Global asymptotic stability of a delay differential equation model for SARS-CoV-2 virus infection mediated by ACE2 receptor protein

Abstract

The COVID-19 pandemic has brought a serious threat to human life safety worldwide. SARS-CoV-2 virus mainly binds to the target cell surface receptor ACE2 (Angiotensin-converting enzyme 2 ) through the S protein expressed on the surface of the virus, resulting in infection of target cells. During this infection process, the target cell ACE2 receptor plays a very important mediating role. In this paper, a delay differential equation model containing the mediated effect of target cell receptor is established according to the mechanism of SARS-CoV-2 virus invasion of target cells, and the global stability of the infection-free equilibrium and the infected equilibrium of the model is obtained by using the basic reproduction number ${ℛ}_0$ and constructing the appropriate Lyapunov functional. The expression of the basic reproduction number ${ℛ}_0$ intuitively gives the dependence on the expression ratio of the target cell surface ACE2 receptor, which is helpful for the understanding of the mechanism of SARS-CoV-2 virus infection.

1. Introduction

Usually, free virions invade target cells by adsorbing virions on the surface of target cells and then entering the target cells. In the infected target cell, the replication of the virus needs to go through steps such as transcription, translation, genome replication, assembly, etc., and finally the virion particles are released from the target cell. In this process, each stage of blockade may prevent the replication of the virus, providing a selective target for the effect of vaccines and drugs. After the virion binds to the target cell receptor, it undergoes membrane fusion and other related steps. For example, HCV virions attack normal cells by binding to target cell receptor through the E2 protein [1]. SARS-CoV-2 invades target cells by binding to the target cell ACE2 receptor [2]. Similarly, in immunotherapy of tumor cells, when PD-1 on the surface of $T$ cells interacts with PD-L1 on neighboring tumor cells or innate immune cells, $T$ cells become dysfunctional and lose the ability to kill target cells. As a result, monoclonal antibody therapies targeting the PD-1 and PD-L1 pathways, known as immune checkpoint inhibitors, have been breakthrough developed [3].

In recent years, differential equation models have been widely used to study the evolutionary behavior of the complex mechanism between viral infection and immune response [4], [5], [6]. [7] used KP differential equation model to describe the interaction between tumor cells, immune effector cells and IL-2, and then explained the short-term oscillation of tumor size or long-term recurrence. In view of the problem of tumor growth and siRNA therapy, the literature [8] uses differential equation models to describe the complex relationship between tumor cells, immune effector factors/cells, the immuno-stimulatory IL-2 and suppressive cytokines TGF-$\beta$. Based on a partial differential equation model, [9] describes the relationship between tumor cells, dendritic cells, CD$4^{+}$ and CD8${}^{+}$T cells, IL-12 and IL-2, vaccine-produced GM-CSF, and a $T$ cell checkpoint inhibitor associated with PD-1. The efficacy of the two drugs was explored when used alone and in combination. For the infection of MERS-CoV virus, the differential equation model of [10] considered the mediating effect of DPP4 protein in the process of viral infection. Recently, [11] used differential equation models to simulate the complex dynamics of oncolytic virus therapy, immune checkpoint inhibition (PD-1 and PD-L1), and innate and adaptive immunity in the treatment of glioblastoma. In particular, on the basis of [12], in [13], according to the infection mechanism of SARS-CoV-2, a virus dynamics model coupled with the concentration of target cell receptor was established under the background of different drug action, and the synergistic effect of drugs was discovered.

Since 2019, COVID-19 has brought a huge threat to the safety of human life worldwide. However, it is encouraging to note that although the continuous mutation of the structure of S protein on the surface of SARS-CoV-2 virus has brought difficulties to the prevention and control of the disease, a large number of effective vaccines and targeted drugs have been rapidly developed. The main objective of this paper is to provide a theoretical understanding of the transmission and control strategies of COVID-19 by constructing a differential equation model based on the mechanism of SARS-CoV-2 invasion mediated by target cells ACE2 receptor [2].

