Levy noise impact on a stochastic epidemic model with healing, relapse and vaccination

In this work, we propose and analyze a stochastic Susceptible-Infected-Recovered-Healed (SIRH) epidemic model with healing, relapse, and jump effects to investigate the impact of vaccination on the disease dynamics. The stochastic perturbations are introduced into the model to account for the inherent randomness in the disease transmission and recovery processes. The existence and uniqueness of the positive global solution for the stochastic model are established. Further, we study the conditions for the extinction and persistence of the disease. The impact of vaccination is incorporated into the model by considering the vaccination rate as a control parameter. Numerical simulations are carried out to illustrate the dynamical behavior of the stochastic SIRH model and to analyze the effects of vaccination on the disease spread. The graphical representations of the approximate solutions for the susceptible, infected, recovered, and healed populations are provided under different vaccination scenarios.