The transmission dynamics of an infectious disease model in fractional derivative with vaccination under real data.

The resurgence of monkeypox causes considerable healthcare risks needing efficient immunization programs. This work investigates the monkeypox disease dynamics in the UK, focusing on the impact of vaccination under real data. The key difficulty is to correctly predict the spread of the disease and evaluate the success of immunization efforts. We construct a mathematical model for monkeypox infection and extend it to the fractional case considering the Caputo derivative. The analysis ensures the positivity, boundedness, and uniqueness of the solution for the non-integer system. We conduct a local asymptotical stability analysis (LAS) at the disease-free equilibrium (DFE) D, showing the result for R<1. Additionally, we demonstrate the existence of multiple endemic equilibria and provide conditions for backward bifurcation, which are illustrated graphically. Using real case data from the UK, we estimate model parameters via the nonlinear least square method. Our results show that, without vaccination, R≈0.8, whereas vaccination reduces it to R=0.48. We perform sensitivity analysis to identify key parameters influencing disease elimination, presenting the outcomes through graphs. To solve numerically the fractional model, we outline a numerical scheme and provide detailed results under various parameter assumptions. Our findings suggest that high vaccine efficacy, a low waning rate of the vaccines, and increased vaccination of the infected people can significantly reduce the future cases of monkeypox in the UK. The present study offers a comprehensive framework for monkeypox dynamics and informs public health strategies for effective disease control and prevention.