Optimal Control and Model Analysis of The Spread of Pneumonia in Toddlers in East Java-Indonesia Using The Pontryagin’s Minimum Principle

Abstract

Pneumonia is  a  type  of  acute  respiratory  infection   (ARI) that  attacks  the  lungs and  is  caused   by  various microorganisms, such  as bacteria, viruses, parasites,  fungi, exposure to chemicals, or physical damage  to the lungs. Pneumonia is  included in  the list  of  10 diseases  with  the highest  number   of  cases  according to the Indonesian Ministry of Health reported  in April 2023. Pneumonia is the biggest cause of death in toddlers  aged 12-59  months,  reaching  12.5%. Therefore,   to  reduce  the  spread   of  pneumonia,  this  research  will  discuss providing  optimal control using the mathematical model  of  SEIR (Susceptible-Exposed-Infected-Recovered). The model  used is a pneumonia spreading model  with  implementing control in the form of first stage treatment and second  stage treatment. The results  of the stability analysis show  that at the disease-free  equilibrium point and  the endemic  equilibrium point,  the system  is  stable  respectively. Based  on  controllability analysis, it  is obtained  that the system  is controlled so that the system  can be controlled. In addition, based on the results  of the analysis of the optimal control  problem  with  Pontryagin’s Minimum Principle simulated with  Runge Kutta order  4, it shows  that the first  stage of treatment control  (u1)  and  the second  stage of treatment  (u2)  are very effective   in   reducing  the  number   of  individuals  infected   with   mild  pneumonia and   severe   pneumonia respectively.