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. 2022 Apr 9;38(2):282–303. doi: 10.1007/s10255-022-1075-7

Dynamical Behavior of SEIR-SVS Epidemic Models with Nonlinear Incidence and Vaccination

Xiao-mei Feng 1,2,✉, Li-li Liu 3,4, Feng-qin Zhang 1
PMCID: PMC8994021  PMID: 35431376

Abstract

For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model. For the age-structured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.

Keywords: vaccination, age-structured epidemic model, delay differential equation model, stability, Lyapunov functional

Footnotes

This paper is supported by The National Natural Science Foundation of China [12026236, 12026222, 12061079, 11601293, 12071418], Science and Technology Activities Priority Program for Overseas Researchers in Shanxi Province [20210049], The Natural Science Foundation of Shanxi Province [201901D211160, 201901D211461, 201901D111295].

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