Numerical Error Comparison of Euler, Heun, and Fourth-Order Runge–Kutta Methods for a SICR Tuberculosis Transmission Model in South Batipuh, West Sumatera

Source abstract

Tuberculosis (TB) remains a major infectious-disease problem and requires quantitative approaches for understanding transmission and treatment dynamics. This study formulates a SICR (Susceptible–Infected–Care–Recovered) compartmental model for TB transmission in South Batipuh District, Tanah Datar Regency, West Sumatra, Indonesia, and compares the numerical accuracy of the Euler, Heun, and fourth-order Runge–Kutta (RK4) methods. Secondary data for 2020–2024 were obtained from population records and TB case records from the local health service. The model consists of susceptible, infected, care, and recovered compartments and is represented by a nonlinear ordinary differential equation system. Model parameters were estimated by curve fitting, and all three numerical schemes were implemented using the same initial conditions and time-step setting. The observed data show that the infected population increased from 69 cases in 2020 to 166 cases in 2023 before decreasing to 132 cases in 2024, while the care and recovered compartments also fluctuated. Numerical simulations reproduce the qualitative dynamics of declining susceptible individuals, an initial increase in infected individuals, a subsequent increase in the care compartment, and growth of the recovered compartment. The average relative errors were 0.10843 for Euler, 0.08342 for Heun, and 0.08317 for RK4. Therefore, RK4 produced the smallest average error and was the most accurate method among the three methods for the studied SICR system. The findings support the use of higher-order numerical integration when accurate approximation of nonlinear TB transmission dynamics is required.