https://doi.org/10.65770/BCLG4677
ABSTRACT
Mathematical modeling, particularly the SEIR (Susceptible, Exposed, Infected, Recovered) compartmental model, has become an essential tool for understanding the dynamics of infectious disease spread. However, the resulting system of nonlinear ordinary differential equations (ODE) generally lacks analytical solutions, thus requiring numerical approaches. This study analyzes the numerical stability limit and the positivity condition of the fourth-order Runge-Kutta method (RK4) when applied to the SEIR epidemiological model with vaccination intervention. Through a local linearization approach using the Jacobian matrix, the dominant eigenvalues of the system are computed to formulate the RK4 stability limit. Additionally, a more restrictive positivity condition is formulated to prevent the emergence of negative populations, which are biologically meaningless. Numerical simulations are conducted using epidemiological parameters representing an aggressive disease transmission and various time step sizes. The results show that the RK4 numerical stability limit provides a relatively loose upper bound on the step size, whereas the positivity condition yields a much stricter bound. Simulations with small step sizes produce stable and positive solutions; however, when the step size exceeds the positivity limit, sharp oscillations emerge, eventually leading to negative population values, causing the solution to lose its biological meaning. This study concludes that satisfying the numerical stability limit alone is insufficient, and the positivity condition must be prioritized as a criterion in selecting the time step size for epidemiological simulations using the RK4 method.
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