Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks

Haijun Hu; Xupu Yuan; Lihong Huang; Chuangxia Huang; Haijun Hu; Xupu Yuan; Lihong Huang; Chuangxia Huang

doi:10.3934/mbe.2019286

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5729-5749. doi: 10.3934/mbe.2019286

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Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks

1.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan, 410114, P. R. China
2.
College of Arts and Science, National University of Defense Technology, Changsha, Hunan, 410073, P.R. China
3.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, Hunan, 410114, P. R. China

Received: 29 December 2018 Accepted: 28 April 2019 Published: 19 June 2019

In this paper, by taking full consideration of demographics, transfer from infectious to susceptible and contact heterogeneity of the individuals, we construct an improved Susceptible-Infected-Removed-Susceptible (SIRS) epidemic model on complex heterogeneous networks. Using the next generation matrix method, we obtain the basic reproduction number $\mathcal{R}_0$ which is a critical value and used to measure the dynamics of epidemic diseases. More specifically, if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable; if $\mathcal{R}_0>1$, then there exists a unique endemic equilibrium and the permanence of the disease is shown in detail. By constructing an appropriate Lyapunov function, the global stability of the endemic equilibrium is proved as well under some conditions. Moreover, the effects of three major immunization strategies are investigated. Finally, some numerical simulations are carried out to demonstrate the correctness and validness of the theoretical results.
- SIRS model,
- heterogeneous network,
- basic reproduction number,
- global dynamics,
- immunization strategy
Citation: Haijun Hu, Xupu Yuan, Lihong Huang, Chuangxia Huang. Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5729-5749. doi: 10.3934/mbe.2019286

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Abstract

In this paper, by taking full consideration of demographics, transfer from infectious to susceptible and contact heterogeneity of the individuals, we construct an improved Susceptible-Infected-Removed-Susceptible (SIRS) epidemic model on complex heterogeneous networks. Using the next generation matrix method, we obtain the basic reproduction number $\mathcal{R}_0$ which is a critical value and used to measure the dynamics of epidemic diseases. More specifically, if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable; if $\mathcal{R}_0>1$, then there exists a unique endemic equilibrium and the permanence of the disease is shown in detail. By constructing an appropriate Lyapunov function, the global stability of the endemic equilibrium is proved as well under some conditions. Moreover, the effects of three major immunization strategies are investigated. Finally, some numerical simulations are carried out to demonstrate the correctness and validness of the theoretical results.

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