Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps

- A delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps is studied.
- We establish sufficient conditions for persistence and extinction of the disease.
- We find that a large noise intensity has the effect of suppressing the epidemic to extinction.
- Persistence and extinction have close relationship with Lévy noise and vaccination.

In this paper, we discuss the persistence and extinction of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps. Firstly, we study the existence and uniqueness of the global positive solution with any positive initial value. Then we establish sufficient conditions for persistence and extinction of the disease. Moreover, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it goes to extinction. Results show that the persistence and extinction of the disease have a very closed relationship with the intensity of Lévy noise and the validity period of the vaccination. Some examples and numerical simulations are carried out to show the effectiveness and feasibility of the theoretical results.