Cost benefit and optimal control analysis of temperature dependent pneumonia transmission in different age groups in Kenya

Pneumonia remains one of the leading causes of morbidity and mortality worldwide, particularly among children and older adults, with transmission increasingly influenced by environmental factors such as temperature. This study develops an age-structured optimal control model to investigate pneumonia transmission under seasonal temperature variation across five age groups in Kenya. The developed compartmental model incorporates a temperature-dependent transmission function together with three time dependent control strategies: public health awareness on temperatures, treatment, and childhood vaccination. Pontryagin’s Maximum Principle is applied to derive the necessary conditions for optimality, leading to the corresponding Hamiltonian, adjoint equations, and optimal control characterizations. The resulting optimality system is solved numerically using the forward–backward sweep method with a fourth-order Runge–Kutta algorithm. Numerical simulations demonstrate that temperature awareness through temperature awareness and treatment should be implemented intensively during the early stages of an outbreak before gradually declining as disease prevalence decreases. For children aged 0 − 5 years, vaccination provides sustained long-term protection by continuously reducing the susceptible population and complementing temperature awareness and treatment efforts. Across all age groups, the combined intervention strategies substantially reduce exposed, mildly infected, severely infected, and disease-induced mortality compared with uncontrolled scenarios. Controlled simulations show rapid suppression of transmission and progression towards disease elimination, whereas uncontrolled simulations exhibit sustained transmission, prolonged epidemics, and significantly higher infection burdens. The results further indicate that early implementation of integrated intervention strategies is more cost effective and biologically realistic compared to late implementation. Overall, the findings demonstrate that combining temperature based public health awareness, prompt treatment, and targeted childhood vaccination provides the most effective strategy for minimizing pneumonia transmission and reducing the overall disease burden in Kenya. These findings provide valuable evidence for policymakers in designing climate responsive and age specific pneumonia control programmes.