A nonstandard finite difference scheme for a multi-group epidemic model with time delay
- Jinhu Xu^{1}Email author and
- Yan Geng^{2}
https://doi.org/10.1186/s13662-017-1415-8
© The Author(s) 2017
Received: 3 August 2017
Accepted: 30 October 2017
Published: 9 November 2017
Abstract
In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if \(\mathfrak{R}_{0}\leq1\), then the disease-free equilibrium \(P_{0}\) is globally asymptotically stable; if \(\mathfrak{R}_{0}>1\), then the infection equilibrium \(P_{*}\) is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model.
Keywords
1 Introduction
Here \(S_{k}\), \(E_{k}\), \(I_{k}\) and \(R_{k}\) (\(k=1, 2, \ldots, m\)) denote the numbers of susceptible, exposed, infectious and recovered individuals at time t in the kth group, respectively. The parameters \(d_{k}^{S}\), \(d_{k}^{E}\), \(d_{k}^{I}\), \(d_{k}^{R}\) are the natural death rates of \(S_{k}\), \(E_{k}\), \(I_{k}\) and \(R_{k}\) compartments in the kth group, respectively. \(\Lambda_{k}\) represents influx of individuals into the kth group; \(\delta_{k}\) is the rate of becoming infectious after a latent period; \(\gamma_{k}\) denotes per capita recovery rate in kth group. The nonnegative constant \(\beta_{kj}\) is the transmission rate due to the contact of susceptible individuals in the kth group with infectious individuals in the jth group. \(\tau _{j}\geq0\) denotes the time delay. For more details on model (1), one can refer to [2]. The global stability of the equilibria of system (1) is investigated in [2] by making use of the method of Lyapunov functionals [3–5]. For more information on multi-group models, one can refer to [6–11] and the references therein.
The global asymptotic stability of the equilibria for the continuous model (1) can be obtained by constructing Lyapunov functionals [2]. Thus, a natural question is whether the discrete model (2) can efficiently preserve the global asymptotic stability of the equilibria for the corresponding continuous model. In this paper, we will deal with this problem. The organization of the paper is as follows. We present some preliminaries including the positivity and boundedness of the solution of model (2) in Section 2. In Section 3, we establish the global stability of the equilibria of model (2) by constructing Lyapunov functions. A brief conclusion ends the paper.
2 Preliminaries
It follows from (4) that all solutions of system (2) subject to initial condition (3) remain nonnegative for all \(n\in\mathbb{N}\).
Theorem 2.1
All solutions of system (2) subject to initial condition (3) remain nonnegative and bounded for all \(n\in\mathbb{N}\).
The existence and global asymptotic stability of the equilibria for the corresponding continuous model of (5) can be directly deduced from the obtained results in [2]. Particularly, if \(\mathfrak{R}_{0}> 1\) and \(B=(\beta_{kj})_{m\times m}\) is irreducible, then model (5) has at least one endemic equilibrium.
3 Global stability
In this section, we establish the global stability of the equilibria of system (5) by constructing Lyapunov functions.
Theorem 3.1
Assume that \(B=(\beta_{kj})_{m\times m}\) is irreducible. For any \(h>0\), if \(\mathfrak{R}_{0}\leq1\), then the disease-free equilibrium \(P_{0}\) is globally asymptotically stable.
Proof
Since \(B=(\beta_{kj})_{m\times m}\) is irreducible, we know that matrix \(M_{0}\) is also irreducible and has a positive left eigenvector \(\omega=(\omega_{1},\ldots,\omega_{m})\) corresponding to the spectral radius \(\mathfrak{R}_{0}=\rho(M_{0})>1\). Let \(I_{n}=(I_{1_{n}},\ldots,I_{m_{n}})\), \(S^{0}=(S_{1}^{0},\ldots,S_{m}^{0})\) and \(c_{k}=\frac{\omega_{k}\delta_{k}}{(d_{k}^{E}+\delta_{k})(d_{k}^{I}+\gamma_{k})}\).
