Mathematical modeling of poliomyelitis virus with vaccination and post-paralytic syndrome dynamics using Caputo and ABC fractional derivatives

Abstract

In this study, using Caputo and ABC derivatives, we present a mathematical analysis of two fractional models for poliomyelitis, considering the presence of vaccination (V) and a post-paralytic class (A). The existence and uniqueness of solutions are proved. The basic reproduction number $R_0$ is computed. Local and global stability of the disease-free stationary state, depending on the threshold $R_0$, is provided, along with conditions for the existence of an endemic stationary state. Moreover, we performed a sensitivity analysis to study the influence of all biological parameters on $R_0$. We concluded our study with numerical simulations to illustrate the models' dynamics and to compare the trajectories of Caputo and ABC solutions. We found that the Caputo and ABC operators are both convenient for the modelization of the poliomyelitis disease. However, the ABC operator not only refined the Caputo operator by removing singularity from the kernel expression but also brought out heredity and memory in the model's characteristics.