Dynamics and modeling of Malaria disease with vector mortality rate and host transmission by using Piecewise Fractional Operator

Abstract

Malaria remains one of the most persistent vector-borne diseases, requiring advanced modeling tools to capture its complex transmission dynamics. This study introduces a novel piecewise Caputo fractional operator with a singular kernel to model malaria transmission between humans and mosquito vectors. The proposed operator allows the system to switch between different fractional dynamics across subintervals, thereby capturing sudden behavioral or environmental changes an ability not present in traditional fractional- or integer-order models. The existence and uniqueness of the systems solutions are rigorously established using the Arzelà–Ascoli and Schauder fixed-point theorems, ensuring mathematical validity. The basic reproduction number is derived via the next-generation matrix approach and analyzed through sensitivity indices to identify key epidemiological parameters influencing transmission. Furthermore, the generalized Ulam–Hyers stability confirms robustness under small perturbations. Numerical simulations based on the Newton polynomial scheme reveal crossover behavior between subintervals and demonstrate that lower fractional orders intensify memory effects, leading to delayed but more stable epidemic responses. Overall, the piecewise Caputo framework enhances the modeling of malaria dynamics by integrating memory-dependent and regime-switching properties, offering a more biologically realistic approach to designing and timing intervention strategies. © 2025 The Authors