# Analysis of an *SIRS* epidemic model with time delay on heterogeneous network

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## Abstract

We discuss a novel epidemic *SIRS* model with time delay on a scale-free network in this paper. We give an equation of the basic reproductive number \(R_{0}\) for the model and prove that the disease-free equilibrium is globally attractive and that the disease dies out when \(R_{0}<1\), while the disease is uniformly persistent when \(R_{0}>1\). In addition, by using a suitable Lyapunov function, we establish a set of sufficient conditions on the global attractiveness of the endemic equilibrium of the system.

### Keywords

epidemic spreading scale-free network basic reproductive number global attractiveness time delay## 1 Introduction

Following both the seminal work on small-world network phenomena by Watts and Strogatz [1] and the scale-free network, in which the probability of \(p(k)\) for any node with *k* links to other nodes is distributed according to the power law \(p(k)=Ck^{-\gamma}\) (\(2<\gamma \le3\)), suggested by Barabási and Albert [2], the spreading of an epidemic disease on heterogeneous networks, *i.e.*, scale-free networks, has been studied by many researchers [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].

Realistic epidemic models should include some past states of the system, and time delay plays an important role in the spreading process of the epidemic. For instance, it can simulate the incubation period of the infectious disease, the infectious period of patients, and the immunity period of recovery of the disease with time delay [21, 23]. However, only little attention has been given to the epidemic models with time delays on heterogeneous networks. Liu and Xu presented a functional differential equation *SEIRS* epidemic model, in which time delay represents the latent period and the immune period [19]. Liu and Deng *et al.* discussed a functional differential equation *SIS* model, in which time delay represents the average infectious period [20], obtained the basic reproduction number, and discussed the persistence of the disease. Wang and Wang *et al.* presented a functional differential equation *SIR* model, in which time delay represents the incubation period, and discussed the global stability of equilibria of the system [21]. Kang and Fu also established a functional differential equation *SIS* model with an infective vector and analyzed the global stability of the endemic equilibrium, the disease-free equilibrium [22], and so on.

Considering the fact that the immune individual may become the susceptible individual, Chen and Sun discussed an *SIRS* epidemic model without delay [15]. In [21], Wang and Wang *et al.* presented a delayed *SIR* model, in which time delay represents the incubation period during which the infectious agents develop in the vector, and discussed the global stability of equilibria of the system. Motivated by the work of Chen [15] and Wang [21], we will present a novel functional differential equation *SIRS* model in which time delay represents the incubation period of an infective vector to investigate the epidemic spreading on a heterogeneous network.

*N*during the period of epidemic spreading, we also suppose that the degree of each node is time-invariant and \(p(k)\) denotes the degree distribution of the network. Let \(S_{k}(t)\), \(I_{k}(t)\), and \(R_{k}(t)\) be the relative density of susceptible nodes, infected nodes, and recovered nodes of connectivity

*k*at time

*t*, respectively, where \(k=m, m+1,\ldots, n\), in which

*m*and

*n*are the minimum and maximum degree in network topology, respectively. Since the number of total nodes with degree

*k*is a constant \(p(k)N\) during the period of epidemic spreading, the following normalization conditions hold:

*τ*during which the infectious agents develop in the vector and the infected vector becomes itself infectious after the delay. At the same time, the vector’s usual activities are in a limited range, and the vector population size is large enough such that at any time

*t*a number of the infectious vector population in the vicinity of the infected nodes with degree

*k*(\(k=m, m+1,\ldots, n\)) at any time

*t*is simply proportional to the number of the infected nodes with degree

*k*. The dynamical equations for the density \(S_{k}(t)\), \(I_{k}(t)\), and \(R_{k}(t)\), at the mean-field level, satisfy the following set of functional differential equations when \(t>0\) [20, 21, 24]:

*r*is the recovery rate from the infected nodes to the recovered nodes because the infected nodes (patients) are cured.

*δ*is the removal rate from the susceptible nodes to the recovered nodes because the susceptible nodes acquire temporary immunity.

