Mathematical analysis and numerical simulation of a fractional-order SIR-SI model for malaria transmission dynamics

This study investigates the complex transmission dynamics of malaria, a critical global health challenge, with a focus on the African continent. We introduce a novel approach that employs Fractional Differential Equations (FDEs) to advance the understanding of malaria spread and control. Specifically, we develop a new SIR-SI model using the Caputo fractional operator, which captures the memory effects and time-delay characteristics inherent in real-world epidemiological systems. A detailed analysis of the model's solvability and uniqueness is conducted using fixed-point theory. To obtain an analytical solution, the system is solved via the Laplace transform method, with solutions expressed in closed form using the Mittag-Leffler function. The model accounts for the non-local nature of disease spread, underscoring the significant influence of past disease states on current transmission dynamics. Furthermore, we present a modified governing model, analyze its stability and convergence, and employ an efficient iterative Grünwald-Letnikov (GL) method for numerical solution. Simulations demonstrate the impact of key parameters on disease dynamics. The results indicate that the fractional-order model offers a more accurate and comprehensive representation of malaria.