Advanced Fractional Reaction-Diffusion Modeling for Spatio-Temporal Dynamics of Poliovirus Transmission with Disability Outcomes and Vaccination Impacts

Abstract

This work presents a novel fractional reaction-diffusion model to analyze the spatio-temporal dynamics of poliovirus transmission. Polio, a highly contagious viral infection that primarily affects children under five and can lead to permanent disability, spreads through fecal-oral and airborne transmission, often exacerbated by poor sanitation and environmental conditions. Traditional polio models, predominantly based on ordinary differential equations (ODEs), have assumed spatial uniformity an oversimplification of real-world scenarios influenced by population density, environmental heterogeneity, and mobility. Our study extends classical models by incorporating spatial heterogeneity and memory effects using Caputo fractional derivatives and reaction-diffusion dynamics. The model divides the population into seven epidemiological compartments: Susceptible (S), Vaccinated (V), Exposed (E), Non-paralytic Infected (\(N_p\)), Paralytic Infected (P), Recovered (R), and Post-paralytic (A), with corresponding diffusion terms to capture spatial mobility. The inclusion of fractional derivatives accounts for the memory-dependent nature of disease progression, offering a more realistic depiction of poliovirus dynamics. Key contributions include proving the existence and uniqueness of positively bounded solutions, identifying equilibrium points, and assessing their local and global stability using the basic reproduction number (\(R_0\)) and Lyapunov functions under fractional dynamics. Sensitivity analysis is conducted to find the most sensitive parameters using the direct differentiation method. Sensitivity analysis highlights critical parameters influencing disease propagation, while numerical simulations validate the theoretical findings. Graphical results demonstrate the impact of fractional order (\(\alpha\)) and vaccination on disease spread, illustrating how memory effects influence convergence to steady states. This fractional reaction-diffusion framework provides valuable insights into poliovirus transmission, offering a robust tool for predicting outbreaks and guiding effective intervention strategies in diverse populations.