Analysis of an SIRS epidemic model with time delay on heterogeneous network
- Qiming Liu^{1}Email author,
- Meici Sun^{1} and
- Tao Li^{1, 2}
https://doi.org/10.1186/s13662-017-1367-z
© The Author(s) 2017
Received: 15 June 2016
Accepted: 19 September 2017
Published: 4 October 2017
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
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 [3–22].
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.
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 \(D_{0}\).
Lemma 1
[28]
- (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
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 \(\tilde{D}_{0}\).
3 Numerical simulations and discussions
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.
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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