Highlights A stochastic SVIS epidemic model with standard incidence and vaccination is established. The existence and uniqueness of an ergodic stationary distribution is obtained under If, we derive the exact expression of density function of the stochastic model around the quasi-stable equilibrium. Some criteria for the disease extinction are obtained. Parameter analyses are studied to provide several effective measures to prevent and control the infectious disease.

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020 ) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value corresponding to the basic reproduction number of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if . Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under . It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.

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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.

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbation. First, we obtain a stochastic threshold scriptR0P related to the basic reproduction number scriptR0. A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if scriptR0P>1. Next, by solving the corresponding Fokker‐Planck equation, the exact expression of probability density function of the stochastic model is obtained. Moreover, we establish the sufficient condition scriptR0Q<1 for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results.