Analysis of a Stochastic SICR Epidemic Model Associated with the Lévy Jump

We propose and study a Susceptible-Infected-Confined-Recovered (SICR) epidemic model. For the proposed model, the driving forces include (for example) the Brownian motion processes and the jump Lévy noise. Usually, in the existing literature involving epidemiology models, the Lévy noise perturbations are ignored. However, in view of the presence of strong fluctuations in the SICR dynamics, it is worth including these perturbations in SICR epidemic models. Quite frequently, this results in several discontinuities in the processes under investigation. In our present study, we consider our SICR model after justifying its used form, namely, the component related to the Lévy noise. The existence and uniqueness of a global positive solution is established. Under some assumptions, we show the extinction and the persistence of the infection. In order to give some numerical simulations, we illustrate a new numerical method to validate our theoretical findings.