions

$$ \begin{aligned} &u(x,\theta)=\phi_{1}(x, \theta)\geq 0, \qquad w(x,\theta)=\phi_{2}(x, \theta)\geq0, \\ &v(x,\theta)=\phi_{3}(x,\theta)\geq0, \qquad z(x,\theta)= \phi_{4}(x, \theta)\geq0, \quad x\in[0,1], \theta\in[-\tau,0]. \end{aligned} $$
(9)

In model (2), we choose β, c, q, \(\tau_{1}\), \(\tau_{2}\) as free parameters, and all remaining parameters are fixed as in Table 1.

Table 1 List of parameters
Full size table

In Figures 1, 2 and 3, (a), (b), (c) and (d) denote time-series figures of \(u(x,t)\), \(w(x,t)\), \(v(x,t)\) and \(z(x,t)\).

Figure 1
figure 1

Taking \(\pmb{\beta=0.01}\) , \(\pmb{c=0.1}\) , \(\pmb{q=0.12}\) , \(\pmb{\tau _{1}=10}\) , \(\pmb{\tau_{2}=6}\) , we have \(\pmb{R_{0}=0.2066<1}\) , the infection-free equilibrium \(\pmb{E_{0}(1000,0,0,0)}\) is globally asymptotically stable.

Full size image
Figure 2
figure 2

Taking \(\pmb{\beta=0.2}\) , \(\pmb{c=0.01}\) , \(\pmb{q=0.2}\) , \(\pmb{\tau _{1}=5}\) , \(\pmb{\tau_{2}=15}\) , we have \(\pmb{R_{0}=3.9696>1}\) , \(\pmb{R_{1}=0.9227<1}\) , the immune-free equilibrium \(\pmb{E_{1}(30.5250,18.4547,2.1179,0)}\) is globally asymptotically stable.

Full size image
Figure 3
figure 3

Taking \(\pmb{\beta=0.3}\) , \(\pmb{c=0.01}\) , \(\pmb{q=0.15}\) , \(\pmb{\tau _{1}=10}\) , \(\pmb{\tau_{2}=5}\) , we have \(\pmb{R_{0}=6.2597>1}\) , \(\pmb{R_{1}=1.2303>1}\) , the infection equilibrium only with CTL immune response \(\pmb{E_{2}(10.9402,15,1.7214,0.1272)}\) is globally asymptotically stable.

Full size image

5 Discussion

In this paper, we propose a delayed virus infection model with diffusion, CTL immune responses and Crowley-Martin incidence rate. We have showed the global asymptotic stability of infection-free equilibrium \(E_{0}\), immune-free equilibrium \(E_{1}\) and interior equilibrium \(E_{2}\). By the analysis, the infection-free equilibrium \(E_{0}\) is globally asymptotically stable if \(R_{0}\leq1\); in such circumstances, the viruses are cleared and the infection dies out. If \(R_{1}\leq1< R_{0}\), the immune-free equilibrium \(E_{1}\) is globally asymptotically stable, it is shown that immune response would not be activated and virus infection becomes vanished. If \(R_{1}>1\), model (2) has an interior equilibrium \(E_{2}\), which is globally asymptotically stable. Mathematically, in order to find treatment strategies, we have to find the analytical conditions under which trajectories of the model will converge to infection-free equilibrium. This condition is given in Theorem 3.1, that is, the infection-free equilibrium is globally asymptotically stable if \(R_{0}=\frac{\beta k \lambda}{\mu h(d+a \lambda) e^{m\tau_{1}+n\tau_{2}}}<1\). This relation relies on parameters β, \(\tau_{1}\) and \(\tau_{2}\). That is to say, the variation of these three parameters plays a vital role in disease treatment, and doctors have to adjust these parameters according to the above relation in order to eliminate virus infection. The parameters β, \(\tau_{1}\) and \(\tau_{2}\) might be changed as a consequence of disease treatment. The above analysis shows that \(R_{0}\) increases proportionally to parameter β and decreases proportionally to parameters \(\tau_{1}\) and \(\tau_{2}\). Thus, in order to eliminate the infection, we need to try to increase the value of \(\tau_{1}\) or \(\tau_{2}\). By decreasing infection rate β, immune-free and interior equilibria disappear and infection-free equilibrium will become stable, which implies the virus eradication and that the patient is cured.

Another important issue from medical point of view is to investigate the factors which cause growth of virus. Mathematically, the relations for existence and stability of interior equilibrium need to be investigated. This matter is solved in Theorem 3.3, which gives possible conditions for the existence of interior equilibrium and describes criteria for the stability of interior equilibrium.

Mathematical models can help physicians to choose a suitable dosage and check the effects of their therapeutic action. Virus infection progression has many variations from patient to patient, which is difficult to obtain by an ordinary differential equation. By choosing a relevant diffusive coefficient, the partial differential equation can be varied to best fit the real data according to the progression of different patients. Consequently, the clinician can recommend administration of drugs or treatment strategies to each individual by using the information from the delayed model with the relevant diffusive coefficient.

Our model considers a four-dimensional diffusive virus infection model with intracellular delay, virus replication delay and Crowley-Martin infection rate. It is considered whether the results also can be extended to five-dimensional diffusive virus infection model with mitosis transmission and immune delay. We leave this for the future work.