Stochastic differential equations for SIR malaria model

Abstract

This study develops a stochastic extension of the classical SIR malaria model to more realistically capture the dynamics of malaria transmission under environmental uncertainty. By incorporating Brownian motion into the deterministic framework, the model accounts for random fluctuations in transmission and recovery processes affecting both human and mosquito populations. The stochastic differential equations (SDEs) are solved using three numerical methods: Euler–Maruyama, Milstein, and Stochastic Runge–Kutta. Each method is evaluated for accuracy and reliability using error metrics such as Mean Absolute Deviation (MAD), Root Mean Square Error (RMSE), and the Anderson–Darling goodness-of-fit test. The Euler–Maruyama method demonstrates superior performance in approximating human infection and recovery dynamics, while the Milstein method provides a balanced estimate for both human and vector compartments. The study further introduces a novel convergence metric–Okwomi C∗–based on the Geweke diagnostic, enabling robust assessment of convergence in stochastic simulations. Simulation results show that stochastic models capture extinction and variability effects not evident in deterministic models, even when the basic reproduction number exceeds one. Overall, the integration of stochastic processes improves epidemic forecasting and informs more resilient malaria control strategies under uncertainty.