The Fig. 1 shows the invasion of target cells by SARS-CoV-2 mediated by the target cell ACE2 receptor, and the completion of this process mainly depends on the effective binding of S protein on the surface of the virus to the target cell ACE2 receptor . For the convenience of model construction, we consider a framework diagram Fig. 2, where the state variables $T\left(t\right),I\left(t\right)$, $v\left(t\right)$ and $D\left(t\right)$ denote, respectively, the concentrations of per unit volume of uninfected target cells, infected target cells, free virions and ACE2 receptor carried by uninfected target cells at time $t$. The following is the main process of model construction. The Eq. (1.1) represents the change in the concentration of uninfected target cells at time $t$. Target cells proliferate and die with rate constants $\lambda$ and $d_1$, respectively. Mediated by the ACE2 receptor , the fusion of free virions with uninfected target cells leads to a reduction of uninfected target cells at the amount of $\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)$ [10,12,13], where $\beta$ is the second order rate constant of the infection by free virions, $v$, of target cells expressing excess ACE2 receptor [13], $f\left(D\left(t\right)\right)$ represents the probability of successful entry of the virion into the target cell mediated by the ACE2 receptor. When the concentration of the target cell ACE2 receptor is lower (higher), there are $f\left(D\left(t\right)\right)\sim 0(\sim 1)$. Usually $f\left(D\right)$ is chosen as the classic Hill function: $f\left(D\right)=\frac{D^n}{D_1^n+D^n}$, where $D_1$ is the half-saturation constant and $n>0$ is the Hill coefficient.

$$\frac{dT\left(t\right)}{dt}=\underset{\text{Proliferation of target cells}}{\underbrace{\lambda }}-\underset{\text{Reduction of target cells by free virions and ACE2}}{\underbrace{\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)}}-\underset{\text{Death of target cells}}{\underbrace{d_{1}T\left(t\right).}}$$ (1.1)

Fig. 1.

Mechanism of SARS-CoV-2 virus invasion of target cells.

Fig. 2.

Flow chart of SARS-CoV-2 virions invasion.

The Eq. (1.2) represents the change in the concentration of infected target cells at time $t$. The proliferation of infected cells is $e^{-d_2\tau }\beta f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )$, and die at a rate $d_2$. The constant $\tau$ is called the time delay, and the term $e^{-d_2\tau }$ is the probability of survival of infected cells [5], [6].

$$\frac{dI\left(t\right)}{dt}=\underset{\text{Production by target cells}}{\underbrace{e^{-d_2\tau }\beta f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )}}-\underset{\text{Death of infected target cells}}{\underbrace{d_{2}I\left(t\right).}}$$ (1.2)

The Eq. (1.3) represents the change in the concentration of free virions at time $t$. The rate of free virions released after rupture of infected target cells is $d_{2}NI\left(t\right)$, and degradation at a rate $c$, $d_2$ is a ratio constant and $N$ is a positive integer.

$$\frac{dv\left(t\right)}{dt}=\underset{\text{Production by infected target cells}}{\underbrace{d_{2}NI\left(t\right)}}-\underset{\text{Death of free virions}}{\underbrace{cv\left(t\right).}}$$ (1.3)

The Eq. (1.4) represents the change in the concentration of ACE2 receptor carried on the surface of uninfected target cells at time $t$. The proliferation rate of ACE2 receptor is ${\lambda }_1$ and the degradation rate is $d_3$. Note that the term $\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)$ indicates a decrease in uninfected target cells (due to free virions), and the average number of ACE2 receptor carried by each uninfected target cell is $D\left(t\right)/T\left(t\right)$. Thus, the decrease in ACE2 receptor due to the decrease in uninfected target cells (caused by free virions) is $k\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)\times D\left(t\right)/T\left(t\right)=k\beta f\left(D\left(t\right)\right)v\left(t\right)D\left(t\right)$, where $k$ is a constant ratio. All the parameters in the model (1.1)–(1.4) are assumed to positive constants.

