A reaction-diffusion pneumonia model with Holling type II transmission dynamics and spatial heterogeneity

Pneumonia remains a leading global health challenge, with transmission dynamics significantly influenced by spatial heterogeneity and saturation effects at high infection densities. This study develops a novel spatiotemporal reaction-diffusion model that incorporates Holling type II incidence and spatially heterogeneous transmission to overcome the limitations of conventional homogeneous models. The model divides the population into six compartments and incorporates diffusion to capture geographic spread. We establish the well-posedness of the system, derive the spatial reproduction number R₀^S, and analyze the global stability of disease-free and endemic equilibria. Furthermore, we formulate a multi-objective optimal control problem with three time- and space-dependent interventions: prevention, vaccination, and isolation. Using Pontryagin’s maximum principle, we characterize optimal control strategies that balance epidemiological and economic costs. Numerical simulations demonstrate that the combined triple-control strategy achieves the highest reduction in peak infection prevalence (up to 78%) and the fastest containment, with interventions optimally targeted to high-transmission urban centers. The findings provide a mathematical framework for designing spatially targeted, cost-effective pneumonia control policies.