Studying the stability of the cholera fractional model and its simulation using generalized hat basis functions

The use of fractional derivatives in epidemiological mathematical modeling offers several advantages over classical integer-order models, including enhanced realism, flexibility, and accuracy. Cholera dynamics, influenced by environmental reservoirs, may exhibit power-law decay, a phenomenon that fractional calculus naturally describes. In this paper, we first introduce a fractional-order SEIRB model for cholera in the Caputo sense. We describe the transmission dynamics of the disease among different compartments. The resulting system of fractional differential equations from the proposed model has a unique solution. We identify a positive invariant set for the solution and derive the equilibrium points and the basic reproduction number. Proving that the disease-free equilibrium always exists, whereas the endemic equilibrium exists if and only if the basic reproduction number is greater than one. We establish the locally asymptotically stability of the equilibrium points using the Routh–Hurwitz criterion, while for globally asymptotically stability analysis, we introduce appropriate Lyapunov functions in the Caputo sense. To analyze the cholera outbreak, we perform a sensitivity analysis of the model by introducing a sensitivity index. In the numerical section, we employ a new and accurate method based on generalized hat functions. Finally, we discuss and analyze the obtained results from both numerical and theoretical perspectives, providing some useful recommendations.