Global dynamics of a network-based SIQS epidemic model with nonmonotone incidence rate

Abstract

The risk of propagation of infectious diseases such as avian influenza and COVID-19 is generally controlled or reduced by quarantine measures. Considering this situation, a network-based SIQS (susceptible-infected-quarantined-susceptible) infectious disease model with nonmonotone incidence rate is established and analyzed in this paper. The psychological impact of the transmission of certain diseases in heterogeneous networks at high levels of infection may be characterized by the related nonmonotone incidence rate. The expressions of the basic reproduction number and equilibria of the model are determined analytically. We demonstrate in detail the uniform persistence of system and the global asymptotic stability of the disease-free equilibrium. The global attractivity of the unique endemic equilibrium is discussed by using monotone iteration technique. We obtain that the endemic equilibrium is globally asymptotically stable under certain conditions by constructing appropriate Lyapunov function. In addition, numerical simulations are performed to indicate the theoretical results.

1. Introduction

It has been known that infectious diseases not only threaten the health of human beings but can even bring great disasters to the national economy and people’s livelihood. Therefore, it is very important to investigate the transmission mechanisms and development trends of infectious diseases and then adopt appropriate measures to control their prevalence. Dynamic modeling is one of the efficient tools to research the transmission of infectious diseases in the population, not only to characterize the intrinsic transmission mechanism of infectious diseases but also to qualitatively and quantitatively study the transmission pattern, which is important in the formulation and evaluation of prevention and control measures. Traditional models, which are based on populations that are uniformly mixed, are inadequate to describe the transmission process of infectious diseases in large-scale social contact networks with distinct heterogeneity. As a matter of fact, transmission of infectious diseases at the group level occurs mainly through social contact networks, consequently, it is more realistic to use complex networks for investigating the spread of epidemics in populations.

In recent years, an increasing number of scholars have begun to focus on the dynamics of infectious diseases on complex networks [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Pastor-Satorras and Vespignani [1], [2] first established the SIS model in scale-free networks using a mean-field approximation. The absence of prevalence threshold in these networks was the most surprising finding. The stability of the SIS model was later determined [4], [5]. These outstanding achievements have sparked several of the related projects [11], [12], [13], [14], [15], [16], [17].

How to propose effective measures of disease control is an important research theme for the complex network infectious diseases. Quarantine of symptomatic individuals is an intervention program for controlling the propagation of infectious diseases, and this method has been used to prevent diseases such as cholera and COVID-19 [18], [19], [20], [21], [22]. The most direct method of controlling the transmission of epidemic may be the quarantine of infected persons, because it disrupts the links that connect the infected persons to the susceptible persons. In particular, quarantining infected persons is probably the most efficient measure for controlling the transmission of disease until associated vaccines and valid therapy for the infectious disease are available. A novel coronavirus (COVID-19) epidemic has been reported in Wuhan, Hubei Province, China, since December 2019. On March 11, 2020, the World Health Organization (WHO) declared COVID-19 as a global pandemic. The Chinese government has controlled the spread of the epidemic by quarantining infected individuals, close contacts, and susceptible individuals as well as other measures. However, there are relatively few studies on the impacts of quarantine on transmission of epidemic on complex networks [22], [23], [24], [25]. Li et al. [23] presented an SIQRS infectious disease model on the scale-free networks, and theoretical results suggest that the prevalence threshold is actually related to the networks’ topology. In addition, the epidemic threshold goes up and the final number of infected persons decreases as quarantine rate increases. Recently, Li et al. [25] proposed the SIQS infectious disease model on complex networks:

$$\{\begin{array}{c}{\displaystyle \frac{d{S}_k\left(t\right)}{dt}=A-\beta k{S}_k\left(t\right)\Theta \left(t\right)-d{S}_k\left(t\right)+\gamma I_k\left(t\right)+\epsilon Q_k\left(t\right),}\hfill \\ {\displaystyle \frac{d{I}_k\left(t\right)}{dt}=\beta k{S}_k\left(t\right)\Theta \left(t\right)-(d+\gamma )I_k\left(t\right)-\delta I_k\left(t\right),}\hfill \\ {\displaystyle \frac{d{Q}_k\left(t\right)}{dt}=\delta I_k\left(t\right)-(d+\epsilon )Q_k\left(t\right),k=1,2,\dots ,n,}\hfill \end{array}$$ (1.1)

where $S_k\left(t\right),I_k\left(t\right),Q_k\left(t\right)$ represent the relative density of susceptible, infected and quarantined individuals with degree $k$ at time $t$, respectively. $\beta$ is the rate of transmission from infected to susceptible individuals. The infected nodes are quarantined with rate $\delta$. A non-quarantined infected node recovers as a susceptible node with probability $\gamma$. $\epsilon$ is the recovery rate of each quarantined infected node. In reality, with improvements in the medical environment, patients who are quarantined are generally more likely to recover than those who are not, which makes $\epsilon >\gamma$. Based on the biological meaning, the parameters $\beta ,\delta ,\gamma$ and $\epsilon$ are all positive constants. The constant $A$ is the birth rate, $d$ is the natural mortality rate. $\Theta \left(t\right)$ denotes the probability that any given edge points to an infected node on networks.

The incidence rate is significant for the investigation of mathematical epidemiology. In the majority of classical disease transmission models, one generally uses bilinear incidence rate $g\left(I\right)S=kIS$, where $g\left(I\right)=kI$. In recent years, many researchers considered nonlinear incidence rates to describe the transmission of infectious diseases. For example, as a result of the research of the 1973 cholera pandemic in Bari, in order to avoid infinite of the contact rate, Capasso and Serio [26] proposed a saturated incidence rate $g\left(I\right)S=\frac{kIS}{1+\alpha I}$, in which $g\left(I\right)=\frac{kI}{1+\alpha I}$, $\alpha >0$, and as $I$ becomes larger, $g\left(I\right)$ converges to saturation level. To describe the psychological effect of the spread of some diseases spread at high infective levels, Xiao and Ruan [27] presented an incidence rate:

$${\displaystyle g\left(I\right)S=\frac{kIS}{1+\alpha I^2},}$$ (1.2)

where the incidence function $g\left(I\right)=\frac{kI}{1+\alpha I^2}$ is nonmonotone when $I\ge 0$ (see Fig. 1 ). Obviously, as $I$ is small, $g\left(I\right)$ gets larger, and as $I$ increases, $g\left(I\right)$ gets smaller. It implies that during an initial stage of an outbreak, the susceptible persons are not aware of the severity of the outbreak and the transmission rate probably increases with increasing numbers of infected persons. However, in cases where there are numerous infected individuals, one usually reduces contact with others in a unit of time. Thus, the transmission rate is likely to reduce with increasing numbers of infected persons in the case of extremely large numbers of infected individuals. For example, during the outbreak of COVID-19, aggressive measures such as mask-wearing and quarantine effectively reduced transmission rate. As a matter of reality, as a result of quarantining infected persons or protective behaviors of susceptible persons, the incidence rate decreases at higher levels of infection. In [27], They analyzed the global dynamics of the SIRS model with (1.2) as well as demonstrated that as time progresses, numbers of infected persons converge toward zero or the disease is persistent. Chen and Wen [28] used (1.2) to characterize the transmission between human and poultry, which conform the effect of human intervention.

Fig. 1.

Nonmonotone incidence function $g\left(I\right)$ with $\alpha =1.5$.

The incidence rate in current models of complex network infectious diseases is generally considered to be a bilinear function. Recently, to analyze effect of nonmonotone incidence rate on complex network infectious disease models, Li [29] investigated a network-based SIS infectious disease model with nonmonotone incidence rate and demonstrated that the disease will become extinct when the transmission rate is below a threshold, otherwise it will be permanent. In [30], Wei et al. proved the global stability and attractivity of the endemic equilibrium in [29]. Liu et al. [31] investigated the network-based SIRS infectious disease model with vaccination and nonmonotone incidence rate and demonstrated global stability of endemic equilibrium.

Based on the above discussions, combining (1.1) and (1.2), we establish the following SIQS epidemic model with nonmonotone incidence rate as follows:

$$\{\begin{array}{c}{\displaystyle \frac{d{S}_k\left(t\right)}{dt}=A-\beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-d{S}_k\left(t\right)+\gamma I_k\left(t\right)+\epsilon Q_k\left(t\right),}\hfill \\ {\displaystyle \frac{d{I}_k\left(t\right)}{dt}=\beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma )I_k\left(t\right)-\delta I_k\left(t\right),}\hfill \\ {\displaystyle \frac{d{Q}_k\left(t\right)}{dt}=\delta I_k\left(t\right)-(d+\epsilon )Q_k\left(t\right),k=1,2,\dots ,n,}\hfill \end{array}$$ (1.3)

where $\alpha$ is a parameter that indicates the psychological or inhibitory effect. The nonmonotone incidence rate gets bilinear and (1.3) simplifies to (1.1) when $\alpha =0$. The term $A$ indicates the number of newly born nodes is the same for all different degrees per unit time, and each newly born node is susceptible. The term $\beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}$ represents when there is a large number of infected individuals in the network, the number of effective contacts between infected individuals and susceptible individuals will decrease due to the quarantine of infected individuals or the protection measures by the susceptible individuals. Throughout this paper, we assume that complex networks are uncorrelated. Therefore, $\Theta \left(t\right)$ can be written as

$${\displaystyle \Theta \left(t\right)=\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)I_k\left(t\right),}$$ (1.4)

where $P\left(k\right)$ is the probability that a randomly chosen node has degree $k$ (i.e., the degree distribution) and thus ${\sum }_{k=1}^{n}P\left(k\right)=1$, $<k>={\sum }_{k=1}^n\mathit{kP}\left(k\right)$ is the average degree of the network.

For system (1.3), we assume $A=d$, which implies that births and deaths are balanced. In addition, since the network’s removing nodes and links due to births and deaths just account for a tiny portion of the total, each node’s degree is assumed to be constant. The two hypotheses mentioned above are also provided in [32], [33].

The remaining part is organized in the following: Section 2 investigates the positivity and boundedness of solutions of the SIQS model. In addition, we get equilibria and epidemic threshold. We demonstrate the global asymptotical stability of disease-free equilibrium and discuss the global dynamics of the endemic equilibrium are discussed in detail in Section 3. In Section 4, we show numerical simulations to confirm all the theoretical results. Conclusions are presented in Section 5.

