An age-structured differential equations model for transmission dynamics of pneumonia with treatment and nutrition intervention

Pneumonia is the leading infectious disease that threatens the lives of children under five and elders over 65. It is an infection that is commonly caused by Streptococcus pneumoniae. In this study, an age-structured (children and elders) model for pneumonia was formulated and analyzed to determine the impact of treatment and proper nutrition on the transmission dynamics of the disease in the two age groups. The effective reproduction number ($R_e$) was determined using the next-generation method. The disease-free equilibrium point was determined and found locally and globally asymptotically stable if $R_e<1$. Sensitivity analysis of the model parameters was performed using the normalized forward sensitivity index method, and the findings show that transmission rates are the most positive parameters to the effective reproduction number, while proper nutrition was the most negatively sensitive parameter. Additionally, numerical simulations were performed, and it was observed that the combination of proper nutrition and treatment was more effective in reducing the number of pneumonia-infected individuals. The study encourages the joint use of proper nutrition and treatment to control pneumonia transmission among children and elders, especially in the developing world, where economic constraints, infrastructure, and distribution challenges limit vaccine availability.