Stability and Hopf Bifurcation in Delayed Age-Structured SVIR Epidemic Model with Vaccination and Incubation

Abstract

An age-structured SVIR epidemic model with vaccination and incubation in two delays effects is investigated. In the model, one delay expresses the incubation period of disease and another delay expresses the minimum infection age of which the infected individuals possesses the infection ability. For the stringency of analysis, the model is firstly transformed into a non-densely defined abstract Cauchy problem. The nonnegativity and boundedness of solutions, basic reproduction number R 0 and the existence of equilibria are established. The linearized systems and characteristic equations at equilibria are calculated from the corresponding abstract Cauchy problem. When R 0 < 1, local and global stability of the disease-free equilibrium is proved, and hence the disease will be deracinated. When R 0 > 1, local stability and the existence of Hopf bifurcation at the endemic equilibrium are established. Particularly, two different cases of Hopf bifurcation are discussed in detail. These results show that the transmission of disease with the incubation period and the minimum infection age have very complex dynamical behavior at the endemic equilibrium. Finally, numerical examples and simulations are presented to verify the theoretical results and further exhibit the existence of stability switch.

Note:

Funding Information: This work was supported by the Natural Science Foundation of China (Grant No. 12061079), the Natural Science Foundation of Xinjiang (Grant No. 2019D01C206), the China Postdoctoral Science Foundation (Grant No. 2020M683714XB), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2021JM-445), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2022JM-023).

Declaration of Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Keywords: SVIR epidemic model, age-structured, delay, Hopf bifurcation