2. Positivity, boundedness and equilibria

Considering the actual situation, the initial conditions of system (1.3) satisfy

$$I_k\left(0\right),Q_k\left(0\right)\ge 0,S_k\left(0\right)=1-I_k\left(0\right)-Q_k\left(0\right)>0,\text{and}\Theta \left(0\right)>0.$$ (2.1)

Therefore, it follows that $S_k\left(t\right)+I_k\left(t\right)+Q_k\left(t\right)\equiv 1$, $t\ge 0$ for $k=1,2,\dots ,n$, then it suffices to discuss the following system

$$\{\begin{array}{c}{\displaystyle \frac{d{I}_k\left(t\right)}{dt}=\beta k[1-I_k\left(t\right)-Q_k\left(t\right)]\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma +\delta )I_k\left(t\right),}\hfill \\ {\displaystyle \frac{d{Q}_k\left(t\right)}{dt}=\delta I_k\left(t\right)-(d+\epsilon )Q_k\left(t\right),k=1,2,\dots ,n.}\hfill \end{array}$$ (2.2)

We just need to explore the dynamical properties of subsystem (2.2) to obtain the dynamical behaviors of (1.3).

The following lemma establishes the positivity of solutions.

Lemma 2.1

Let$(S_1\left(t\right),I_1\left(t\right),Q_1\left(t\right),\dots ,S_n\left(t\right),I_n\left(t\right),Q_n\left(t\right))$be the solution of(1.3)with initial conditions(2.1). Then for any$t>0$and$k=1,2,\dots ,n$, one has$0<S_k\left(t\right),I_k\left(t\right),Q_k\left(t\right)<1$and$0<\Theta \left(t\right)<1$.

Proof

We will start by proving that $\Theta \left(t\right)>0$ for every $t>0$. It follows from the second equation of (1.3) that

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\Theta \left(t\right)}{dt}}& {\displaystyle =\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)\frac{d{I}_k\left(t\right)}{dt}}\hfill \\ & {\displaystyle =\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)\left[{\displaystyle \beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma +\delta )I_k\left(t\right)}\right]}\hfill \\ & =\Theta \left(t\right)\left[{\displaystyle \frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}P\left(k\right)\frac{S_k\left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma +\delta )}\right].\hfill \end{array}$$ (2.3)

By integrating (2.3) from 0 to $t$ yields

$$\Theta \left(t\right)=\Theta \left(0\right)\text{exp}\left\{{\displaystyle \frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}P\left(k\right)\frac{S_k\left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma +\delta )}\right\}.$$

It’s obvious to observe that for any $t>0$ there is $\Theta \left(t\right)>0$ due to $\Theta \left(0\right)>0$.

Note that $S_k\left(0\right)>0$ for $k=1,2,\dots ,n$. It follows from the first equation of (1.3) and continuity of $S_k\left(t\right)$ that a small enough $\zeta >0$ exists leading to $S_k\left(t\right)>0$ when $t\in (0,\zeta )$. Then we want $S_k\left(t\right)>0$ holds for any $t>0$. Suppose that when $t>0$, $S_k\left(t\right)$ may be equal to zero. There is $h\in \{1,2,\dots ,n\}$ and the first time $t^1\ge \zeta >0$ resulting in $S_h\left(t^1\right)=0$ and $S_h\left(t\right)>0$ for $t\in (0,t^1)$. Together with the second equation of (1.3), it follows that when $t\in (0,t^1)$ yields $\frac{d{I}_h\left(t\right)}{dt}+(d+\gamma +\delta )I_h\left(t\right)>0$. It is clear that there is $I_h\left(t\right)>I_h\left(0\right)e^{-(d+\gamma +\delta )t}\ge 0$ for $t\in (0,t^1)$.

This gives $\frac{d{Q}_h\left(t\right)}{dt}+(d+\epsilon )Q_h\left(t\right)>0$ with respect to $t\in (0,t^1)$ from the third equation of (1.3), which implies that with respect to $t\in (0,t^1)$ there is $Q_h\left(t\right)>Q_h\left(0\right)e^{-(d+\epsilon )t}\ge 0$. Obviously, by continuity of $Q_h\left(t\right)$ we obtain $Q_h\left(t^1\right)\ge 0$. Accordingly, the first equation of (1.3) represents that $\frac{d{S}_h\left(t^1\right)}{d{t}^1}=A+\gamma I_h\left(t^1\right)+\epsilon Q_h\left(t^1\right)>0$. Then $S_h\left(t\right)<S_h\left(t^1\right)=0$ for $t\in (t^1-\kappa ,t^1)\subset (0,t^1)$, where $\kappa$ ia an arbitrary positive constant. Clearly, it gives rise to a contradiction. As a result, $S_k\left(t\right)>0$ for any $t>0$.

Analogously, for any $t>0$, we get $I_k\left(t\right)>0$ by the second equation of (1.3). Consequently, we derive $Q_k\left(t\right)>0$ for every $t>0$ from the last equation of (1.3). Notice that $S_k\left(t\right)+I_k\left(t\right)+Q_k\left(t\right)=1$. Therefore, one can easily demonstrated that for any $t>0$ and all $k=1,2,\dots ,n$, there are $0<S_k\left(t\right),I_k\left(t\right),Q_k\left(t\right)<1$ and $0<\Theta \left(t\right)<1$. □

Lemma 2.2

For system(1.3), the following results are obtained.

(1) There always exists a disease-free equilibrium$E_0=(1,0,0,\dots ,1,0,0)$.

(2) If$R_0>1$, system admits a unique epidemic equilibrium$E_{*}={\{S_k^{*},I_k^{*},Q_k^{*}\}}_{k=1}^n$.

Here

$$\begin{array}{cc}\hfill S_k^{*}& {\displaystyle =\frac{(d+\gamma +\delta )(1+\alpha {\left({\Theta }^{*}\right)}^2)}{\beta k{\Theta }^{*}}{I}_k^{*},Q_k^{*}=\frac{\delta }{d+\epsilon }{I}_k^{*},}\hfill \\ \hfill I_k^{*}& {\displaystyle =\frac{(d+\epsilon )\beta k{\Theta }^{*}}{(d+\gamma +\delta )(d+\epsilon )+(d+\epsilon +\delta )\beta k{\Theta }^{*}+\alpha (d+\gamma +\delta )(d+\epsilon ){\left({\Theta }^{*}\right)}^2},}\hfill \end{array}$$ (2.4)

with${\Theta }^{*}=\frac{1}{〈k〉}{\sum }_{k=1}^{n}kP\left(k\right)I_k^{*}$.

Proof

It is obvious that the disease-free equilibrium $E_0$ of (1.3) always exists. To compute the endemic equilibrium $E_{*}$, suppose the right sides of (1.3) is zero. In other words, the ${\{S_k^{*},I_k^{*},Q_k^{*}\}}_{k=1}^n$ should satisfy

$$\{\begin{array}{cc}& {\displaystyle \beta k{S}_k^{*}\frac{{\Theta }^{*}}{1+\alpha {\left({\Theta }^{*}\right)}^2}-(d+\gamma +\delta )I_k^{*}=0,}\hfill \\ & \delta I_k^{*}-(d+\epsilon )Q_k^{*}=0,\hfill \\ & S_k^{*}+I_k^{*}+Q_k^{*}=1,k=1,2,\dots ,n,\hfill \end{array}$$ (2.5)

where ${\Theta }^{*}=\frac{1}{〈k〉}{\sum }_{k=1}^{n}kP\left(k\right)I_k^{*}$. We get from (2.5)

$$I_k^{*}{\displaystyle =\frac{(d+\epsilon )\beta k{\Theta }^{*}}{(d+\gamma +\delta )(d+\epsilon )+(d+\epsilon +\delta )\beta k{\Theta }^{*}+\alpha (d+\gamma +\delta )(d+\epsilon ){\left({\Theta }^{*}\right)}^2}.}$$ (2.6)

For convenience, we replace ${\Theta }^{*}$ with $\Theta$. Substituting (2.6) into ${\Theta }^{*}$, we obtain a self-consistency equality

$${\displaystyle \Theta =\frac{1}{〈k〉}\sum _{k=1}^n\frac{(d+\epsilon )\beta k^{2}P\left(k\right)\Theta }{(d+\gamma +\delta )(d+\epsilon )+(d+\epsilon +\delta )\beta k\Theta +\alpha (d+\gamma +\delta )(d+\epsilon ){\Theta }^2}=f\left(\Theta \right).}$$ (2.7)

As can be easily seen, $\Theta =0$ is a solution of (2.7), and $f\left(1\right)<1$. Next, the conditions for (2.7) to have nontrivial solutions in the interval $0<\Theta <1$ are needed to be found. That is, it satisfies the inequality

$${\displaystyle \frac{\mathit{df}\left(\Theta \right)}{d\Theta }{|}_{\Theta =0}=\frac{d}{d\Theta }{\left({\displaystyle \frac{1}{〈k〉}\sum _{k=1}^n\frac{(d+\epsilon )\beta k^{2}P\left(k\right)\Theta }{(d+\gamma +\delta )(d+\epsilon )+(d+\epsilon +\delta )\beta k\Theta +\alpha (d+\gamma +\delta )(d+\epsilon ){\Theta }^2}}\right)}_{\Theta =0}>1.}$$

As a result, the basic reproduction number $R_0$ is

$${\displaystyle R_0=\frac{\beta 〈k^2〉}{(d+\gamma +\delta )〈k〉},}$$

where $〈k^2〉={\sum }_{k=1}^n{k}^{2}P\left(k\right)$.

Substituting the nontrivial solution of (2.7) into (2.6), we derive $I_k^{*}$. From (2.5), (2.4) is established as well as $0<S_k^{*},I_k^{*},Q_k^{*}<1$ for $k=1,2,\dots ,n$. As a result, a unique epidemic equilibrium $E_{*}$ exists when $R_0>1$. □

Remark 2.1

Epidemic thresholds determine the presence of endemic equilibrium, as shown in Lemma 2.2. In the case where $\alpha =0$, system (1.3) simplies to (1.1), and $R_0>1$ can be simplified to $\beta >{\beta }_c$, where ${\beta }_c=\frac{(d+\gamma +\delta )〈k〉}{〈k^2〉}$, which agrees with that given in [25]. That is to say, the nonmonotone incidence rate (1.2) does not affect the epidemic threshold.

