Stochastic Modeling of Nonlinear SIS Epidemic Dynamics with Jump Perturbations

This research rigorously investigates a stochastic Susceptible-Infectious-Susceptible (SIS) epidemic model incorporating a generalized nonlinear incidence rate. The study further expands the model scope by introducing white and Lévy noise components. Including jumps and noises enhances the likelihood of disease extinction within the population. Initially, we utilized a comprehensive mathematical analysis to establish the global stability of the equilibrium state devoid of the disease. Our findings demonstrate that the system maintains stability even in the presence of the disease. This serves as a fundamental insight into the dynamics of disease transmission. Moreover, we define a collection of adequate criteria that contribute to the continuous presence of the ailment, thereby guaranteeing its non-complete eradication. By identifying the critical factors that contribute to the sustained presence of the disease, we enhance our understanding of the long-term behavior of the epidemic. To validate our theoretical findings, we conduct numerical simulations. These simulations effectively demonstrate the accuracy and reliability of the presented results. By comparing the simulated data with our analytical predictions, we can confidently confirm the validity of our model and its applicability in practical scenarios.