A stochastic Susceptible Vaccinees Infected Recovered epidemic model with three types of noises

Epidemics pose a serious threat to public health, and effective disease control measures are essential. To address this challenge, we propose a novel stochastic Susceptible Vaccinees Infected Recovered (SVIR) epidemic model that incorporates temporary immunities, media coverage, and three types of noises: white noise, telegraph noise, and L vy noise. Our model is more realistic than traditional epidemic models, as it accounts for the random fluctuations in disease transmission and the impact of media coverage on public behavior. We use Lyapunov functions and stopping times to prove the existence of a global positive solution to our model. We also derive sufficient conditions for the extinction and persistence of the infection. Furthermore, we develop a numerical scheme to validate our theoretical findings and to analyze the influence of various parameters on the model's dynamics. Our model can be used to guide policymakers in developing and implementing effective disease control measures.