$$\frac{dD\left(t\right)}{dt}=\underset{\text{Proliferation of ACE2 receptor }}{\underbrace{{\lambda }_1}}-\underset{\text{Decrease in ACE2 receptor }}{\underbrace{k\beta f\left(D\left(t\right)\right)v\left(t\right)D\left(t\right)}}-\underset{\text{Degradation of ACE2 receptor}}{\underbrace{d_{3}D\left(t\right).}}$$ (1.4)

The main purpose of this paper is to study the global stability of the equilibria of the model (1.1)–(1.4). We assume that the function $f\left(D\right)$ is continuously differentiable and strictly monotonically increasing on $[0,+\infty )$, and satisfies $f\left(0\right)=0$.

The initial condition for the model (1.1)–(1.4) is

$$T\left(\theta \right)={\phi }_1\left(\theta \right),I\left(\theta \right)={\phi }_2\left(\theta \right),v\left(\theta \right)={\phi }_3\left(\theta \right),D\left(\theta \right)={\phi }_4\left(\theta \right),\theta \in [-\tau ,0],$$

where, $\phi =({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4)\in C([-\tau ,0],R_{+}^4)$. $C\left([-\tau ,0],R_{+}^4\right)$ is the Banach space of continuous functions mapping from $[-\tau ,0]$ to $R_{+}^4$. By using the existence and uniqueness theorems of the solutions of functional differential equations [14], it is easy to show that the solution of model (1.1)–(1.4) with the initial condition is existent, unique and non-negative for $t\ge 0$.

The structure of this paper is as follows. In the second section, the calculation of the basic reproduction number as well as the complete classification of the infection-free equilibrium and the infected equilibrium will be given. In the third section, by constructing Lyapunov functionals (e.g., see [14], [15], [16], [17], [18] and the references cited therein), the global stability of the infection-free equilibrium and the infected equilibrium is demonstrated. Finally in the fourth section, a brief discussion is given.

2. The existence of equilibrium

First of all, it is easy to find that the model (1.1)–(1.4) always has an infection-free equilibrium $E_0=(T_0,0,0,D_0)$, where $T_0=\frac{\lambda }{d_1}$ and $D_0=\frac{{\lambda }_1}{d_3}$. Then, using the next-generation matrix method [19], [20], it is not difficult to get the basic reproduction number as

$${ℛ}_0=\frac{e^{-d_2\tau }\beta N{T}_0}{c}f\left(D_0\right).$$

Theorem 2.1

The model (1.1) – (1.4) has a unique infected equilibrium $E^{\ast }=(T^{\ast },I^{\ast },v^{\ast },D^{\ast })$ if and only if ${ℛ}_0>1$ .

Proof

An infected equilibrium $(T,I,v,D)$ of the model (1.1)–(1.4) satisfies the following system of equations

$$\left\{\begin{array}{c}0=\lambda -\beta f\left(D\right)vT-d_{1}T,\hfill \\ 0=e^{-d_2\tau }\beta f\left(D\right)vT-d_{2}I,\hfill \\ 0=d_{2}NI-cv,\hfill \\ 0={\lambda }_1-k\beta f\left(D\right)vD-d_{3}D.\hfill \end{array}\right)$$ (2.1)

Therefore, we obtain

$$T=\frac{\lambda -d_2{e}^{d_2\tau }I}{d_1},v=\frac{N{d}_{2}I}{c},D=\frac{{\lambda }_1}{k{d}_2{e}^{d_2\tau }I/T+d_3}.$$ (2.2)

Substituting (2.2) into the second equation of (2.1), we have

$$e^{-d_2\tau }\beta f\left(\frac{{\lambda }_1}{e^{d_2\tau }k{d}_{2}I/T+d_3}\right)\frac{\lambda -d_2{e}^{d_2\tau }I}{d_1}\frac{N{d}_{2}I}{c}-d_{2}I=0.$$

