Analysis of a stochastic SVIR model with time‐delayed stages of vaccination and Lévy jumps

Abstract

The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long‐period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible‐Vaccinated‐Infected‐Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy‐type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well‐posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping‐time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics.