SIR Epidemic Model with General Nonlinear Incidence Rate and Lévy Jumps

Abstract

This article proposes a stochastic SIR model with general nonlinear incidence and Lévy jumps, which is used to describe diseases spreading in human populations. The model takes into account the randomness and sublinearity of diseases and can more accurately describe the disease transmission process. Firstly, we prove that this stochastic SIR model has a unique global positive solution. Then, sufficient conditions for the extinction of the disease are given. We also discuss the case that the disease persists in the model. In addition, we study the asymptotic behavior of the solution of the stochastic SIR model relative to the equilibrium points of the deterministic SIR model. These results allow us to understand the trends and dynamic changes of diseases in human populations, providing theoretical support for developing more scientific and effective disease control strategies and prevention measures. Finally, we give some examples and numerical simulations to demonstrate the effectiveness and feasibility of the theoretical results.