3. Global dynamics analysis

In this section, a qualitative analysis of (2.2) is presented. For the purpose of studying the global dynamical behavior of (2.2), the following lemma is first given to ensure that the solutions of system are nonnegative. Set $I_k\left(t\right)=x_k\left(t\right)$ and $Q_k\left(t\right)=x_{n+k}\left(t\right)$ for $k=1,2,\dots ,n$. After that, we investigate (2.2) for $x:=(x_1,x_2,\dots ,x_{2n})\in \Omega$, where

$$\Omega =\{x\in R^{2n}:x_i\ge 0\text{for}1\le i\le 2n,\text{and}{x}_j+x_{n+j}\le 1\text{for}\text{all}1\le j\le n\}.$$ (3.1)

Firstly, the following lemma is provided.

Lemma 3.1

The set$\Omega$is positively invariant with respect to system(2.2).

Proof

It is to be shown that if $x\left(0\right)\in \Omega$, there is $x\left(t\right)\in \Omega$ for all $t>0$. Define

$$\begin{array}{c}\hfill \partial {\Omega }_1=\{x\in \Omega |x_i=0\},\partial {\Omega }_2=\{x\in \Omega |x_{n+i}=0\},\\ \hfill \partial {\Omega }_3=\{x\in \Omega |x_i+x_{n+i}=1\},i=1,2,\dots ,n.\end{array}$$

Let the outer normals to the $3n$ hyperplanes be

$$\begin{array}{cc}\hfill {\eta }_i^1& =(0,\dots ,\stackrel{i}{\overbrace{-1}},\dots ,0,0,\dots ,0)\left(ithposition\right),\hfill \\ \hfill {\eta }_i^2& =(0,\dots ,0,0,\dots ,\stackrel{n+i}{\overbrace{-1}},\dots ,0)\left((n+i)thposition\right),\hfill \\ \hfill {\eta }_i^3& =(0,\dots ,\stackrel{i}{\overbrace{+1}},\dots ,0,0,\stackrel{n+i}{\overbrace{+1}},\dots ,0)\left(ithand(n+i)thposition\right).\hfill \end{array}$$

Through the results of Nagumo [34], it suffices to demonstrate that for $i=1,2,\dots ,n$,

$$\begin{array}{cc}\hfill {\displaystyle (\frac{dx}{dt}{|}_{x\in \partial {\Omega }_1}·{\eta }_k^1)}& {\displaystyle =-\beta i(1-x_{n+i})\frac{\tilde{\Theta }}{1+\alpha {\tilde{\Theta }}^2}\le 0,(\frac{dx}{dt}{|}_{x\in \partial {\Omega }_2}·{\eta }_k^2)=-\delta x_i\le 0,}\hfill \\ \hfill {\displaystyle (\frac{dx}{dt}{|}_{x\in \partial {\Omega }_3}·{\eta }_k^3)}& {\displaystyle =-(d+\gamma )x_i-(d+\epsilon )x_{n+i}\le 0,\text{where}\tilde{\Theta }=\frac{1}{〈k〉}\sum _{\genfrac{}{}{0pt}{}{k=1}{k\ne i}}^{n}kP\left(k\right)x_k\ge 0.}\hfill \end{array}$$

It is clear that any solutions starting from $\partial {\Omega }_1\cup \partial {\Omega }_2\cup \partial {\Omega }_3$ will stay inside $\Omega$. Accordingly, the set $\Omega$ is positively invariant. □

Next, we are ready to consider the local asymptotic stability of the disease-free equilibrium $E_0$.

Theorem 3.1

If$R_0<1$, the disease-free equilibrium$E_0$of system(1.3)is locally asymptotically stable, while it is unstable if$R_0>1$.

Proof

System (2.2) can be rewritten as

$${\displaystyle \frac{dz\left(t\right)}{dt}=Bz+G\left(z\right),}$$ (3.2)

which initial condition satisfies $z\left(0\right)\in \Omega$. $Bz$ is the linear part with $B$ being the Jacobian matrix of (2.2) evaluated at $E_0$ is shown below

$$B={\left(\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right)}_{2n\times 2n},$$

$$B_{11}=\left(\begin{array}{cccc}{l}_1{q}_1-(d+\gamma +\delta )& l_1{q}_2& \cdots & l_1{q}_n\\ l_2{q}_1& l_2{q}_2-(d+\gamma +\delta )& \cdots & l_2{q}_n\\ ⋮& ⋮& \ddots & ⋮\\ l_n{q}_1& l_n{q}_2& \cdots & l_n{q}_n-(d+\gamma +\delta )\end{array}\right),$$

where $l_i=\frac{\beta i}{〈k〉}$, $q_j=jP\left(j\right)$, $(1\le i,j\le n)$, $B_{12}=O_n$, $B_{21}=\delta E_n$, $B_{22}=-(d+\epsilon )E_n$. Here, $O_n$ and $E_n$ denote the $n$th-order zero and identity matrix, respectively. It follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{I}_k\left(t\right)}{dt}}& {\displaystyle =\beta k[1-I_k\left(t\right)-Q_k\left(t\right)]\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma +\delta )I_k\left(t\right)}\hfill \\ & {\displaystyle =[\beta k\Theta \left(t\right)-(d+\gamma +\delta )I_k\left(t\right)]-\beta k\Theta \left(t\right)\frac{I_k\left(t\right)+Q_k\left(t\right)+\alpha {\Theta }^2\left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}.}\hfill \end{array}$$

As a result, the nonlinear part $G\left(z\right)=-{(g_1,g_2,\dots ,g_n,{\overbrace{0,0,\dots ,0}}^n)}^T$, $g_k=\beta k\Theta \frac{I_k+Q_k+\alpha {\Theta }^2}{1+\alpha {\Theta }^2}$, for $k=1,2,\dots ,n$.

By mathematical induction, the characteristic equation of the matrix $B$ can be calculated as

$$\begin{array}{cc}\hfill \text{det}(\lambda E_{2n}-B)& ={(\lambda +d+\epsilon )}^n·\text{det}(\lambda E_n-B_{11})\hfill \\ & ={(\lambda +d+\epsilon )}^n{(\lambda +d+\gamma +\delta )}^{n-1}\{\lambda +(d+\gamma +\delta )-\sum _{k=1}^n{l}_k{q}_k\}=0.\hfill \end{array}$$ (3.3)

Eq. (3.3) has a negative eigenvalue $-(d+\epsilon )$ with multiplicity $n$ and a negative eigenvalue $-(d+\gamma +\delta )$ with multiplicity $n-1$. The stability of $E_0$ is only determined by

$${\displaystyle \lambda +(d+\gamma +\delta )-\sum _{k=1}^n{l}_k{q}_k=\lambda +(d+\gamma +\delta )-\frac{\beta 〈k^2〉}{〈k〉}=0,i.e.,\lambda =(d+\gamma +\delta )(R_0-1).}$$

If $R_0<1$, one has $\lambda <0$; and if $R_0>1$, one has $\lambda >0$. Hence, $E_0$ is locally asymptotically stable if $R_0<1$, whereas it is unstable if $R_0>1$. □

We now proceed to examine the global stability of $E_0$.

Theorem 3.2

If$R_0<1$, the disease-free equilibrium$E_0$of(1.3)is globally asymptotically stable.

Proof

Rewrite (1.3) as

$$\{\begin{array}{c}{\displaystyle \frac{d{S}_k\left(t\right)}{dt}=A-\beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-d{S}_k\left(t\right)+\gamma I_k\left(t\right)+\epsilon (1-S_k\left(t\right)-I_k\left(t\right)),}\hfill \\ {\displaystyle \frac{d{I}_k\left(t\right)}{dt}=\beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}-(d+\gamma +\delta )I_k\left(t\right),k=1,2,\dots ,n.}\hfill \end{array}$$ (3.4)

Considering a non-negative solution ${\{S_k\left(t\right),I_k\left(t\right)\}}_{k=1}^n$ of (3.4). Firstly, we prove that $\underset{t\to +\infty }{lim}{I}_k\left(t\right)=0$. From the first equation of (3.4), we derive

$${\displaystyle \frac{d{S}_k\left(t\right)}{dt}\le (A+\epsilon )-(d+\epsilon )S_k\left(t\right).}$$

It can be deduced that

$${\displaystyle \underset{t\to +\infty }{lim\; sup}{S}_k\left(t\right)\le \frac{A+\epsilon }{d+\epsilon }=1=:S_k^0.}$$ (3.5)

Accordingly, for arbitrarily sufficiently small ${\sigma }_1>0$, there is $T_1>0$ resulting in $S_k\left(t\right)\le S_k^0+{\sigma }_1$ for $t>T_1$. If $t>T_1$, we get

$${\displaystyle \frac{d{I}_k\left(t\right)}{dt}<\beta k(S_k^0+{\sigma }_1)\Theta \left(t\right)-(d+\gamma +\delta )I_k\left(t\right).}$$

Take into account the following auxiliary system

$${\displaystyle \frac{d{I}_k\left(t\right)}{dt}=\beta k(S_k^0+{\sigma }_1)\Theta \left(t\right)-(d+\gamma +\delta )I_k\left(t\right).}$$ (3.6)

Then it suffices to demonstrate that the positive solutions of system converge to zero as $t$ tends to infinity. We construct the following Lyapunov function

$$V\left(t\right)=\sum _{k=1}^n{a}_k{I}_k\left(t\right),$$

where $a_k=\frac{kP\left(k\right)}{(d+\gamma +\delta )〈k〉}>0$. Calculating the derivative of $V\left(t\right)$ along solutions of (3.6), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{dV\left(t\right)}{dt}}& =\sum _{k=1}^n{a}_k[\beta k(S_k^0+{\sigma }_1)\Theta \left(t\right)-(d+\gamma +\delta )I_k\left(t\right)]\hfill \\ & =\sum _{k=1}^n\left[{\displaystyle \frac{\beta k^{2}P\left(k\right)}{(d+\gamma +\delta )〈k〉}(S_k^0+{\sigma }_1)\Theta \left(t\right)-\frac{kP\left(k\right)}{〈k〉}{I}_k\left(t\right)}\right]\hfill \\ & =\Theta \left(t\right)\left({\displaystyle R_0+\frac{\beta 〈k^2〉}{(d+\gamma +\delta )〈k〉}{\sigma }_1-1}\right).\hfill \end{array}$$

Since $R_0<1$, we can select an ${\sigma }_1>0$ sufficiently small such that $R_0+\frac{\beta 〈k^2〉}{(d+\gamma +\delta )〈k〉}{\sigma }_1<1$. This guarantees that $R_0<1,\frac{dV\left(t\right)}{dt}\le 0$ for all $I_k\left(t\right)\ge 0$, and that $\frac{dV\left(t\right)}{dt}=0$ if and only if $I_k\left(t\right)=0$ for $k=1,2,\dots ,n$.