Consider

$$H\left(I\right)=\frac{e^{-d_2\tau }\beta N}{c}f\left(\frac{{\lambda }_1}{k{d}_2{e}^{d_2\tau }I/T+d_3}\right)\frac{\lambda -d_2{e}^{d_2\tau }I}{d_1}-1.$$

Obviously, we can get

$$H\left(0\right)=e^{-d_2\tau }\frac{\beta N}{c}f\left(\frac{{\lambda }_1}{d_3}\right)\frac{\lambda }{d_1}-1={ℛ}_0-1,\underset{I\to \frac{\lambda }{d_2}{e}^{-d_2\tau }}{lim}H\left(I\right)=-1<0,$$

and

$$\frac{d}{dI}\left[f\left(\frac{{\lambda }_1}{e^{d_2\tau }k{d}_{2}I/T+d_3}\right)\right]=-\frac{k\lambda {\lambda }_1{e}^{d_2\tau }{d}_1{d}_2}{{\left(\lambda d_3-I{d}_2{d}_3{e}^{d_2\tau }+kI{d}_1{d}_2{e}^{d_2\tau }\right)}^2}{f}_I\left(\frac{{\lambda }_1}{e^{d_2\tau }k{d}_{2}I/T+d_3}\right)\equiv {\rm Y}<0.$$

Thus, we have

$$\frac{dH\left(I\right)}{dI}={\rm Y}\frac{N{e}^{-d_2\tau }\beta }{c}\frac{\lambda -d_2{e}^{d_2\tau }I}{d_1}-\frac{N{d}_2}{c{d}_1}\beta f\left(D\right)<0.$$

Hence, it follows that a unique infected equilibrium $E^{\ast }=(T^{\ast },I^{\ast },v^{\ast },D^{\ast })$ exists only when ${ℛ}_0>1$, where $T^{\ast }\in \left(0,\frac{\lambda }{d_1}\right)$, $I^{\ast }\in \left(0,\frac{\lambda }{d_2}{e}^{-d_2\tau }\right)$, $v^{\ast }>0$ and $D^{\ast }\in \left(0,\frac{{\lambda }_1}{d_3}\right)$. □

3. Global stability of equilibria

For the global stability of the infection-free equilibrium $E_0$ and the infected equilibrium $E^{\ast }$, we have

Theorem 3.1

$\left(i\right)$ If ${ℛ}_0\le 1$ , then the infection-free equilibrium $E_0$ is globally asymptotically stable. $\left(ii\right)$ If ${ℛ}_0>1$ , then the infected equilibrium $E^{\ast }$ is globally asymptotically stable.

Proof

(i) Let ${ℛ}_0\le 1$. We construct Lyapunov functional $W\left(\phi \right)$ as follows,

$$W\left(\phi \right)=\left({\phi }_1\left(0\right)-T_0-T_{0}ln\frac{{\phi }_1\left(0\right)}{T_0}\right)+e^{d_2\tau }{\phi }_2\left(0\right)+e^{d_2\tau }\frac{1}{N}{\phi }_3\left(0\right)+\frac{T_0}{k{D}_0}\left({\phi }_4\left(0\right)-D_0-{\int }_{D_0}^{{\phi }_4\left(0\right)}\frac{f\left(D_0\right)}{f\left(\xi \right)}d\xi \right)+{\int }_{-\tau }^0\beta f\left({\phi }_4\left(s\right)\right){\phi }_3\left(s\right){\phi }_1\left(s\right)ds.$$

