Global stability analysis of an SVEIR epidemic model with general incidence rate

Da-Peng Gao; Nan-Jing Huang; Shin Min Kang; Cong Zhang

doi:10.1186/s13661-018-0961-7

Global stability analysis of an SVEIR epidemic model with general incidence rate

Bound Value Probl. 2018;2018(1):42. doi: 10.1186/s13661-018-0961-7. Epub 2018 Mar 27.

Authors

Da-Peng Gao¹, Nan-Jing Huang², Shin Min Kang³, Cong Zhang⁴

Affiliations

¹ School of Mathematics and Information, China West Normal University, Nanchong, P.R. China.
² Department of Mathematics, Sichuan University, Chengdu, P.R. China.
³ Department of Mathematics and the RINS, Gyeongsang National University, Jinju, Korea.
⁴ School of Management, Sichuan University of Science and Engineering, Zigong, P.R. China.

Abstract

In this paper, a susceptible-vaccinated-exposed-infectious-recovered (SVEIR) epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated, assuming that the horizontal transmission is governed by an unspecified function $f (S, I)$ . The role that temporary immunity (vaccinated-induced) and treatment of infected people play in the spread of disease, is incorporated in the model. The basic reproduction number $R_{0}$ is found, under certain conditions on the incidence rate and treatment function. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. By constructing a suitable Lyapunov function, it is observed that the global asymptotic stability of the disease-free equilibrium depends on $R_{0}$ as well as on the treatment rate. If $R_{0} > 1$ , then the endemic equilibrium is globally asymptotically stable with the help of the Li and Muldowney geometric approach applied to four dimensional systems. Numerical simulations are also presented to illustrate our main results.

Keywords: Epidemic model; Geometric approach; Global stability; Lyapunov function; Reproduction number; Susceptible–Vaccinated–Exposed–Infectious–Recovered.