Analysis of SIR Model Dynamics with Vaccination Strategies and Graph-Based Ethnomathematical Representation

Abstract

Tuberculosis (TB) ranks Indonesia second globally in terms of the highest case burden. This study aims to develop a modified Susceptible-Infected-Recovered (SIR) model with vaccination strategies to evaluate and optimise national TB control efforts. Model parameters were adjusted based on the latest epidemiological data, including under-reporting rates and case incidence in Daerah Istimewa Yogyakarta (DIY) from the 2023 Indonesian Tuberculosis Inventory Study Report. Mathematical analysis focused on determining key model parameters and equilibrium stability to identify the conditions that define the boundary between disease sustainability and potential elimination. Numerical simulations compared in detail the effectiveness of constant and periodic vaccination, highlighting the optimal scheme capable of reducing the number of infected cases (I) more quickly. The uniqueness of this research lies in the integration of science and culture through ethnomathematics, where the dynamic patterns of the SIR phase space model are processed into distinctive and meaningful batik motifs. This research supports the development of more effective TB vaccination policies and examines the dynamic relationship between epidemiology and the richness of Indonesian visual arts.