Mathematical model of cholera spread based on SIR: Optimal control

Noer Hidayati; Eminugroho Ratna Sari; Nur Hadi Waryanto

doi:10.21831/pg.v16i1.35729

Mathematical model of cholera spread based on SIR: Optimal control

Noer Hidayati, MTsS Arul Kumer, Mathematics, Indonesia
Eminugroho Ratna Sari, Department of Mathematics Education, Universitas Negeri Yogyakarta, Indonesia
Nur Hadi Waryanto, Department of Mathematics Education, Universitas Negeri Yogyakarta, Indonesia

10.21831/pg.v16i1.35729

Abstract

The bacterium Vibrio cholerae is the cause of cholera. Cholera is spread through the feces of an infected individual in a population. From a mathematical point of view, this problem can be brought into a mathematical model in the form of Susceptible-Infected-Recovered (SIR), which considers the birth rate. Because outbreaks that occur easily spread if not treated immediately, it is necessary to control the susceptible individual population by vaccination. The vaccine used is Oral Vibrio cholera. For this reason, the purposes of this study were to establish a model for the spread of cholera without vaccination, analyze the stability of the model around the equilibrium point, form a model for the spread of cholera with vaccination control, and describe the simulation results of numerical model completion. Based on the analysis of the stability of the equilibrium point of the model, it indicates that if the contact rate is smaller than the sum of the birth rate and the recovery rate, cholera will disappear over time. If the contact rate is greater than the sum of the birth rate and the recovery rate, then cholera is still present, or in other words, the disease can still spread. Because the spread is endemic, optimal control of the population of susceptible individuals is needed, in this case, control by vaccination, so that the population of susceptible individuals becomes minimum and the population of recovered individuals increases.

Keywords

SIR model; cholera; vaccination; optimal control

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DOI: https://doi.org/10.21831/pg.v16i1.35729