<![CDATA[This research aims to apply optimal control to model the pneumonia disease using a system of ordinary differential equations. We developed based on the SIR pneumonia epidemic model. We divided the population into two groups: children and the elderly. The optimal controls illustrate the effectiveness of vaccination and treatment in preventing new infections among both children and the elderly. The Pontryagin’s maximum principle is used to define the optimal controls, and the resulting optimality system is formulated and solved numerically. Numerical simulations have been performed. It was found that combining vaccination and treatment effectively reduces the number of pneumonia infections. In fact, when we change the cost scenario, we see changes in the control variables. This study supports the joint implementation of vaccination and therapeutic interventions to control pneumonia transmission among children and the elderly, particularly in developing countries where economic limitations, inadequate infrastructure, and distribution barriers restrict vaccine access.]]>
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