Obviously, $W\left(\phi \right)$ is positive definite with respect to the infection-free equilibrium $E_0$. In addition, along the solution $(T\left(t\right),I\left(t\right),v\left(t\right),D\left(t\right))$ of the model (1.1)–(1.4), the derivative of $W$ with respect to $t$ is, for $t\ge 0$,

$$\frac{dW}{dt}=\left(1-\frac{T_0}{T\left(t\right)}\right)(\lambda -\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)-d_{1}T\left(t\right))+\beta f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )-e^{d_2\tau }{d}_{2}I\left(t\right)+e^{d_2\tau }\frac{1}{N}(d_{2}NI\left(t\right)-cv\left(t\right))+\frac{T_0}{k{D}_0}\left(1-\frac{f\left(D_0\right)}{f\left(D\left(t\right)\right)}\right)({\lambda }_1-k\beta f\left(D\left(t\right)\right)v\left(t\right)D\left(t\right)-d_{3}D\left(t\right))+\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)-\beta f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )=\left(1-\frac{T_0}{T\left(t\right)}\right)(\lambda -\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)-d_{1}T\left(t\right))+\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)-c{e}^{d_2\tau }\frac{1}{N}v\left(t\right)+\frac{T_0}{k{D}_0}\left(1-\frac{f\left(D_0\right)}{f\left(D\left(t\right)\right)}\right)({\lambda }_1-k\beta f\left(D\left(t\right)\right)v\left(t\right)D\left(t\right)-d_{3}D\left(t\right))=\lambda \left(2-\frac{T_0}{T\left(t\right)}-\frac{T\left(t\right)}{T_0}\right)+\frac{c{e}^{d_2\tau }}{N}\left(\frac{e^{-d_2\tau }N{T}_0\beta f\left(D_0\right)}{c}-1\right)v\left(t\right)+T_0\beta v\left(t\right)(f\left(D\left(t\right)\right)-f\left(D_0\right))+\frac{T_0\beta f\left(D\left(t\right)\right)v\left(t\right)D\left(t\right)}{D_0}\left(\frac{f\left(D_0\right)}{f\left(D\left(t\right)\right)}-1\right)+\frac{T_0{\lambda }_1}{k{D}_0}\left(1-\frac{f\left(D_0\right)}{f\left(D\left(t\right)\right)}\right)+\frac{T_0{d}_{3}D\left(t\right)}{k{D}_0}\left(\frac{f\left(D_0\right)}{f\left(D\left(t\right)\right)}-1\right)=\lambda \left(2-\frac{T_0}{T\left(t\right)}-\frac{T\left(t\right)}{T_0}\right)+e^{d_2\tau }\frac{c}{N}({ℛ}_0-1)v\left(t\right)+\left(\frac{T_0{d}_3}{k{D}_0}+\frac{\beta v\left(t\right)T_{0}f\left(D\left(t\right)\right)}{D_0}\right)(D\left(t\right)-D_0)\left(\frac{f\left(D_0\right)}{f\left(D\left(t\right)\right)}-1\right)\le 0.$$

Therefore, the infection-free equilibrium $E_0$ is stable [14]. In addition, the use of LaSalle’s invariance principle yields that the infection-free equilibrium $E_0$ is globally attractive [14]. Hence, the infection-free equilibrium $E_0$ is globally asymptotically stable.

(ii) Let ${ℛ}_0>1$ and $g\left(x\right)=x-1-lnx(x>0)$. Construct the following Lyapunov functional,

$$L\left(\phi \right)=e^{-d_2\tau }\left({\phi }_1\left(0\right)-T^{\ast }-T^{\ast }ln\frac{{\phi }_1\left(0\right)}{T^{\ast }}\right)+\left({\phi }_2\left(0\right)-I^{\ast }-I^{\ast }ln\frac{{\phi }_2\left(0\right)}{I^{\ast }}\right)+\frac{1}{N}\left({\phi }_3\left(0\right)-v^{\ast }-v^{\ast }ln\frac{{\phi }_3\left(0\right)}{v^{\ast }}\right)+\frac{T^{\ast }}{k{D}^{\ast }}{e}^{-d_2\tau }\left({\phi }_4\left(0\right)-D^{\ast }-{\int }_{D^{\ast }}^{{\phi }_4\left(0\right)}\frac{f\left(D^{\ast }\right)}{f\left(\theta \right)}d\theta \right)+d_2{I}^{\ast }{\int }_{-\tau }^{0}g\left(\frac{f\left({\phi }_4\left(\theta \right)\right){\phi }_3\left(\theta \right){\phi }_1\left(\theta \right)}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }}\right)d\theta .$$

