Application of Runge Kutta Fehlberg (RKF45) Method as a Numerical Analysis to SIR Model of Tuberculosis Transmission in Central Java

Abstract

This study uses a three-compartment SIR model to describe the spread of tuberculosis in Central Java. Changes in individuals who are detected, recover, die naturally, and die from tuberculosis affect disease transmission. Numerical simulations are used to validate the analytical results and identify the key parameters that contribute most to disease transmission among susceptible, infected, quarantined, and recovered individuals. The numerical method used is the Runge-Kutta-Fehlberg method. Using this method, a quantitative description of the numbers of susceptible, infected, and recovered individuals is obtained, which can assist the Central Java Health Office in efforts to prevent and control the spread of tuberculosis. The SIR model obtained from the parameter determination is then solved using the Runge-Kutta-Fehlberg method. The results obtained using data from 2021–2023 show initial values of 111,120,397 for Susceptible, 157,024 for Infected, and 38,452 for Recovered, with a birth rate parameter of 0.013043, a natural mortality rate of 0.001287, a tuberculosis mortality rate of 0.041376, a transmission rate from Susceptible to Infected of 0.001411, and a recovery rate from Infected to Recovered of 0.24488 individuals. In the 50th year, there are 35,073,325 Susceptible individuals, the number of Infected individuals is 0.04, and the number of Recovered individuals is 74,774. The number of tuberculosis infection cases decreases from year to year.