*μ*is the removal rate from the recovered nodes to the susceptible nodes because the recovered nodes lose temporary immunity. The dynamics of

*n*groups of

*SIRS*subsystems are coupled through the function \(\Theta(t)\), which represents the probability that any given link points to an infected node at time

*t*. Considering the uncorrelated network [3, 11], we have

*k*-dependent) infection rate, such as

*λk*, \(\lambda c(k)\) [11], and so on. \(\varphi(k)\) denotes an infected node, with degree

*k*, occupied edges which can transmit the disease, while \(\varphi(k)=ak^{\alpha}/(1+bk^{\alpha})\), \(0\leq \alpha<1\), \(a>0\), \(b\geq0\) [7]. With different parameters, \(\varphi(k)\) can be divided into different cases, such as \(\varphi(k)=k\) [3] when \(\alpha=1\), \(b=0\), and \(a=1\); \(\varphi(k)=A\) [5] when \(a=A\), \(\alpha=0\), and \(b=0\); \(\varphi(k)=k^{\alpha}\) [6] when \(a=1\) and \(b=0\). Specially, if \(b\neq0\), \(\varphi(k)\) will become gradually saturated as degree

*k*increases,

*i.e.*, \(\lim_{k\rightarrow\infty}\varphi(k)=b/a\).

*C*denotes the Banach space \(C([-\tau, 0], R^{3(n-m+1)})\) with the norm

It is well known, by the fundamental theory of functional differential equations [25], that system (1) has a unique solution \((S_{m}(t), \ldots, S_{n}(t), I_{m}(t), \ldots, I_{n}(t), R_{m}(t), \ldots, R_{n}(t))\) satisfying the initial conditions (3). It is easy to show that all solutions of system (1) with initial conditions (3) are defined on \([0, +\infty)\) due to the boundedness of \(S_{k}\), \(I_{k}\), and \(R_{k}\). In addition, using similar arguments as in [26], it is easy to show that all solutions of system (1) with initial conditions (3) remain positive for all \(t\geq0\).

The rest of this paper is organized as follows. The dynamical behaviors of the *SIRS* model are discussed in Section 2. Numerical simulations and discussions are given to demonstrate the main results in Section 3. Finally, the main conclusions of this work are summarized in Section 4.

## 2 Dynamical behaviors of the model

### Theorem 1

*System* (4) *always has a disease*-*free equilibrium* \(E_{0}( {\frac{\mu}{\mu+\delta }},\ldots, {\frac{\mu}{\mu+\delta}},0,\ldots,0)\). *System* (4) *has a unique endemic equilibrium* \(E_{*}(S_{m}^{*}, S_{m+1}^{*},\ldots , S_{n}^{*}, I_{m}^{*},I_{m+1}^{*},\ldots,I_{n}^{*})\) *when* \(R_{0}>1\).

### Proof

### Remark 1

We know that the basic reproductive number for system (1) is \(R_{0}\). If \(\varphi(k)=k\) and \(\tau=0\), system (1) reduces to the *SIRS* model (1.2) without delay in [15], and \(R_{0}\) in this paper consists/coincides with one of the model in [15].

### Theorem 2

*If* \(R_{0}<1\), *the disease*-*free equilibrium* \(E_{0}\) *of system* (4) *is globally attractive*.

### Proof

Obviously, we need only to discuss the global attractiveness of system (4) in \(D_{0}\).

*η*is a constant to be determined.

### Lemma 1

[28]

*Consider the following equation*:

*where*\(a_{1},a_{2},\tau>0\); \(x(t)>0\)

*for*\(-\tau\leq t\leq0\).

*We have*

- (i)
*if*\(a_{1}< a_{2}\),*then*\(\lim_{t\rightarrow+\infty}x(t)=0\), - (ii)
*if*\(a_{1}>a_{2}\),*then*\(\lim_{t\rightarrow+\infty}x(t)=+\infty\).

### Theorem 3

*For system* (4), *if* \(R_{0}>1\), *the disease*-*free equilibrium* \(E_{0}\) *is unstable*, *and the disease is uniformly persistent*, i.e., *there exists a positive constant* *ϵ* *such that* \(\lim_{t\rightarrow+\infty}\inf I_{k}(t)\geq\epsilon \), \(k=m,m+1,\ldots,n\).

