A New Stochastic Split-Step θ-Nonstandard Finite Difference Method for the Developed SVIR Epidemic Model with Temporary Immunities and General Incidence Rates

In this paper, an SVIR epidemic model with temporary immunities and general incidence rates is constructed and analyzed. By utilizing Lyapunov functions, we prove the existence and uniqueness of the positive global solution of the constructed model, as well as the sufficient conditions of extinction and persistence of disease, are provided. Due to the difficulty of obtaining the analytical solution to our model, we construct two numerical schemes to generate an approximate solution to the model. The first one is called the split-step (Formula presented.) -Milstein (SSTM) method, and the second one is called the stochastic split-step (Formula presented.) -nonstandard finite difference (SSSNSFD) method, which is designed by merging split-step (Formula presented.) method with stochastic nonstandard finite difference method for the first time in this paper. Further, we prove the positivity, boundedness, and stability of the SSSTNSFD method. By employing the two mentioned methods, we support the validity of the studied theoretical results, as well, the effect of the length of immunity periods, parameters values of the incidence rates, and noise on the dynamics of the model are discussed and simulated. The increase in the size of time step size plays a vital role in revealing the method that preserves positivity, boundedness, and stability. To this end, a comparison between the proposed numerical methods is carried out graphically.