Obviously, $L\left(\phi \right)$ is positive definite with respect to the infected equilibrium $E^{\ast }$. We now calculate the derivative of $L$ along any positive solution $(T\left(t\right),I\left(t\right),v\left(t\right),D\left(t\right))$ of the model  (1.1)–(1.4) , it follows that, $t\ge 0$,

$$\frac{dL}{dt}=e^{-d_2\tau }\left(1-\frac{T^{\ast }}{T\left(t\right)}\right)(\beta f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }-\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)+d_1(T^{\ast }-T\left(t\right)))+\left(1-\frac{I^{\ast }}{I\left(t\right)}\right)(e^{-d_2\tau }\beta f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )-d_{2}I\left(t\right))+\frac{1}{N}\left(1-\frac{v^{\ast }}{v\left(t\right)}\right)(d_{2}NI\left(t\right)-cv\left(t\right))+\frac{T^{\ast }}{k{D}^{\ast }}{e}^{-d_2\tau }\left(1-\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}\right)(k\beta f\left(D^{\ast }\right)v^{\ast }{D}^{\ast }-k\beta f\left(D\left(t\right)\right)v\left(t\right))(D\left(t\right)+d_3{D}^{\ast }-d_{3}D\left(t\right))+d_2{I}^{\ast }\frac{f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }}-d_2{I}^{\ast }\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }}+d_2{I}^{\ast }ln\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )}{f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)}.$$

According to (2.1), we have

$$\beta f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }=e^{d_2\tau }{d}_2{I}^{\ast },c{v}^{\ast }=d_{2}N{I}^{\ast }.$$

And then, we can get, for $t\ge 0$,

$$\frac{dL}{dt}=-\frac{d_1{e}^{-d_2\tau }}{T\left(t\right)}{(T\left(t\right)-T^{\ast })}^2+d_2{I}^{\ast }\left(4-\frac{v^{\ast }I\left(t\right)}{v\left(t\right)I^{\ast }}-\frac{T^{\ast }}{T\left(t\right)}-\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}-\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )I^{\ast }}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }I\left(t\right)}\right)+e^{-d_2\tau }{T}^{\ast }\beta f\left(D\left(t\right)\right)v\left(t\right)-\frac{c}{N}v\left(t\right)+e^{-d_2\tau }{d}_3\frac{T^{\ast }}{k{D}^{\ast }}{D}^{\ast }-e^{-d_2\tau }{d}_3\frac{T^{\ast }}{k{D}^{\ast }}D\left(t\right)+k\beta e^{-d_2\tau }\frac{T^{\ast }}{k{D}^{\ast }}f\left(D^{\ast }\right)v\left(t\right)D\left(t\right)-e^{-d_2\tau }k\beta \frac{T^{\ast }}{k{D}^{\ast }}f\left(D\left(t\right)\right)v\left(t\right)D\left(t\right)+d_3{e}^{-d_2\tau }\frac{T^{\ast }}{k{D}^{\ast }}\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}D\left(t\right)-d_3{e}^{-d_2\tau }\frac{T^{\ast }}{k{D}^{\ast }}\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}{D}^{\ast }+d_2{I}^{\ast }\frac{f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }}-e^{-d_2\tau }\beta f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)+e^{-d_2\tau }\beta f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )-d_2{I}^{\ast }\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }}+d_2{I}^{\ast }ln\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )}{f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)}.$$

