Mathematical modeling of human metapneumovirus transmission dynamics with optimal control analysis

Human metapneumovirus (HMPV) is a pathogen that causes severe respiratory infections, especially in children, the elderly, and immunocompromised people. Deterministic epidemiological models cannot capture memory effects or stochastic variability of the environment and thus limit the formulation of effective control measures. In this paper, we developed a stochastic optimal control problem, based on a fractional-order stochastic dynamic, to promote vaccination and treatment solutions to the disease burden. In this study, we designed a modified fractional-order stochastic SEIR model with a Brownian motion to consider long-range time memory effects and stochastic environmental variation. We verified the existence, uniqueness, and positivity of the model, as well as identified disease-free equilibrium, the basic reproduction number $R_0$, and endemic equilibrium. Local and global stability of both states were examined, and a bifurcation analysis disclosed a forward transcritical bifurcation at the point $R_0=1$. Sensitivity analysis highlighted the transmission rate $\beta$ (+1.000), progression rate $\sigma$ (+0.4545), and recovery rate $\gamma$ (-0.6818) as critical parameters affecting $R_0$. We proposed an optimal control analysis that uncovered time-dependent optimal vaccination and treatment techniques. Under the optimal treatment-focused strategy, decreasing the fractional order $\alpha$ from 1.0 to 0.7 reduced peak infections by $36.1\mathrm{\%}$ (from 244 to 156 cases) and delayed the epidemic peak by $57.8\mathrm{\%}$ (from 26.8 to 42.3 days). A strong linear correlation ( $R^2=0.96$) disclosed that lower fractional orders $\alpha$ impose earlier but less intensive interventions. This adaptive control framework offers the public health authorities a tool for designing an outbreak response that is resilient to real-life uncertainty.