Next, we will prove $\underset{t\to \infty }{lim}{S}_k\left(t\right)=S_k^0$. Since $\underset{t\to \infty }{lim}{I}_k\left(t\right)=0$, for arbitrary ${\sigma }_2>0$ small enough, there exists $T_2>0$ leading to $0\le I_k\left(t\right)\le {\sigma }_2$ for $t>T_2$. From the first equation of (3.4), it follows that

$${\displaystyle \frac{d{S}_k\left(t\right)}{dt}\ge (A+\epsilon )-(\epsilon -\gamma ){\sigma }_2-(d+\epsilon +W{\sigma }_2)S_k\left(t\right),}$$

where $W=\beta k$. We have $\underset{t\to +\infty }{lim\; inf}{S}_k\left(t\right)\ge \frac{A+\epsilon -(\epsilon -\gamma ){\sigma }_2}{d+\epsilon +W{\sigma }_2}$. Taking ${\sigma }_2\to 0$, one obtains

$${\displaystyle \underset{t\to +\infty }{lim\; inf}{S}_k\left(t\right)\ge \frac{A+\epsilon }{d+\epsilon }=1=:S_k^0.}$$ (3.7)

Combining (3.5) and (3.7), it is obvious that $\underset{t\to +\infty }{lim}{S}_k\left(t\right)=S_k^0=1$.

Finally, since $Q_k\left(t\right)=1-S_k\left(t\right)-I_k\left(t\right)$, we obtain $\underset{t\to +\infty }{lim}{Q}_k\left(t\right)=0$. This shows that $E_0$ of (1.3) is globally attractive when $R_0<1$ and it can be deduced from Theorem 3.1 that $E_0$ is globally asymptotically stable. □

The following result shows that (2.2) is uniformly persistent when $R_0>1$, that is, the disease persists in the population. We will apply the following two lemmas.

Lemma 3.2 [35] —

Let$B$be an irreducible$n\times n$matrix. If$b_{ij}\ge 0$whenever$i\ne j$, then there is an eigenvector$u$of$B$such that$u>0$, and the corresponding eigenvalue is$s\left(B\right)=\max \text{Re}{\lambda }_i,i=1,2,\dots ,n$, where${\lambda }_i$are the eigenvalues of$B$, and$\text{Re}$represents the real part of the eigenvalues.

Lemma 3.3 [16] —

Consider the following system

$${\displaystyle \frac{dx\left(t\right)}{dt}=\tilde{B}x+\tilde{G}\left(x\right),x={(x_1,x_2,\dots ,x_{2n})}^T\in F\subset R^{2n},}$$ (3.8)

where$\tilde{B}$is an$2n\times 2n$matrix and$\tilde{G}\left(x\right)$is continuously differentiable in$F$. Suppose that

• (i)a compact convex set$D\subset F$is positively invariant for system(3.8), with$\mathbf{0}\in D$;
• (ii)there is a positive integer$m\le 2n$resulting in${\text{lim}}_{x\to 0}\parallel \tilde{G}\left(x\right)\parallel /\sqrt{{\sum }_{i=1}^m{x}_i^2}=0$;
• (iii)there are a positive number$\tilde{r}$and a real eigenvector$v$corresponding to a positive eigenvalue of${\tilde{B}}^T$resulting in$(x,v)\ge \tilde{r}\sqrt{{\sum }_{i=1}^m{x}_i^2}$for all$x\in D$;
• (iv)$(\tilde{G}\left(x\right),v)\le 0$is true for all$x\in D$.

Therefore, the solution$\varphi (t,x_0)$of(3.8)admits${\text{liminf}}_{t\to \infty }\parallel \varphi (t,x_0)\parallel \ge {ϵ}_0$for any$x_0\in D-\{\mathbf{0}\}$, where${ϵ}_0>0$has no relation to the initial value$x_0$. Further, there is a constant solution of(3.8),$x=x^{*}$with$x^{*}\in D-\{\mathbf{0}\}$.

In the following, system (3.2) will be verified to satisfy all assumptions of Lemma 3.3. By Lemma 3.1, condition (i) is established with respect to (3.2) by choosing $D=\Omega$. Obviously, ${\text{lim}}_{z\to 0}\parallel G\left(z\right)\parallel /\sqrt{{\sum }_{i=1}^n{z}_i^2}=0$

$\sqrt{{\sum }_{i=1}^n{z}_i^2}=0$, so condition (ii) holds.

With respect to condition (iii), it is noticed that $B_{11}^T={\left(b_{ji}\right)}_{n\times n}$ is irreducible and $b_{ji}>0$ when $j\ne i$, as a result, by Lemma 3.2 and Eq. (3.3), there exists an eigenvector $\tilde{v}={(v_1,v_2,\dots ,v_n)}^T$ of $B_{11}^T$ such that $v_i>0$ for all $i=1,2,\dots ,n$, and the corresponding eigenvalue is $s\left(B_{11}^T\right)=s\left(B\right)=(d+\gamma +\delta )(R_0-1)$. If $R_0>1$, then $s\left(B_{11}^T\right)=:{\lambda }_0>0$. Set $v_{n+1}=\dots =v_{2n}=0$ and $v={(v_1,v_2,\dots ,v_{2n})}^T$, we get $B^{T}v={\lambda }_{0}v$, which means that $v$ is the eigenvector corresponding to a positive eigenvalue ${\lambda }_0$ of $B^T$. If one defines $\tilde{r}={\text{min}}_{1\le i\le n}{v}_i>0$, it follows that $(z,v)\ge \tilde{r}\sqrt{{\sum }_{i=1}^n{z}_i^2}$ for all $z\in \Omega$, i.e., condition (iii) follows.

The condition (iv) is also confirmed since $(G\left(z\right),v)=-\Theta {\sum }_{k=1}^n\beta k{v}_k\frac{z_k+z_{n+k}+\alpha {\Theta }^2}{1+\alpha {\Theta }^2}\le 0$ for all $z\in D$. As a result, all the assumptions of Lemma 3.3 are valid.

These results are summarized in the following theorem.

Theorem 3.3

If$R_0>1$, system(2.2)is uniformly persistent, that is, there exists a constant$0<{ϵ}_0\ll 1$such that

$$\underset{t\to +\infty }{lim\; inf}{I}_k\left(t\right)\ge {ϵ}_0,\underset{t\to +\infty }{lim\; inf}{Q}_k\left(t\right)\ge {ϵ}_0,k=1,2,\dots ,n,$$

for any solution of(2.2)with(2.1).

Remark 3.1

Theorem 3.3 shows that when $R_0>1$, the disease is permanent. In addition, according to Lemma 3.3, there is a constant solution of (2.2) when $R_0>1$, which is the endemic equilibrium of (2.2). This is consistent with the conclusion of Lemma 2.2.

Next, the global attractivity of $E_{*}$ is discussed.

Theorem 3.4

Assume that$(I_k\left(t\right),Q_k\left(t\right))$is a solution of system(2.2)satisfying(2.1). If$R_0>1$and$\delta >d+\epsilon$, then$\underset{t\to +\infty }{lim}(I_k\left(t\right),Q_k\left(t\right))=(I_k^{*},Q_k^{*})$, where$(I_k^{*},Q_k^{*})$is the unique positive equilibrium of(2.2)satisfying(2.4)for$k=1,2,\dots ,n$.

Proof

Without loss of generality, fix $k$ to be any integer of the set $\{1,2,\dots ,n\}$. In accordance with Theorem 3.3, there is a small enough constant ${\phi }_0$ $(0<{\phi }_0\ll 1)$ and a large enough constant $T_0$ resulting in $I_k\left(t\right)\ge {\phi }_0$ for $t>T_0$. Accordingly,

$${\displaystyle \Theta \left(t\right)=\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)I_k\left(t\right)\ge {\phi }_0=:{\psi }_0>0.}$$

From the first equation of (2.2), one obtains

$${\displaystyle \frac{d{I}_k\left(t\right)}{dt}\le \beta k[1-I_k\left(t\right)]-(d+\gamma +\delta )I_k\left(t\right)=\beta k-(\beta k+d+\gamma +\delta )I_k\left(t\right).}$$

According to the comparison principle, for any given small constant $0<{\psi }_1^{\left(1\right)}<\frac{d+\gamma +\delta }{2(\beta k+d+\gamma +\delta )}$, there is a $t_1>T_0$ resulting in $I_k\left(t\right)\le X_k^{\left(1\right)}-{\psi }_1^{\left(1\right)}$ for $t>t_1$, where

$${\displaystyle X_k^{\left(1\right)}:=\frac{\beta k}{\beta k+d+\gamma +\delta }+2{\psi }_1^{\left(1\right)}<1.}$$ (3.9)

From the second equation of (2.2), it follows that

$${\displaystyle \frac{d{Q}_k\left(t\right)}{dt}=\delta [1-S_k\left(t\right)-Q_k\left(t\right)]-(d+\epsilon )Q_k\left(t\right)\le \delta -(\delta +d+\epsilon )Q_k\left(t\right).}$$

Analogously, for any given small constant $0<{\psi }_1^{\left(2\right)}<\text{min}\{\frac{1}{2},{\psi }_1^{\left(1\right)},\frac{d+\epsilon }{2(\delta +d+\epsilon )}\}$, there is a $t_2>t_1$ resulting in $Q_k\left(t\right)\le Y_k^{\left(1\right)}-{\psi }_1^{\left(2\right)}$ for $t>t_2$, where

$${\displaystyle Y_k^{\left(1\right)}:=\frac{\delta }{\delta +d+\epsilon }+2{\psi }_1^{\left(2\right)}<1.}$$ (3.10)

In view of the fact that $0<{\psi }_0\le \Theta \left(t\right)<1$. Substituting $Q_k\left(t\right)\le Y_k^{\left(1\right)}$ into the first equation of (2.2) yields

$${\displaystyle \frac{d{I}_k\left(t\right)}{dt}\ge \beta k(1-Y_k^{\left(1\right)})\frac{{\psi }_0}{1+\alpha }-\left({\displaystyle \frac{\beta k{\psi }_0}{1+\alpha }+d+\gamma +\delta }\right)I_k\left(t\right),t>t_2.}$$

