An Age-Structured Analytics Model for Pneumonia Transmission and Vaccination Strategies

Pneumonia remains a major public health concern worldwide, with millions of new cases reported each year. In this study, we develop a modified mathematical model to describe the transmission dynamics of pneumonia with vaccination and analyze the spread of infection across three age groups: under 5 years, 5--64 years, and above 64 years. First, we compute the basic reproduction number R₀ = 0.298, which indicates that the disease cannot sustain transmission within the population across all age groups. In addition, we study the local and global stability and perform a global sensitivity analysis in terms of R₀, which shows how key parameters are important in preventing disease transmission. We examine the model by comparing system dynamics under two scenarios: with vaccination and without vaccination. An optimal control framework is introduced with two control variables representing a balanced diet and improved hygiene practices. By applying Pontryagin's Maximum Principle, we derive the optimal control strategies and characterize the corresponding optimal control profiles for the three age groups. Numerical simulations show graphically that vaccination combined with additional control measures can significantly reduce pneumonia incidence, demonstrate how the parameters vary, and compare the cost-effectiveness of the two introduced control measures. The results suggest that vaccination alone is particularly effective for children under 5 years, whereas for 5--64 years and above 64 years, the combined implementation of vaccination and the two control strategies provides the greatest reduction in disease burden. These findings highlight the importance of targeted immunization programs, along with supportive health interventions such as improved nutrition and hygiene, in reducing pneumonia prevalence and preventing future outbreaks.