Stochastic analysis of Mpox epidemiology with vaccination strategies and environmental persistence

In the context of global preparedness and response strategies for Mpox outbreaks, we develop a comprehensive mathematical model to investigate its transmission dynamics, integrating critical factors such as vaccination coverage and environmental contamination. The model is analytically examined to ensure well-posedness, including the positivity of solutions and the existence of a biologically invariant region. Stability analysis reveals that the disease-free equilibrium is both locally and globally stable when the basic reproduction number [Formula: see text], while a unique endemic equilibrium exists when [Formula: see text] Parameter values are estimated using the least squares method, utilizing U.S. outbreak data from May 10 to December 31, 2022 to capture the most intense phase of Mpox transmission and burden. The model is validated by comparing simulations against reported daily, cumulative, and death cases, and residuals are analyzed to assess model accuracy. A global sensitivity analysis, supported by contour and surface plots, identifies critical parameters affecting disease transmission. Bifurcation analysis reveals complex system behavior near threshold conditions. Furthermore, we formulate and analyze an optimal control problem incorporating vaccination and sanitation efforts to evaluate intervention effectiveness. The findings underscore that a combined approach is essential for optimal disease mitigation- vaccination provides direct individual protection, while sanitation curtails environmental sources of reinfection. Additionally, we employ CTMC and SDE frameworks to capture stochastic effects, offering a more realistic understanding of outbreak dynamics beyond deterministic predictions.