As a result, for any given small constant $0<{\psi }_1^{\left(3\right)}<\text{min}\{\frac{1}{3},{\psi }_1^{\left(2\right)},\frac{\beta k(1-Y_k^{\left(1\right)}){\psi }_0}{2[\beta k{\psi }_0+(d+\gamma +\delta )(1+\alpha )]}\}$, there exists a $t_3>t_2$ such that $I_k\left(t\right)\ge x_k^{\left(1\right)}+{\psi }_1^{\left(3\right)}$ for $t>t_2$, where

$${\displaystyle x_k^{\left(1\right)}:=\frac{\beta k(1-Y_k^{\left(1\right)}){\psi }_0}{\beta k{\psi }_0+(d+\gamma +\delta )(1+\alpha )}-2{\psi }_1^{\left(3\right)}>0.}$$ (3.11)

It follows from the second equation of (2.2) that $\frac{d{Q}_k\left(t\right)}{dt}\ge \delta x_k^{\left(1\right)}-(d+\epsilon )Q_k\left(t\right)$, $t>t_3$. In a similar way, for any given small constant $0<{\psi }_1^{\left(4\right)}<\text{min}\{\frac{1}{4},{\psi }_1^{\left(3\right)},\frac{\delta x_k^{\left(1\right)}}{2(d+\epsilon )}\}$, there is a $t_4>t_3$ resulting in $Q_k\left(t\right)\ge y_k^{\left(1\right)}+{\psi }_1^{\left(4\right)}$ for $t>t_4$, where

$${\displaystyle y_k^{\left(1\right)}:=\frac{\delta x_k^{\left(1\right)}}{d+\epsilon }-2{\psi }_1^{\left(4\right)}>0.}$$ (3.12)

Since ${\phi }_0$ is sufficiently small constant, it holds that $0<x_k^{\left(1\right)}\ll 1$ and $0<y_k^{\left(1\right)}\ll 1$. It can be seen from the above discussion that $0<x_k^{\left(1\right)}<X_k^{\left(1\right)}<1$ and $0<y_k^{\left(1\right)}<Y_k^{\left(1\right)}<1$ for $t>t_4$. Consequently, it can be seen that

$$0<u_1<\Theta \left(t\right)<U_1<1,$$ (3.13)

where ${\displaystyle u_1=\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)x_k^{\left(1\right)}}$ and ${\displaystyle U_1=\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)X_k^{\left(1\right)}}$. Once again, by (2.2), we obtain

$${\displaystyle \frac{d{I}_k\left(t\right)}{dt}\le \beta k[1-I_k\left(t\right)-y_k^{\left(1\right)}]\frac{U_1}{1+\alpha u_1^2}-(d+\gamma +\delta )I_k\left(t\right),t>t_4.}$$

Therefore, for any given constant $0<{\psi }_2^{\left(1\right)}<\text{min}\{\frac{1}{5},{\psi }_1^{\left(4\right)},\frac{(d+\gamma +\delta )(1+\alpha u_1^2)+\beta k{y}_k^{\left(1\right)}{U}_1}{\beta k{U}_1+(d+\gamma +\delta )(1+\alpha u_1^2)}\}$, there exists a $t_5>t_4$ such that

$${\displaystyle I_k\left(t\right)\le X_k^{\left(2\right)}:=\frac{\beta k(1-y_k^{\left(1\right)})U_1}{\beta k{U}_1+(d+\gamma +\delta )(1+\alpha u_1^2)}+2{\psi }_2^{\left(1\right)}<1.}$$ (3.14)

By the second equation of (2.2), one gets $\frac{d{Q}_k\left(t\right)}{dt}\le \delta X_k^{\left(2\right)}-(d+\epsilon )Q_k\left(t\right)$, $t>t_5$. Consequently, for any given constant $0<{\psi }_2^{\left(2\right)}<\text{min}\{\frac{1}{6},{\psi }_2^{\left(1\right)}\}$, there is a $t_6>t_5$ such that

$$Q_k\left(t\right)\le Y_k^{\left(2\right)}:=\text{min}\left\{{\displaystyle Y_k^{\left(1\right)}-{\psi }_1^{\left(2\right)},\frac{\delta X_k^{\left(2\right)}}{d+\epsilon }+{\psi }_2^{\left(2\right)}}\right\},t>t_6.$$ (3.15)

Combining (3.9), (3.14) and (3.15), we know that $X_k^{\left(2\right)}<X_k^{\left(1\right)}$ and $Y_k^{\left(2\right)}<Y_k^{\left(1\right)}$.

Again from system (2.2), it is clear that

$${\displaystyle \frac{d{I}_k\left(t\right)}{dt}\ge \beta k[1-I_k\left(t\right)-Y_k^{\left(2\right)}]\frac{u_1}{1+\alpha U_1^2}-(d+\gamma +\delta )I_k\left(t\right),t>t_6.}$$

As a result, for any given constant $0<{\psi }_2^{\left(3\right)}<\text{min}\{\frac{1}{7},{\psi }_2^{\left(2\right)},\frac{\beta k(1-Y_k^{\left(2\right)})u_1}{\beta k{u}_1+(d+\gamma +\delta )(1+\alpha U_1^2)}\}$, there exists a $t_7>t_6$ such that $I_k\left(t\right)\ge x_k^{\left(2\right)}$, where

$$x_k^{\left(2\right)}:=\text{max}\left\{{\displaystyle x_k^{\left(1\right)}+{\psi }_1^{\left(2\right)},\frac{\beta k(1-Y_k^{\left(2\right)})u_1}{\beta k{u}_1+(d+\gamma +\delta )(1+\alpha U_1^2)}-{\psi }_2^{\left(3\right)}}\right\},t>t_7.$$ (3.16)

By (2.2), we arrive at $\frac{d{Q}_k\left(t\right)}{dt}\ge \delta x_k^{\left(2\right)}-(d+\epsilon )Q_k\left(t\right)$, $t>t_7$. Therefore, for any given constant $0<{\psi }_2^{\left(4\right)}<\text{min}\{\frac{1}{8},{\psi }_2^{\left(3\right)},\frac{\delta x_k^{\left(2\right)}}{d+\epsilon }\}$, there is a $t_8>t_7$ such that

$${\displaystyle Q_k\left(t\right)\ge y_k^{\left(2\right)}:=\frac{\delta x_k^{\left(2\right)}}{d+\epsilon }-{\psi }_2^{\left(4\right)},t>t_8.}$$ (3.17)

By repeating the above process, four sequences: $X_k^{\left(i\right)},Y_k^{\left(i\right)},x_k^{\left(i\right)},y_k^{\left(i\right)},i=1,2,\dots$ are derived. Through generalization, as we can see the first two sequences are monotonically decreasing and the last two are monotonically increasing. Therefore, there is a large enough positive integer $N$ such that for $n\ge N$,

$$\begin{array}{cc}{\displaystyle X_k^{\left(n\right)}=\frac{\beta k(1-y_k^{(n-1)})U_{n-1}}{\beta k{U}_{n-1}+(d+\gamma +\delta )(1+\alpha u_{n-1}^2)}+{\psi }_n^{\left(1\right)},}& {\displaystyle Y_k^{\left(n\right)}=\frac{\delta X_k^{\left(n\right)}}{d+\epsilon }+{\psi }_n^{\left(2\right)},}\\ {\displaystyle x_k^{\left(n\right)}=\frac{\beta k(1-Y_k^{\left(n\right)})u_{n-1}}{\beta k{u}_{n-1}+(d+\gamma +\delta )(1+\alpha U_{n-1}^2)}-{\psi }_n^{\left(3\right)},}& {\displaystyle y_k^{\left(n\right)}=\frac{\delta x_k^{\left(n\right)}}{d+\epsilon }-{\psi }_n^{\left(4\right)}.}\end{array}$$ (3.18)

It is evident that

$$0\le x_k^{\left(n\right)}\le I_k\left(t\right)\le X_k^{\left(n\right)}<1,0\le y_k^{\left(n\right)}\le Q_k\left(t\right)\le Y_k^{\left(n\right)}<1,t>T_n^{\left(4\right)}.$$ (3.19)

As the sequential limits of (3.18) exist, let $\underset{t\to \infty }{lim}{M}_k^{\left(n\right)}=M_k$, where $M_k^{\left(n\right)}=(X_k^{\left(n\right)},Y_k^{\left(n\right)},x_k^{\left(n\right)},y_k^{\left(n\right)},U_n,u_n)$ $U_n,u_n)$ and $M_k=(X_k,Y_k,x_k,y_k,U,u)$. Note that ${\displaystyle 0<{\psi }_n^{\left(i\right)}<\frac{1}{4n+i-4}(i=1,2,3,4,n>1)}$, then ${\psi }_n^{\left(i\right)}\to 0$ as $n\to \infty$. Consequently, assuming $n\to \infty$, It can be deduced from (3.18) that

$$\begin{array}{cc}{\displaystyle X_k=\frac{\beta k(1-y_k)U}{\beta \mathit{kU}+(d+\gamma +\delta )(1+\alpha u^2)},}& {\displaystyle Y_k=\frac{\delta X_k}{d+\epsilon },}\\ {\displaystyle x_k=\frac{\beta k(1-Y_k)u}{\beta \mathit{ku}+(d+\gamma +\delta )(1+\alpha U^2)},}& {\displaystyle y_k=\frac{\delta x_k}{d+\epsilon },}\end{array}$$ (3.20)

where ${\displaystyle u=\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)x_k,U=\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)X_k,0<u\le U<1}$. In addition, by (3.20) one gets

$$\begin{array}{cc}{H}_k{X}_k=\beta \mathit{kU}\left[{\displaystyle \frac{\beta \mathit{ku}}{(1+\alpha U^2)(1+\alpha u^2)}+\frac{d+\gamma +\delta }{1+\alpha u^2}-\frac{\beta \mathit{ku}\delta }{(1+\alpha U^2)(1+\alpha u^2)(d+\epsilon )}}\right],& \\ H_k{x}_k=\beta \mathit{ku}\left[{\displaystyle \frac{\beta \mathit{kU}}{(1+\alpha U^2)(1+\alpha u^2)}+\frac{d+\gamma +\delta }{1+\alpha U^2}-\frac{\beta \mathit{kU}\delta }{(1+\alpha U^2)(1+\alpha u^2)(d+\epsilon )}}\right],\end{array}$$ (3.21)

where

$${\displaystyle H_k=\frac{[\beta kU+(d+\gamma +\delta )(1+\alpha u^2)][\beta ku+(d+\gamma +\delta )(1+\alpha U^2)]{(d+\epsilon )}^2-{\beta }^2{k}^2{\delta }^{2}Uu}{(1+\alpha U^2)(1+\alpha u^2){(d+\epsilon )}^2}.}$$

For this, we declare $H_k\ne 0$. Noticing that $X_k$ is the unique nonzero value decided by (3.20). If $H_k=0$, then ${\displaystyle \frac{\beta ku}{1+\alpha U^2}+(d+\gamma +\delta )=\frac{\beta ku\delta }{(1+\alpha U^2)(d+\epsilon )}}$. According to the symmetry, we get $\frac{\beta kU}{1+\alpha u^2}+(d+\gamma +\delta )=\frac{\beta kU\delta }{(1+\alpha u^2)(d+\epsilon )}$. It is apparent that $\frac{\beta ku}{1+\alpha U^2}-\frac{\beta kU}{1+\alpha u^2}=\frac{\beta ku\delta }{(1+\alpha U^2)(d+\epsilon )}-\frac{\beta kU\delta }{(1+\alpha u^2)(d+\epsilon )}$, i.e., $d+\epsilon =\delta$. This contradicts the assumptions in the theorem. Thus, $H_k\ne 0$.

