Stationary distribution and density function analysis of stochastic susceptible-vaccinated-infected-recovered (SVIR) epidemic model with vaccination of newborns

Birth vaccinations are becoming more common in society. In this paper, we describe the developed stochastic susceptible-vaccinated-infected-recovered (SVIR) epidemic model with vaccination of newborns that enable us to concern the stationary distribution and further density function. By constructing a series suitable Lyapunov function, we derive the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution. More importantly, under the same conditions, we creatively find further the density function which is based on solving corresponding Fokker–Planck equation. The results of numerical simulation, which is supported by pertussis disease data, show that our conclusion accords with reality. The density function throws light on the property of an epidemic after being stationary and furnishes more information about the disease.