Mathematical Model of Diphtheria Transmission with Quarantine and Vaccination Effects Visualized through Batik Motifs

Abstract

Diphtheria is an infectious disease that can spread rapidly through direct contact between individuals. In this study, an SIQR (Susceptible, Infected, Quarantined, Recovered) mathematical model was developed to analyze the spread of diphtheria, taking into account the effects of vaccination and quarantine as control measures. This model is expressed in the form of a nonlinear differential equation system, then an analysis is performed on the disease-free and endemic equilibrium points and their stability using the basic reproduction number (R₀). The results of the analysis show that there are two equilibrium conditions, namely a stable disease-free state (E₀) when R₀ < 1, and a stable endemic state (E₁) when R₀> 1 under certain conditions. Based on numerical simulations, an increase in the vaccination rate above 0.884 and the quarantine rate above 0.049 will reduce the R₀ value below one, which means that diphtheria will gradually disappear from the population. As an additional visualization, the results of this model simulation are presented in the form of mathematical batik motifs, where the patterns and color gradations represent the dynamics of each population compartment (S, I, Q, R). The motifs formed reflect the balance between disease spread and control—like the harmonious relationship between mathematics and local cultural arts. This integration not only clarifies the meaning of the model visually, but also shows that the beauty of batik can represent the stability of the system in the context of infectious disease spread.