Combining (3.21) with the formulas of $U$ and $u$, we obtain

$$\begin{array}{cc}{\displaystyle 1=\frac{1}{〈k〉}\sum _{k=1}^n\mathit{kP}\left(k\right)\frac{\beta k}{H_k}\left[{\displaystyle \frac{\beta \mathit{ku}}{(1+\alpha U^2)(1+\alpha u^2)}+\frac{d+\gamma +\delta }{1+\alpha u^2}-\frac{\beta \mathit{ku}\delta }{(1+\alpha U^2)(1+\alpha u^2)(d+\epsilon )}}\right],}& \\ {\displaystyle 1=\frac{1}{〈k〉}\sum _{k=1}^n\mathit{kP}\left(k\right)\frac{\beta k}{H_k}\left[{\displaystyle \frac{\beta \mathit{kU}}{(1+\alpha U^2)(1+\alpha u^2)}+\frac{d+\gamma +\delta }{1+\alpha U^2}-\frac{\beta \mathit{kU}\delta }{(1+\alpha U^2)(1+\alpha u^2)(d+\epsilon )}}\right].}\end{array}$$ (3.22)

By subtracting the two equations above, it yields

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)\frac{\beta k}{H_k}\{\frac{\beta k}{(1+\alpha U^2)(1+\alpha u^2)}(U-u)\left({\displaystyle 1-\frac{\delta }{d+\epsilon }}\right)}\hfill \\ & & +(d+\gamma +\delta )\left({\displaystyle \frac{1}{1+\alpha U^2}-\frac{1}{1+\alpha u^2}}\right)\}\hfill \\ & & {\displaystyle =\frac{1}{〈k〉}\sum _{k=1}^{n}kP\left(k\right)\frac{\beta k}{H_k}(U-u)\{\frac{\beta k}{(1+\alpha U^2)(1+\alpha u^2)}\left({\displaystyle 1-\frac{\delta }{d+\epsilon }}\right)}\hfill \\ & & -(d+\gamma +\delta )\left({\displaystyle \frac{\alpha (U+u)}{(1+\alpha U^2)(1+\alpha u^2)}}\right)\}=0.\hfill \end{array}$$ (3.23)

Since $d+\epsilon <\delta$, this means that $U=u$. Thus, we get $\frac{1}{〈k〉}{\sum }_{k=1}^{n}kP\left(k\right)(X_k-x_k)=0$, which means that $X_k=x_k$ for $k=1,2,\dots ,n$. From (3.19) and (3.20), one arrives at $\underset{t\to \infty }{lim}{I}_k\left(t\right)=X_k=x_k$ and $\underset{t\to \infty }{lim}{Q}_k\left(t\right)=Y_k=y_k$. Noticing that when $R_0>1$, Eq. (2.7) has a unique positive solution ${\Theta }^{*}$. Therefore, substituting $U=u$ and $X_k=x_k$ into (3.21), according to (2.4) and (3.20), we derive $X_k=I_k^{*}$ and $Y_k=Q_k^{*}$. Consequently, the endemic equilibrium $E_{*}$ of (2.2) is globally attractive if $R_0>1$ and $\delta >d+\epsilon$. □

Finally, we examine the globally asymptotic stability of the endemic equilibrium.

Theorem 3.5

If$R_0>1$and$\alpha \ge {\alpha }_c:=\frac{{\beta }^2{n}^2{\left(R_0\right)}^4}{512{(d+\gamma )}^2}$or$\alpha \le {\alpha }_l:=\frac{2(d+\gamma )}{n\beta }$, the endemic equilibrium$E_{*}={\{S_k^{*},I_k^{*},Q_k^{*}\}}_{k=1}^n$of system(1.3)is globally asymptotically stable.

Proof

For convenience, $S_k$, $I_k$, $Q_k$ and $\Theta$ are used to instead of $S_k\left(t\right)$, $I_k\left(t\right)$, $Q_k\left(t\right)$ and $\Theta \left(t\right)$ respectively. It follows from Lemma 2.1 that $0<\Theta \left(t\right)<1$ for all $t>0$. We consider a Lyapunov function candidate $V\left(t\right)=V_S+V_Q+V_{\Theta }$, the elements of which are shown below

$${\displaystyle V_S=\frac{1}{2}\sum _{k=1}^n{\xi }_k{(S_k\left(t\right)-S_k^{*})}^2,V_Q=\frac{1}{2}\sum _{k=1}^n{\eta }_k{(Q_k\left(t\right)-Q_k^{*})}^2,\text{and}{V}_{\Theta }=\Theta \left(t\right)-{\Theta }^{*}-{\Theta }^{*}\text{ln}\frac{\Theta \left(t\right)}{{\Theta }^{*}},}$$

where

$${\displaystyle Q_k^{*}=1-S_k^{*}-I_k^{*},{\xi }_k=\frac{kP\left(k\right)}{〈k〉S_k^{*}},k=1,2,\dots ,n,}$$

and ${\eta }_k$ is positive constant, which will be determined later.

In the following, we derive the derivatives of $V_S$, $V_Q$ and $V_{\Theta }$, respectively. We can rewrite the first equation of (3.4) as

$${\displaystyle \frac{d{S}_k\left(t\right)}{dt}=(d+\epsilon )-(d+\epsilon )S_k\left(t\right)+(\gamma -\epsilon )I_k\left(t\right)-\beta k{S}_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}.}$$ (3.24)

In addition, it can be deduced from (2.3) that

$${\displaystyle \frac{d\Theta \left(t\right)}{dt}=-(d+\gamma +\delta )\Theta \left(t\right)+\frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}P\left(k\right)S_k\left(t\right)\frac{\Theta \left(t\right)}{1+\alpha {\Theta }^2\left(t\right)}.}$$ (3.25)

Differentiating $V_S$ and employing the identity $d+\epsilon =(d+\epsilon )S_k^{*}+(\epsilon -\gamma )I_k^{*}+\beta k{S}_k^{*}\frac{{\Theta }^{*}}{1+\alpha {\left({\Theta }^{*}\right)}^2}$, we get

$$\begin{array}{ccc}& & {\displaystyle V_S^{{}^{\prime }}=\sum _{k=1}^n{\xi }_k(S_k-S_k^{*})\frac{d{S}_k}{dt}}\hfill \\ & & =\sum _{k=1}^n{\xi }_k(S_k-S_k^{*})\left({\displaystyle d+\epsilon -(d+\epsilon )S_k+(\gamma -\epsilon )I_k-\beta k{S}_k\frac{\Theta }{1+\alpha {\Theta }^2}}\right)\hfill \\ & & =\sum _{k=1}^n{\xi }_k(S_k-S_k^{*})\{-(d+\epsilon )(S_k-S_k^{*})+(\gamma -\epsilon )(I_k-I_k^{*})+\beta k\left({\displaystyle \frac{S_k^{*}{\Theta }^{*}}{1+\alpha {\left({\Theta }^{*}\right)}^2}-\frac{S_k\Theta }{1+\alpha {\Theta }^2}}\right)\}\hfill \\ & & =-\sum _{k=1}^n{\xi }_k(d+\epsilon ){(S_k-S_k^{*})}^2+\sum _{k=1}^n{\xi }_k(\gamma -\epsilon )(S_k-S_k^{*})(I_k-I_k^{*})\hfill \\ & & {\displaystyle -\frac{\Theta }{1+\alpha {\Theta }^2}\sum _{k=1}^n{\xi }_k\beta k{(S_k-S_k^{*})}^2-\sum _{k=1}^n\frac{{\xi }_k\beta k{S}_k^{*}}{1+\alpha {\left({\Theta }^{*}\right)}^2}(S_k-S_k^{*})(\Theta -{\Theta }^{*})}\hfill \\ & & {\displaystyle +\sum _{k=1}^n{\xi }_k\alpha \beta k{S}_k^{*}(S_k-S_k^{*})\frac{\Theta ({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}.}\hfill \end{array}$$ (3.26)

The last term of right hand side of (3.26) can be expressed as

$$\begin{array}{ccc}& & {\displaystyle \sum _{k=1}^n{\xi }_k\alpha \beta k{S}_k^{*}(S_k-S_k^{*})\frac{\Theta ({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}}\hfill \\ & & =\sum _{k=1}^n{\xi }_k\alpha \beta k{S}_k^{*}(S_k-S_k^{*})\left[{\displaystyle \frac{(\Theta -{\Theta }^{*})({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}+\frac{{\Theta }^{*}({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}}\right]\hfill \\ & & {\displaystyle =\sum _{k=1}^n{\xi }_k\alpha \beta k{S}_k^{*}{S}_k\frac{(\Theta -{\Theta }^{*})({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}-\sum _{k=1}^n\frac{{\xi }_k\alpha \beta k{\left(S_k^{*}\right)}^2(\Theta +{\Theta }^{*})}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}{(\Theta -{\Theta }^{*})}^2}\hfill \\ & & {\displaystyle +\sum _{k=1}^n\frac{{\xi }_k\alpha \beta k{S}_k^{*}{\Theta }^{*}(\Theta +{\Theta }^{*})}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}(S_k-S_k^{*})(\Theta -{\Theta }^{*}).}\hfill \end{array}$$

