The Global Dynamics for a Stochastic $SIR$ Epidemic Model with Vaccination (Journal of Nonlinear Modeling and Analysis, 2024)
A new stochastic SIR epidemic model with vaccination is established and its dynamical behavior is analyzed. Considering the random effectsof vaccination rates and mortality in this model, it is demonstrated that theextinction and persistence of the virus is only correlated with the threshold $R^s_0.$ If $R^s_01,$ thedisease is stochastic persistent in the means with probability one. In addition, the existence and uniqueness of a smooth distribution are proven usingthe Itôs formula, and the sufficiency criterion is obtained using the Lyapunovfunction. Finally, the accuracy and efficiency of the stochastic SIR epidemicmodel with vaccination in predicting disease transmission trends were verifiedthrough simulation. Unlike the singularity of stochastic perturbations in existing infectious disease models, the innovation of this paper is in the additionof multiple stochastic perturbations, especially distinguishing the stochasticperturbations of mortality under vaccination, which are used to study thedynamics of the model.
Keywords: Vaccination; random disturbance; smooth distribution; Lyapunov function; stability.