Using the equality

$$ln\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )}{f\left(D\left(t\right)\right)v\left(t\right)T\left(t\right)}=ln\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )I^{\ast }}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }I\left(t\right)}+ln\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}+ln\frac{v^{\ast }I\left(t\right)}{v\left(t\right)I^{\ast }}+ln\frac{T^{\ast }}{T\left(t\right)},$$

we have, for $t\ge 0$,

$$\frac{dL}{dt}=-d_1{e}^{-d_2\tau }\frac{1}{T\left(t\right)}{(T\left(t\right)-T^{\ast })}^2-d_2{I}^{\ast }g\left(\frac{v^{\ast }I\left(t\right)}{v{I}^{\ast }}\right)-d_2{I}^{\ast }g\left(\frac{T^{\ast }}{T\left(t\right)}\right)-d_2{I}^{\ast }g\left(\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}\right)-d_2{I}^{\ast }g\left(\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )I^{\ast }}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }I\left(t\right)}\right)+\beta e^{-d_2\tau }{T}^{\ast }f\left(D\left(t\right)\right)v\left(t\right)-\frac{c}{N}v\left(t\right)+d_3{e}^{-d_2\tau }\frac{T^{\ast }}{k{D}^{\ast }}(D^{\ast }-D\left(t\right))+k\beta e^{-d_2\tau }\frac{T^{\ast }}{k{D}^{\ast }}v\left(t\right)D\left(t\right)(f\left(D^{\ast }\right)-f\left(D\left(t\right)\right))+d_3{e}^{-d_2\tau }\frac{T^{\ast }}{k{D}^{\ast }}\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}(D\left(t\right)-D^{\ast })=-\frac{d_1{e}^{-d_2\tau }}{T\left(t\right)}{\left(T\left(t\right)-T^{\ast }\right)}^2-\left(\frac{e^{-d_2\tau }\beta T^{\ast }v\left(t\right)}{D^{\ast }}+\frac{e^{-d_2\tau }{d}_3{T}^{\ast }}{f\left(D\left(t\right)\right)k{D}^{\ast }}\right)(D\left(t\right)-D^{\ast })(f\left(D\left(t\right)\right)-f\left(D^{\ast }\right))-d_2{I}^{\ast }\left[g\left(\frac{v^{\ast }I\left(t\right)}{v\left(t\right)I^{\ast }}\right)+g\left(\frac{T^{\ast }}{T\left(t\right)}\right)+g\left(\frac{f\left(D^{\ast }\right)}{f\left(D\left(t\right)\right)}\right)+g\left(\frac{f\left(D(t-\tau )\right)v(t-\tau )T(t-\tau )I^{\ast }}{f\left(D^{\ast }\right)v^{\ast }{T}^{\ast }I\left(t\right)}\right)\right]\le 0.$$

Therefore, the infected equilibrium $E^{\ast }$ is stable [14]. In addition, the infected equilibrium $E^{\ast }$ is globally attractive by using LaSalle’s invariance principle [14]. Hence, the infected equilibrium $E^{\ast }$ is globally asymptotically stable. □

4. Discussion

We know that ACE2, a receptor protein expressed on the surface of target cells, is critical in the process of infection of target cells by SARS-CoV-2. The model (1.1)–(1.4) is based on the mediated effect of ACE2 receptor. Theorem 3.1 implies that virus infection can be removed, if the basic reproduction number ${ℛ}_0$ satisfies the following conditions,

$${ℛ}_0=\frac{e^{-d_2\tau }\beta N}{c}\left(\frac{\lambda }{d_1}\right)f\left(\frac{{\lambda }_1}{d_3}\right)\le 1,\text{or}\tau \ge \frac{1}{d_2}ln\left[\frac{\beta N}{c}\left(\frac{\lambda }{d_1}\right)f\left(\frac{{\lambda }_1}{d_3}\right)\right]\equiv {\tau }_0.$$

Therefore, in order to make ${ℛ}_0\le 1$, it can be achieved by inhibiting the proliferation rate of ACE2 receptor ${\lambda }_1$ or increasing the degradation rate of ACE2 receptor $d_3$, or decreasing the infection ratio constant $\beta$.

Data availability

No data was used for the research described in the article.