Substituting the above equality into (3.26), one gets

$$\begin{array}{cc}\hfill V_S^{{}^{\prime }}=& -\sum _{k=1}^n{\xi }_k(d+\epsilon ){(S_k-S_k^{*})}^2+\sum _{k=1}^n{\xi }_k(\gamma -\epsilon )(S_k-S_k^{*})(I_k-I_k^{*})\hfill \\ & {\displaystyle -\frac{\Theta }{1+\alpha {\Theta }^2}\sum _{k=1}^n{\xi }_k\beta k{(S_k-S_k^{*})}^2}\hfill \\ & {\displaystyle +\sum _{k=1}^n\frac{{\xi }_k\alpha \beta k{S}_k^{*}{\Theta }^{*}(\Theta +{\Theta }^{*})}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}(S_k-S_k^{*})(\Theta -{\Theta }^{*})}\hfill \\ & {\displaystyle -\sum _{k=1}^n\frac{{\xi }_k\alpha \beta k{\left(S_k^{*}\right)}^2(\Theta +{\Theta }^{*})}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}{(\Theta -{\Theta }^{*})}^2}\hfill \\ & -\sum _{k=1}^n{\xi }_k\beta k{S}_k^{*}\left[{\displaystyle \frac{(S_k-S_k^{*})(\Theta -{\Theta }^{*})}{1+\alpha {\left({\Theta }^{*}\right)}^2}-\frac{\alpha S_k(\Theta -{\Theta }^{*})({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)}}\right].\hfill \end{array}$$ (3.27)

Substituting the last equation of (1.3) and the identity $\delta I_k^{*}-(d+\epsilon )Q_k^{*}=0$ into the differential process of $V_Q$, we derive $V_Q^{{}^{\prime }}$ as

$$\begin{array}{cc}\hfill V_Q^{{}^{\prime }}& {\displaystyle =\sum _{k=1}^n{\eta }_k(Q_k-Q_k^{*})\frac{d{Q}_k}{dt}}\hfill \\ & =\sum _{k=1}^n{\eta }_k(Q_k-Q_k^{*})[\delta I_k-(d+\epsilon )Q_k]\hfill \\ & =\sum _{k=1}^n{\eta }_k(Q_k-Q_k^{*})[\delta (I_k-I_k^{*})-(d+\epsilon )(Q_k-Q_k^{*})]\hfill \\ & =\sum _{k=1}^n{\eta }_k(Q_k-Q_k^{*})[-\delta (S_k-S_k^{*})-\delta (Q_k-Q_k^{*})-(d+\epsilon )(Q_k-Q_k^{*})]\hfill \\ & =-\sum _{k=1}^n{\eta }_k(d+\epsilon +\delta ){(Q_k-Q_k^{*})}^2+\sum _{k=1}^n{\eta }_k\delta (S_k-S_k^{*})[(S_k-S_k^{*})+(I_k-I_k^{*})]\hfill \\ & =-\sum _{k=1}^n{\eta }_k(d+\epsilon +\delta ){(Q_k-Q_k^{*})}^2+\sum _{k=1}^n{\eta }_k\delta {(S_k-S_k^{*})}^2+\sum _{k=1}^n{\eta }_k\delta (S_k-S_k^{*})(I_k-I_k^{*}).\hfill \end{array}$$ (3.28)

Similarly, combining (3.25) and the identity ${\displaystyle 1=\frac{\beta }{(d+\gamma +\delta )〈k〉}\sum _{k=1}^n\frac{k^{2}P\left(k\right)S_k^{*}}{1+\alpha {\left({\Theta }^{*}\right)}^2}}$, $V_{\Theta }^{{}^{\prime }}$ can be constructed as follows

$$\begin{array}{cc}\hfill V_{\Theta }^{{}^{\prime }}& {\displaystyle =\frac{\Theta -{\Theta }^{*}}{\Theta }\frac{d\Theta }{dt}}\hfill \\ & =(d+\gamma +\delta )(\Theta -{\Theta }^{*})\left({\displaystyle -1+\frac{\beta }{(d+\gamma +\delta )〈k〉}\sum _{k=1}^n\frac{k^{2}P\left(k\right)S_k}{1+\alpha {\Theta }^2}}\right)\hfill \\ & {\displaystyle =(\Theta -{\Theta }^{*})\frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}P\left(k\right)\left({\displaystyle \frac{S_k}{1+\alpha {\Theta }^2}-\frac{S_k^{*}}{1+\alpha {\left({\Theta }^{*}\right)}^2}}\right)}\hfill \\ & {\displaystyle =\frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}P\left(k\right)\left[{\displaystyle \frac{(S_k-S_k^{*})(\Theta -{\Theta }^{*})}{1+\alpha {\left({\Theta }^{*}\right)}^2}-\frac{\alpha S_k(\Theta -{\Theta }^{*})({\Theta }^2-{\left({\Theta }^{*}\right)}^2)}{(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}}\right].}\hfill \end{array}$$ (3.29)

Due to the fact that ${\displaystyle {\xi }_k=\frac{kP\left(k\right)}{〈k〉S_k^{*}}}$, we observe that the last term of (3.27) is equal to $-V_{\Theta }^{{}^{\prime }}$. Then combining (3.28), and selecting ${\displaystyle {\eta }_k=\frac{\epsilon -\gamma }{\delta }{\xi }_k=\frac{(\epsilon -\gamma )kP\left(k\right)}{\delta 〈k〉S_k^{*}}}$, $V^{{}^{\prime }}$ is given as

$$\begin{array}{ccc}& & V^{{}^{\prime }}=V_S^{{}^{\prime }}+V_Q^{{}^{\prime }}+V_{\Theta }^{{}^{\prime }}\hfill \\ & & =-\sum _{k=1}^n{\xi }_k(d+\gamma ){(S_k-S_k^{*})}^2-\sum _{k=1}^n{\eta }_k(d+\epsilon +\delta ){(Q_k-Q_k^{*})}^2\hfill \\ & & {\displaystyle -\frac{\Theta }{1+\alpha {\Theta }^2}\sum _{k=1}^n{\xi }_k\beta k{(S_k-S_k^{*})}^2}\hfill \\ & & {\displaystyle +\sum _{k=1}^n\frac{{\xi }_k\alpha \beta k{S}_k^{*}{\Theta }^{*}(\Theta +{\Theta }^{*})}{(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}(S_k-S_k^{*})(\Theta -{\Theta }^{*})}\hfill \\ & & {\displaystyle -\sum _{k=1}^n\frac{{\xi }_k\alpha \beta k{\left(S_k^{*}\right)}^2(\Theta +{\Theta }^{*})}{(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}{(\Theta -{\Theta }^{*})}^2}\hfill \\ & & {\displaystyle =-\frac{\Theta }{1+\alpha {\Theta }^2}\sum _{k=1}^n{\xi }_k\beta k{(S_k-S_k^{*})}^2-\sum _{k=1}^n{\eta }_k(d+\epsilon +\delta ){(Q_k-Q_k^{*})}^2}\hfill \\ & & -\sum _{k=1}^n{\xi }_k(d+\gamma )({\tilde{X}}_k^2-b_k{\tilde{X}}_k\tilde{Y}+c_k{\tilde{Y}}^2)\hfill \\ & & {\displaystyle =-\frac{\Theta }{1+\alpha {\Theta }^2}\sum _{k=1}^n{\xi }_k\beta k{(S_k-S_k^{*})}^2-\sum _{k=1}^n{\eta }_k(d+\epsilon +\delta ){(Q_k-Q_k^{*})}^2}\hfill \\ & & {\displaystyle -\sum _{k=1}^n{\xi }_k(d+\gamma ){\left({\displaystyle {\tilde{X}}_k-\frac{b_k}{2}\tilde{Y}}\right)}^2+\sum _{k=1}^n{\xi }_k(d+\gamma )\frac{b_k^2-4c_k}{4}{\tilde{Y}}^2,}\hfill \end{array}$$ (3.30)

where

$${\displaystyle b_k=\frac{\alpha \beta k{S}_k^{*}{\Theta }^{*}(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)},c_k=\frac{\alpha \beta k{\left(S_k^{*}\right)}^2(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)},}$$

and

$${\tilde{X}}_k=S_k-S_k^{*},\tilde{Y}=\Theta -{\Theta }^{*}.$$

In order to ensure that $V^{{}^{\prime }}\le 0$, it is sufficient to show that the following inequality holds.

$$\begin{array}{cc}\hfill b_k^2-4c_k& {\displaystyle ={\left[{\displaystyle \frac{\alpha \beta k{S}_k^{*}{\Theta }^{*}(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}}\right]}^2-4\frac{\alpha \beta k{\left(S_k^{*}\right)}^2(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}}\hfill \\ & {\displaystyle =\frac{\alpha \beta k{\left(S_k^{*}\right)}^2(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}\left[{\displaystyle \frac{\alpha \beta k{\left({\Theta }^{*}\right)}^2(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}-4}\right]\le 0.}\hfill \end{array}$$ (3.31)

It is noted that $0<\Theta ,{\Theta }^{*}<1$. Then when ${\displaystyle \alpha \le \frac{2(d+\gamma )}{n\beta }}$, apparently we can obtain

$${\displaystyle \frac{\alpha \beta k{\left({\Theta }^{*}\right)}^2(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}-4\le \frac{2\alpha \beta n}{d+\gamma }-4\le 0,}$$ (3.32)

which means that (3.31) holds.

