Stability analysis of a discrete SIRS epidemic model with vaccination

ABSTRACT

A discrete SIRS epidemic model with vaccination and a general infection probability function is investigated in this paper. The basic reproduction number ${\mathrm{\Re }}_0$ of the model is calculated. It is proven that the disease-free equilibrium is locally asymptotically stable if ${\mathrm{\Re }}_0<1$ and globally asymptotically stable for some additional conditions, and the endemic equilibrium is locally asymptotically stable if ${\mathrm{\Re }}_0>1$. Numerical simulations are presented to illustrate the stability results, and the sensitivity analysis of the basic reproduction number ${\mathrm{\Re }}_0$ shows that the proportionality coefficient of unvaccinated susceptible, the probability that infected individuals do not recover, and the probability of survival have much more effect on ${\mathrm{\Re }}_0$.

Keywords:

• Discrete SIRS epidemic model
• vaccination
• basic reproduction number
• stability

2010 Mathematics Subject Classifications:

• 37N25
• 37G10

Acknowledgments

The first and second authors especially thank Professor Shigui Ruan, our advisor, for his advice, encouragement, corrections and suggestions. The authors would also like to express their deep gratitude to the referees and the editor for careful reading and helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was partially supported by the National Natural Science Foundation of China (NSFC) [grant number 11871235] and the Fundamental Research Funds for the Central Universities [grant number 30106190481].