A MATHEMATICAL MODEL OF DIPHTHERIA TRANSMISSION DYNAMICS WITH HETEROGENEOUS SUSCEPTIBILITY
Abstract
Despite the availability of vaccines, diphtheria continues to pose a public health risk in Indonesia due to uneven vaccination coverage across regions. Previous models have not distinguished between highly susceptible (unvaccinated) and susceptible (vaccinated) populations, nor have they been calibrated with actual Indonesian epidemiological data. To address this gap, this study develops a five-compartment diphtheria transmission model: Highly susceptible (unvaccinated)-Susceptible (vaccinated)-Exposed-Infectious-Recovered (S_1 S_2 EIR), which incorporates two levels of susceptibility based on vaccination status, using empirical diphtheria case data in Indonesia from 2012 to 2023. The analysis begins by proving the positivity, boundedness, and uniqueness of solutions, followed by the calculation of the basic reproduction number using the Next-Generation Matrix method. The analysis shows that the disease-free equilibrium (DFE) is locally and globally asymptotically stable when R₀<1, while the endemic equilibrium (EE) is globally stable when R₀>1. Simulations indicate that the interaction parameter for the unvaccinated group η₁, strongly accelerates epidemic growth, leading to a higher and earlier infection peak, whereas increased vaccination coverage and recovery rates effectively suppress transmission. This model can be used because the Mean Absolute Percentage Error (MAPE) between the data and the model solution for diphtheria cases in Indonesia is 8.77%. These results highlight the importance of interventions focused on highly susceptible groups to prevent more severe outbreaks. Therefore, this study is significant in strengthening the theoretical understanding of diphtheria transmission, while also providing data-driven insights as recommendations for policymakers to implement effective and efficient outbreak control measures.
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