The global stability of an SIRS model with infection age
Yuming Chen1, Junyuan Yang, Fengqin Zhang
1Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi, China. ychen@wlu.ca.
Abstract:
Infection age is an important factor affecting the transmission of infectious diseases. In this paper, we consider an SIRS model with infection age, which is described by a mixed system of ordinary differential equations and partial differential equations. The expression of the basic reproduction number R0 is obtained. If R0≤1 then the model only has the disease-free equilibrium, while if R0>1 then besides the disease-free equilibrium the model also has an endemic equilibrium. Moreover, if R0<1 then the disease-free equilibrium is globally asymptotically stable otherwise it is unstable; if R0>1 then the endemic equilibrium is globally asymptotically stable under additional conditions. The local stability is established through linearization. The global stability of the disease-free equilibrium is shown by applying the fluctuation lemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.
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