Modelling Infectious Disease Dynamics: A Robust Computational Approach for Stochastic SIRS with Partial Immunity and an Incidence Rate

Abstract

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings.

On the Analytical Solution of the SIRV-Model for the Temporal Evolution of Epidemics for General Time-Dependent Recovery, Infection and Vaccination Rates

“…No attempt has been made to include any spatial spread by diffusion. Future work will be concerned with finding analytical approximations for the more complex set of equations including spatial diffusion [19,[37][38][39][40][41][42][43]. Applying Equation (A3) to the two error functions in Equation (81) then yields Equation (82).…”