Mathematical model of cholera spread based on SIR: Optimal control

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.

Calculation Method and Application of Time-Varying Transmission Rate via Data-Driven Approach

“…Mathematical modeling has been extensively used in many different fields of study including medicine, physics, economics, and so on [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Medical research issues, especially high-level medical research achievements, often rely on the establishment of reasonable medical mathematical models [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. For example, Tripathi et al [7] provided several examples of designing various mathematical models that can help us better comprehend dynamics at the single-cell and population levels.…”

“…The established model with the use of new epidemiological data can provide theoretical support for the prevention and control of infectious diseases, expose the mechanism of transmission, and facilitate the development of national and even international health policies [10]. After the outbreak of infectious diseases such as Ebola, COVID-19, and Plague, several domestic and foreign scholars refined the existing infectious disease models in use to analyze their epidemiological characteristics [11][12][13][14][15][16][17][18][19][20]. For example, a study [11] used two modified susceptible-infected-removed (SIR) models to simulate the transmission dynamics of Ebola, and the results showed that although there are currently no specific drugs to treat Ebola, effective isolation measures can better manage and control its transmission.…”

“…As is well known, suitable models and reliable data are necessary to ascertain the spread mechanism of emerging infectious diseases, and a key point in establishing compartmental models of infection disease is to determine epidemiological characteristics. Though a large number of works, such as [11][12][13][14][15][16][17][18][19][20] and the references therein, have made it possible to research on the development of these disease with the previous models quantitatively or qualitatively, there still remains some uncertainties in the theoretical framework. To assure a vaccine's efficacy and safety, it typically takes 10 to 15 years to produce one to combat emerging infectious diseases.…”

“…The parameters of compartmental models, including the transmission rate, are often chosen based on individual decisions or assumptions. For example, certain infectious disease models may assume that τ remains as a constant [12,13,15,18,20,[22][23][24] or a piecewise constant [25,26] over the whole epidemic. However, a dynamic transmission rate is more reasonable than a constant one.…”

Optimal Control on Cholera Disease Spreading Model With Three Variables Control Variation

“…Furthermore, in 2019 another research was carried out by Hasanah regarding optimal control of preventing the spread of cholera by controlling treatment and chlorination [10]. Research on the spread of cholera using the SIR model was also carried out by Hidayati, et al (2021) with vaccination control [11].…”