The bifurcation analysis of an SIRS epidemic model with immunity age and constant treatment

ABSTRACT

In this paper, an SIRS epidemic model with immunity age is investigated, where the constant treatment rate and the loss of the acquired immunity are incorporated. The well-posedness of the model is verified by changing it into an abstract non-densely defined Cauchy problem, and the conditions for the existence of disease-free equilibrium and the endemic equilibria are found. The theoretic analysis showed that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number is less than unity, and the numerical simulation illustrated that it is asymptotically stable as the number is greater than unity. Combining numerical simulations, the instability and the local stability of different endemic equilibrium, and the existence of saddle-node bifurcation, and Hopf bifurcation are analyzed. Again, we think it is possible that the Bogdanov–Takens bifurcation may occur for the model under some conditions. Both non-periodic and periodic behaviors are shown when the disease persists in population, where the duration that the recovered individual stays in the recovery class plays an important role in the spread of the disease.

Keywords:

• $C_0$-semigroup
• asymptotical stability
• saddle-node bifurcation
• Hopf bifurcation

2010 Mathematics Subject Classifications:

• 92D30
• 34D20
• 47E05

Acknowledgements

We would like to thank the referees very much for the careful review and the valuable comments to this manuscript which improve it greatly.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Hui Cao http://orcid.org/0000-0001-9612-9746

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant numbers 11971281 and 11301314]; by Natural Science Basic Research Plan in Shaanxi Province of China [grant number 2019JM-081]; and by Natural Science Foundation of Shaanxi Provincial Department of Education grant 18JK0092.