Mathematical Analysis of Monkeypox Disease Using Fractional Delay Differential Equations and Vaccination Control

Abstract

This study presents a fractional-order model for the transmission dynamics of Monkeypox, formulated using Caputo fractional order delay differential equations. The model incorporates two types of delays: one representing diagnostic delays in humans and the other associated with rodent populations. The proposed framework further integrates vaccination strategies and assesses the demand for clinical care. Analytically, the existence, uniqueness, non-negativity, and boundedness of solutions have been established. The basic reproduction number R0 has been derived, and a sensitivity analysis identifies the parameters most influencing R0. Specifically, the transmission rates β1 and β3, and the parameter ω1 exhibit the strongest influence, whereas the γ has the least impact. Furthermore, the stability of the disease-free and endemic equilibria has been investigated using the curve-switching method in the presence of time delays. Additionally, graphical simulations demonstrated the effects of delays, vaccination rates, and fractional orders. The results indicate that increasing the vaccination rate reduces the infected population, and large values of the delay parameters of the model produce slower disease trajectories and lower infection peaks at fixed time as compared to delays with lower values. This reflects a slower system response rather than the actual situation. These findings highlight the sensitivity of disease progression to delays.