Complex patterns of an SIR model with a saturation treatment on complex networks: An edge-compartmental approach

Abstract

In this paper, we adopt an edge-compartmental approach to deeply investigate an SIR model with a saturation treatment on complex networks. We find that the system may exhibit the coexistence of multi-endemic equilibria, whose stabilities are determined by signs of tangent slopes of the epidemic curve. Numerical examples illustrate the theoretical results.

Introduction

In [1], Li and Yousef proposed an SIR epidemic model with a saturation treatment function complex network as follows:

$$\left\{\begin{array}{cc}\hfill \frac{d{S}_k\left(t\right)}{dt}=& \Lambda -\beta k{S}_k\left(t\right)\Theta \left(t\right)-\mu S_k\left(t\right),\hfill \\ \hfill \frac{d{I}_k\left(t\right)}{dt}=& \beta k{S}_k\left(t\right)\Theta \left(t\right)-\frac{r{I}_k\left(t\right)}{1+\alpha \Theta \left(t\right)}-(\mu +\gamma )I_k\left(t\right),\hfill \\ \hfill \frac{d{R}_k\left(t\right)}{dt}=& \frac{r{I}_k\left(t\right)}{1+\alpha \Theta \left(t\right)}+\gamma I_k-\mu R_k\left(t\right),\hfill \end{array}\right.k=1,2,\dots ,n,$$

where $S_k,I_k$ and $R_k$ denote densities of susceptible nodes, infected nodes and recovered nodes at time $t$ and degree $k$, respectively. $\Lambda$ represents the input rate; $\beta$ stands for the transmission rate; $\gamma$ is the natural treatment; $\mu$ denotes the natural death rate; $r{I}_k/(1+\alpha \Theta )$ is a form of the saturation treatment with a delay effect $\alpha$. $n$ denotes the maximal degree of a network. Let $N_k\left(t\right)=S_k\left(t\right)+I_k\left(t\right)+R_k\left(t\right)$ represent the subtotal population with degree $k$ at time $t$ and then the total population is denoted by $N\left(t\right)={\sum }_{k=1}^n{N}_k\left(t\right)$ at time $t$. The term $\Theta \left(t\right)$ represents the probability of one infected edge connecting other uninfected edges and it is defined by

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

with average degree $〈k〉={\sum }_{k=1}^{n}kp\left(k\right)$ and the degree distribution $p\left(k\right)=N_k/N$. Since the dimension of model (2) is $3n$, the calculation and analysis processes must be tedious. Actually, epidemic models on complex networks are concerned with the variations of infected edges, which have been presented in forms of the pairwise models [2] and the edge-compartmental models [3]. In [4], Wang and Yang proposed a mixed degree-edge-compartmental model and considered the evolution of susceptible nodes and infected edges. Inspired by those ideas, we transform model (2) into

$$\left\{\begin{array}{cc}\hfill \frac{d{S}_k\left(t\right)}{dt}=& \Lambda -\beta k{S}_k\left(t\right)\Theta \left(t\right)-\mu S_k\left(t\right),\hfill \\ \hfill \frac{d\Theta \left(t\right)}{dt}=& \left(\frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}p\left(k\right)S_k\left(t\right)-\frac{r}{1+\alpha \Theta \left(t\right)}-(\mu +\gamma )\right)\Theta \left(t\right),\hfill \\ \hfill \frac{dR\left(t\right)}{dt}=& \frac{r\Theta }{1+\alpha \Theta \left(t\right)}+\gamma \Theta \left(t\right)-\mu R\left(t\right),\hfill \end{array}\right.k=1,2,\dots ,n,$$

where $R\left(t\right)=\frac{1}{〈k〉}{\sum }_{k=1}^{n}kp\left(k\right)R_k\left(t\right)$. Apparently, the $(n+2)$th equation of system (2) has no relation with other equations. Hence, we can drop it off and reconsider the following system

$$\left\{\begin{array}{cc}\hfill \frac{d{S}_k\left(t\right)}{dt}=& \Lambda -\beta k{S}_k\left(t\right)\Theta \left(t\right)-\mu S_k\left(t\right),\hfill \\ \hfill \frac{d\Theta \left(t\right)}{dt}=& \left(\frac{\beta }{〈k〉}\sum _{k=1}^n{k}^{2}p\left(k\right)S_k\left(t\right)-\frac{r}{1+\alpha \Theta \left(t\right)}-(\mu +\gamma )\right)\Theta \left(t\right),\hfill \end{array}\right.k=1,2,\dots ,n$$

with initial conditions

$$S_k\left(0\right)=S_{k0},\Theta \left(0\right)={\Theta }_0,k=1,2,\dots ,n.$$

Without loss of the generality, we assume that $\Lambda =\mu$. In this case,

$$\Omega =\left\{\varphi \in {\left({\mathbb{R}}_{+}\right)}^{n+1}|0<S_k\le 1,0\le \Theta <1\right\}$$

is a positively invariant set. Define the basic reproduction number of system (3) by

$${\mathcal{R}}_0=\frac{\beta 〈k^2〉}{(\mu +\gamma +r)〈k〉},$$

which characterizes that the average number of infected edges produced by one infected $\left[SI\right]$ edge during its infected period $\frac{1}{\mu +\gamma +r}$ at total size of susceptible nodes. $〈k^2〉={\sum }_{k=1}^n{k}^{2}p\left(k\right)$ denotes the two movement of the network with degree distribution $p\left(k\right)$ and maximal degree $n$.