Numerical study on fractional order nonlinear SIR-SI model for dengue fever epidemics.

This paper focuses on a combined SIR-SI epidemic model to evaluate the transmission dynamics of dengue fever, integrating the susceptible-infected-recovered (SIR) framework for the human population with the susceptible-infected (SI) framework for mosquitoes. The model is formulated as a system of nonlinear differential equations and is further extended by incorporating fractional-order derivatives in the Caputo sense to capture memory effects in disease transmission. A thorough investigation of the disease-free and endemic equilibrium points is conducted, encompassing both local and global stability at the disease-free state. The basic reproduction number, [Formula: see text], is derived, and a sensitivity analysis is performed to identify the key parameters influencing the transmission dynamics. To ensure mathematical rigor, the existence and uniqueness of the model's solutions are also examined. For numerical approximation, the two-step Lagrange polynomial method is applied, enabling simulation of the model under various fractional orders and parameter settings. The results demonstrate that the fractional-order approach offers deeper insights into the dynamics of dengue transmission, highlighting the importance of memory effects. These findings provide valuable guidance for medical professionals, policymakers, and public health authorities in designing more effective control strategies.