### Proof

*X*and \(X^{0}\) are positively invariant sets for \(T(t)\). \(T(t)\) is completely continuous for \(t>0\). Also, it follows from \(0< S_{k}(t), I_{k}(t)\leq1\) for \(t>0\) that \(T(t)\) is point-dissipative. \(E_{0}\) is the unique equilibrium of system (4) on \(\partial X^{0}\) and it is globally stable on \(\partial X^{0}\), \(\tilde{A_{\partial}}=\{E_{0}\}\), and \(E_{0}\) is isolated and acyclic. Finally, the proof will be complete if we prove \(W^{s}(E_{0})\cap X^{0}=\emptyset\), where \(W^{s}(E_{0})\) is the stable manifold of \(E_{0}\). Suppose it is not true. Then there exists a solution \((\bar{S},\bar{I})\) in \(X^{0}\) such that

Hence, the infection is uniformly persistent according to Theorem 2.4 in [27, Chapter 8], *i.e.*, there exists an *ϵ*, being a positive constant, such that \(\lim_{t\rightarrow+\infty}\inf I_{k}(t)\geq \epsilon\) and \(\lim_{t\rightarrow+\infty}\inf S_{k}(t)\geq\epsilon\). The disease-free equilibrium \(E_{0}\) is unstable accordingly. This completes the proof. □

Furthermore, we obtain the following Theorem 4 about the global attractiveness of the endemic equilibrium \(E_{*}\) of system (4) by constructing a suitable Lyapunov function.

### Theorem 4

*If* \(R_{0}>1\), \(\delta< r\), *and* \(I_{k}^{*}<\mu /(\mu+\delta)(\delta/r)\), \(k=m,m+1,\ldots, n\), *then the endemic equilibrium* \(E_{*}\) *of system* (4) *is globally attractive*.

### Proof

For convenience, we still discuss system (1).

Thus we just need to discuss the global attractiveness of system (1) in \(\tilde{D}_{0}\).

*i.e.*,

*k*th diagonal entry of

*B*and \(m\leq k\leq n\) [29, Lemma 2.1].

*i.e.*,

## 3 Numerical simulations and discussions

*τ*.

*C*satisfies \(\sum_{k=m}^{n}p(k)=1\). Assuming the network is finite, the maximum connectivity

*n*of any node is related to the network age, measured as the number of nodes

*N*[5, 8]. We have

Although the time delay *τ* has no effects on both the spreading threshold and the density of infected nodes at the stationary state according to (7) and (25), we find that the delay *τ* has much impact on the density \(I(t)\) of the infected nodes; the slower the relative density of infected nodes converges to the stationary state, the larger *τ* gets. Thus the delay cannot be ignored. The numerical simulations in Figure 1 and Figure 2 verify it.

## 4 Conclusions

An *SIRS* model with time delay on a scale-free network has been proposed, in which time delay describes the incubation period of the infective vector. We obtained the basic reproduction number \(R_{0}\), which is irrelative to *τ*. The disease-free equilibrium is globally attractive and infection may disappear when \(R_{0}<1\), while the infection is uniformly persistent when \(R_{0}>1\). Moreover, the endemic equilibrium is globally attractive if \(R_{0}>1\), \(\delta< r\), and \(I_{k}^{*}<\mu/(\mu+\delta)(\delta/r)\), \(k=m,m+1,\ldots, n\). However, numerical simulations (Figure 3) show that the endemic equilibrium is still globally attractive even if \(\delta< r\) and \(I_{k}^{*}<\mu/(\mu+\delta )(\delta/r)\), \(k=m,m+1,\ldots, n\) do not hold when \(R_{0}>1\). Therefore, improvement of the sufficient condition in Theorem 4 on the global attractiveness of the endemic equilibrium of system (4) is an interesting but challenging problem.

## Notes

### Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions, which greatly led to significant improvement of the original manuscript. This research was supported by the Hebei Provincial Natural Science Foundation of China under Grant No. A2016506002 and the Innovation Foundation of Shijiazhuang Mechanical Engineering College under Grant No. YSCX1201.

### Authors’ contributions

All authors contributed to the expression of the model and the discussion of results. They read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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