Analysis of hybrid fractional integro-differential equations with application to cholera dynamics.

This study establishes the existence of solutions for a class of fractional hybrid integro-differential equations governed by the ϑ-Caputo derivative, subject to slit-strip boundary conditions. The significance of this work lies in the integration of advanced fractional calculus, specifically, the ϑ-Caputo operator, with a biologically meaningful cholera epidemic model. Using Dhage's fixed point theorem, we rigorously prove the existence of solutions, distinguishing this work from existing models based solely on standard fractional derivatives. Two illustrative examples are presented, including a novel cholera model that captures memory effects, environmental feedback, and nonlinear transmission pathways. Numerical simulations, implemented via the Adams-Bashforth-Moulton method, demonstrate how variations in the fractional-order parameter influence the disease dynamics over time. These results underscore the value of fractional modeling in capturing real-world epidemic behaviors and support the use of the ϑ-Caputo framework in future epidemiological studies.