Additionally, It can be deduced from the inequalities $a^2+b^2\ge \frac{1}{2}{(a+b)}^2$ and $a^2+b^2\ge 2ab$ that

$$\begin{array}{cc}\hfill (1+\alpha {\Theta }^2)(1+\alpha {\left({\Theta }^{*}\right)}^2)& =1+\alpha [{\Theta }^2+{\left({\Theta }^{*}\right)}^2]+{\alpha }^2{\Theta }^2{\left({\Theta }^{*}\right)}^2\hfill \\ & \ge 1+\alpha [{\Theta }^2+{\left({\Theta }^{*}\right)}^2]\hfill \\ & {\displaystyle \ge 1+\frac{\alpha }{2}{(\Theta +{\Theta }^{*})}^2}\hfill \\ & \ge \sqrt{2\alpha }(\Theta +{\Theta }^{*}),\hfill \end{array}$$ (3.33)

and

$${\displaystyle I_k^{*}\le \frac{(d+\epsilon )\beta k{\Theta }^{*}}{(d+\gamma +\delta )(d+\epsilon )(1+\alpha {\left({\Theta }^{*}\right)}^2)}\le \frac{\beta k}{2\sqrt{\alpha }(d+\gamma +\delta )},k=1,2,\dots ,n.}$$ (3.34)

From (3.34), we get

$${\displaystyle {\Theta }^{*}={〈k〉}^{-1}\sum _{k=1}^{n}kP\left(k\right)I_k^{*}\le \frac{\beta }{2\sqrt{\alpha }(d+\gamma +\delta )〈k〉}\sum _{k=1}^n{k}^{2}P\left(k\right)=\frac{R_0}{2\sqrt{\alpha }}.}$$ (3.35)

Inserting (3.33) and (3.35) into (3.31) and utilizing the hypothesis $\alpha \ge \frac{{\beta }^2{n}^2{\left(R_0\right)}^4}{512{(d+\gamma )}^2}$ yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{\alpha \beta k{\left({\Theta }^{*}\right)}^2(\Theta +{\Theta }^{*})}{(d+\gamma )(1+\alpha {\left({\Theta }^{*}\right)}^2)(1+\alpha {\Theta }^2)}-4}& {\displaystyle \le \frac{\sqrt{\alpha }\beta k}{\sqrt{2}(d+\gamma )}{\left({\displaystyle \frac{R_0}{2\sqrt{\alpha }}}\right)}^2-4}\hfill \\ & {\displaystyle \le \frac{\beta n{\left(R_0\right)}^2}{4\sqrt{\alpha }(d+\gamma )}-4\le 0,}\hfill \end{array}$$

which indicates that (3.31) is confirmed. Consequently, if $\alpha \ge {\alpha }_c$ or $\alpha \le {\alpha }_l$ then (3.31) holds, and from (3.30) we can obtain

$${\displaystyle V^{{}^{\prime }}\le -\frac{\Theta }{1+\alpha {\Theta }^2}\sum _{k=1}^n{\xi }_k\beta k{(S_k-S_k^{*})}^2-\sum _{k=1}^n{\eta }_k(d+\epsilon +\delta ){(Q_k-Q_k^{*})}^2\le 0.}$$ (3.36)

Moreover, $V^{{}^{\prime }}=0$ if and only if $S_k=S_k^{*}$ and $Q_k=Q_k^{*}$ for $k=1,2,\dots ,n$. By LaSalle’s Invariant Principle [36], we can conclude that $E_{*}={\{S_k^{*},I_k^{*},Q_k^{*}\}}_{k=1}^n$ of system (1.3) is globally asymptotically stable. □

4. Numerical simulations

In this section, we perform some numerical simulations to verify our theoretical results obtained in the previous sections and to understand the effects of parameters on the spread of epidemics, so as to find better control strategies. The following simulations are based on a scale-free network, which satisfy a power-law degree distribution, namely, $P\left(k\right)=C{k}^{-3}$ for $k=1,2,\dots ,500$. The parameter $C$ is chosen to make ${\sum }_{k=1}^{n}P\left(k\right)=1$.

Example 4.1

In Figs. 2 and 3 , we show the outcome of system (1.3) when the basic reproduction number $R_0<1$. Let $d=0.01,\beta =0.01,\delta =0.02,\gamma =0.1$ and $\epsilon =0.2$. In this case, $R_0=0.318<1$. The time series of $I_{100},I_{200},I_{300},I_{400},I_{500}$ are performed in Fig. 2(a) $(\alpha =0.5)$ and Fig. 2(b) $(\alpha =50)$. The initial values are given by $S_k\left(0\right)=0.9,I_k\left(0\right)=0.1$ and $Q_k\left(0\right)=0$ for any degree $k$. From the numerical results, it can be seen that $E_0$ is globally asymptotically stable when $R_0<1$, which means that the disease will eventually become extinct. In addition, when $\alpha =0.5$ and $\alpha =50$, there is a peak in the density of infected nodes before leading to the disease-free equilibrium, however, as $\alpha$ increases, the peak level decreases, that is, a big $\alpha$ can efficiently reduce the epidemic peak during the initial outbreak of the disease. Fig. 3 depicts the trajectories of $I_{350}\left(t\right)$ versus $t$ with eight different initial conditions. The parameters in Fig. 3 are the same as those in Fig. 2. Fig. 3(a) and 3(b) show that the disease-free equilibrium is indeed globally asymptotically stable.

Example 4.2

In Figs. 4 and 5 , we show the outcome of system (1.3) when the basic reproduction number $R_0>1$. The parameters are chosen as $d=0.02,\beta =0.05,\delta =0.04,\gamma =0.05,\epsilon =0.2$. Subsequently, we can obtain $R_0=1.8793>1$ and ${\alpha }_l=0.0056$ and ${\alpha }_c=3108$. By selecting $\alpha =0.005$ and 3500 respectively, the condition $\alpha <{\alpha }_l$ or $\alpha >{\alpha }_c$ in Theorem 3.5 holds. From Fig. 4, it can be seen that the disease will approach to a positive stationary level when $R_0>1$. The trajectories of $I_{350}\left(t\right)$ for eight different initial values are plotted in Fig. 5. In Figs. 4 and 5, we see that the endemic equilibrium $E_{*}$ is globally asymptotically stable when $R_0>1$ and the condition $\alpha <{\alpha }_l$ or $\alpha >{\alpha }_c$ is satisfied, in accord with Theorem 3.5. In addition, as can be seen from Fig. 4, the density of the infected nodes decreases with increasing $\alpha$ when the disease is prevalent, indicating that a larger $\alpha$ can weaken the epidemic level of the disease.

Example 4.3

From Theorem 3.5, we know that the endemic equilibrium $E_{*}$ is globally asymptotically stable when $R_0>1$ and the condition $\alpha <{\alpha }_l$ or $\alpha >{\alpha }_c$ is satisfied, which is shown in Figs. 4 and 5. Nevertheless, in Fig. 6 , we choose $\alpha =500,1000,\dots ,3000$ and other parameters are the same as in Example 4.2, it is clear that ${\alpha }_l<\alpha <{\alpha }_c$ does not satisfy the conditions of Theorem 3.5. Fig. 6 depicts the time evolution of $I\left(t\right)$ with different $\alpha$, where $I\left(t\right)={\sum }_{k=1}^{n}P\left(k\right)I_k\left(t\right)$ is the relative average density of the infected nodes on the entire networks. It can be observed from Fig. 6 that the percentages of infected individuals eventually converge to a positive stationary level.

Fig. 2.

The time series of $I_k\left(t\right)$ with $R_0<1$.

Fig. 3.

The time series of $I_{350}\left(t\right)$ with eight different initial values and $R_0<1$.

Fig. 4.

The time series of $I_k\left(t\right)$ with $R_0>1$.

Fig. 5.

The time series of $I_{350}\left(t\right)$ with eight different initial values and $R_0>1$.

Fig. 6.

The time evolution of $I\left(t\right)={\sum }_{k=1}^{n}P\left(k\right)I_k\left(t\right)$ with $R_0>1$ and different $\alpha$.

Example 4.4. Finally, we display the effect of quarantine by numerical simulations. In Fig. 7 , the initial values are $S_k\left(0\right)=0.3,I_k\left(0\right)=0.7$ and $Q_k\left(0\right)=0$. We fix the parameters $d=0.02,\beta =0.05,\gamma =0.05,\epsilon =0.2$ and $\alpha =100$ to graph the time evolution of $I\left(t\right)$ with six different $\delta$. From the figure, it is observed that by increasing the quarantine rate $\delta$, the level of endemic disease prevalence reduces significantly, indicating that raising the quarantine rate helps in controlling the disease.

Fig. 7.

The time evolution of $I\left(t\right)$ corresponding to different $\delta$.

5. Conclusions

Quarantine strategies remain an extremely efficient method of controlling the outbreaks of epidemics when we suffer from a variety of serious infectious diseases. As a result, we have proposed and investigated a network-based SIQS infectious disease model with nonmonotone incidence rate. The psychological response of individuals during an outbreak may be characterized by the nonmonotone incidence rate, that is, in the case of a great number of infected persons, individuals consciously reduce their contact with others for fear of being infected, and thus infectivity decreases as the number of infected persons goes up. The expression for the basic reproduction number $R_0$ is derived, which is related to the network topology as well as some parameters.

Furthermore, we show that the basic reproduction number $R_0$ determines not only the existence of the endemic equilibrium $E_{*}$ but also the global dynamics of model. More specially, by utilizing the Lyapunov function, it is shown that the disease-free equilibrium $E_0$ is globally asymptotically stable when $R_0<1$, namely, the disease will disappear eventually. On the other hand, the disease is uniformly persistent on the network when $R_0>1$. At the same time, by using a novel monotone iterative scheme, we demonstrate that if $R_0>1$ and $\delta >d+\epsilon$, the unique endemic equilibrium $E_{*}$ is globally attractive. Furthermore, by constructing appropriate Lyapunov function, $E_{*}$ is globally asymptotically stable if $R_0>1$ and the inhibitory effect $\alpha$ is large or small enough.

To facilitate the understanding of theoretical results, several numerical examples are designed and simulated vividly in Section 4. Although the parameter $\alpha$ has no effect on the epidemic threshold, it can be seen that bigger $\alpha$ makes the disease die out faster and the level of prevalence lower, which is confirmed from the numerical consequences. Moreover, it can be seen from Fig. 6 that the average density of infected individuals eventually converges to a positive stationary level when $\alpha$ does not satisfy the conditions in Theorem 3.5. As a result, we conjecture that the endemic equilibrium $E_{*}$ of system (1.3) is globally asymptotically stable only when $R_0>1$. In addition, the results of Example 4.4 indicate that the level of endemic disease prevalence reduces significantly with increasing the quarantine rate $\delta$, which means that quarantine strategies are efficient methods for preventing and controlling epidemic spreading.

It should be noted that, in our model, we assume that the birth rate $A$ is constant, i.e., the number of newly born nodes is the same for all different degrees per unit time. If we suppose the birth rate $A$ is distributed into group $k$ at the probability $r_k(0\le r_k<1)$, one would expect much more complicated dynamics. Furthermore, we only consider a simple epidemic model on static networks. Nevertheless, it is known that the real contact networks are time-varying. Hence, it is also necessary to study the spreading dynamics on real contact networks. We leave these research topics for future work.

CRediT authorship contribution statement

Xinxin Cheng: Conceptualization, Methodology, Investigation, Writing – original draft, Writing – review & editing. Yi Wang: Methodology, Visualization, Writing – review & editing. Gang Huang: Methodology, Visualization, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.