Numerical Methods for Addressing Fractional Order Trypanosomiasis Through the Generalized Adams-Bashforth-Moulton Approach

Trypanosomiasis remains a critical vector-borne disease burden, especially in sub-Saharan African communities where tsetse fly populations thrive and healthcare infrastructure remains limited. Conventional integer-order mathematical frameworks frequently inadequately represent the hereditary characteristics and intricate transmission patterns inherent in vector-borne disease systems. This research presents a fractional-order mathematical framework for examining the epidemiological characteristics of trypanosomiasis transmission, with particular focus on how treatment interventions at different disease stages affect overall transmission dynamics. The primary goal is to explore how variations in treatment efficacy rates and vector-human contact patterns influence disease persistence and spread within affected populations. The framework employs fractional derivatives to more accurately capture the non-Markovian properties of infection progression and immune response mechanisms. The computational results further reveal that strategically implemented treatment protocols can dramatically reduce infection prevalence, potentially driving the system toward disease elimination scenarios. The innovation of this work centers on applying fractional calculus to trypanosomiasis transmission modeling, an approach relatively unexplored in this epidemiological context, while simultaneously incorporating multi-stage treatment interventions and natural immunity waning processes. The model demonstrates superior precision in representing temporal disease progression compared to classical integer-order approaches. This investigation underscores the utility of fractional-order modeling in vector-borne disease research and emphasizes the critical importance of strengthening treatment capacity and vector control measures to effectively manage and eliminate trypanosomiasis transmission in endemic regions.