- (i)
if \(\mathfrak{R}_{0}<1\), \(\lim_{n\rightarrow\infty}(V_{n+1}-V_{n})=0\) is equivalent to \(\lim_{n\rightarrow\infty}S_{k_{n}}=S_{k}^{0}\), \(\lim_{n\rightarrow\infty}I_{k_{n}}=0\). It follows from (5) that \(\lim_{n\rightarrow\infty}I_{k_{n}}=0\) for all \(1\leq k\leq m\).
- (ii)
if \(\mathfrak{R}_{0}=1\), \(\lim_{n\rightarrow\infty}(V_{n+1}-V_{n})=0\) is equivalent to \(\lim_{n\rightarrow\infty}S_{k_{n}}=S_{k}^{0}\). By (5), it can be shown that \(\lim_{n\rightarrow\infty}E_{k_{n}}=0\), \(\lim_{n\rightarrow\infty}I_{k_{n}}=0\) for all \(1\leq k\leq m\).
By the above discussion, it is concluded that if \(\mathfrak{R}_{0}\leq1\), the disease-free equilibrium \(P_{0}\) is globally asymptotically stable. This completes the proof. □
Theorem 3.2
Assume that \(B=(\beta_{kj})_{m\times m}\) is irreducible. For any \(h>0\), if \(\mathfrak{R}_{0}>1\), then there exists a unique endemic equilibrium \(P_{*}\) which is globally asymptotically stable.
Proof
In this part, we show that the endemic equilibrium \(P_{*}\) is globally asymptotically stable when \(\mathfrak{R}_{0}>1\). The method is based on the graph-theoretical approach and Lyapunov functions by Guo et al. [3, 4] and Li and Shuai [5].
This implies that \(G_{n,Q}=0\) for each Q, and \(G_{2}\equiv0\) for all \(I_{1_{n}}, I_{2_{n}},\ldots,I_{m_{n}}>0\). Notice that \(\frac{S_{k}^{*}}{S_{k_{n+1}}}+\frac{S_{k_{n+1}}}{S_{k}^{*}}\geq 2\), the equality holds if and only if \(S_{k_{n}}=S_{k}^{*}\), and \(\varphi(x)=x-1-\ln x\) has global minimum value \(\varphi(1)=0\) defined with all \(x>0\). Hence, we have \(L_{n+1}-L_{n}\leq0\). Thus, \(L_{n}\) is a monotone decreasing sequence. Due to \(L_{n}\geq0\), there is a limit \(\lim_{n\rightarrow\infty }L_{n}\geq0\), which implies that \(\lim_{n\rightarrow\infty}(L_{n+1}-L_{n})=0\). Furthermore, similar to [2, 21], we can show that the only compact invariant subset of \(\{\lim_{n\rightarrow\infty }(L_{n+1}-L_{n})=0\}\) is the singleton \(\{P_{*}\}\), which implies that the endemic equilibrium \(P_{*}\) is globally asymptotically stable. This completes the proof. □
4 Numerical simulations
In this section, some numerical simulations are carried out to demonstrate our theoretical results. To this end, we just need to simulate (5) for simplicity. We do simulations for \(k=1, 2\). We select \(\tau_{1}=5\), \(\tau_{2}=10\) in the following simulations.
5 Conclusions
In this paper, a discrete multi-group epidemic model with time delay has been constructed by applying a nonstandard finite difference (NSFD) scheme to a class of continuous multi-group model. The advantage of the NSFD scheme is that the global properties of the solutions for the corresponding continuous model can be preserved. A crucial observation regarding the advantage of the NSFD scheme is that the discrete model has equilibria which are exactly the same as those of the original continuous model, and the conditions for their stability are identical in case of both the continuous and discrete models. It is shown that the global stability of the equilibria is completely determined by \(\mathfrak{R}_{0}\): if \(\mathfrak{R}_{0}\leq1\), then the disease-free equilibrium \(P_{0}\) is globally asymptotically stable; if \(\mathfrak{R}_{0}>1\), then the infection equilibrium \(P_{*}\) is globally asymptotically stable.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (# 11701445, #11702214, #11501443).
Authors’ contributions
The authors 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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