## MATHEMATICAL MODELS OF CONTROL MEASURES IN THE SPREAD OF CHOLERA

ABSTRACT

Although cholera has existed for ages, it has continued to plague many parts of the world. In this study, a mathematical model for cholera epidemic is presented and analyzed in order to determine the effects of the control measures. We modify a compartmental SIBR model for cholera dynamics proposed by Codeco, introducing control measures such as vaccination, provision of clean water and treatment into the model one after the other. The epidemic threshold known as the basic reproduction number and equilibra for the models are determined. Stability analysis of the models is carried out. Analysis of the models reveals that introduction of control measure reduces the rate of the spread of cholera. We also considered a situation where the control measures were introduced simultaneously into the model. Further analysis shows that the combined control measures yield a better result, that is, it reduces the rate of the spread of cholera more than when only one control measure is introduced. The analytical predictions were confirmed by numerical simulation results. Finally, we performed the sensitivity analysis of the basic reproduction number for the combined control model.

TABLE OF CONTENT

Title page
Abstract
Table of content

CHAPTER ONE
1.         INTRODUCTION
1.1       The Epidemiology of Cholera
1.2       Definition of terms

CHAPTER TWO
2.         LITERATURE REVIEW
2.1       Review of Mathematical Models
2.2       SIBR Model

CHAPTER THREE
3.         CONTROL MODELS FOR CHOLERA EPIDEMICS
3.1       Vaccination model
3.2       Provision of clean water model
3.3       Treatment model
3.4       Combined control measure model
3.5       Global Stability of the Combined Control Model

CHAPTER FOUR
4.         NUMERICAL SIMULATIONS
4.1       Explanation of Simulation
4.2       Sensitivity analysis
4.3       Numerical Simulation graph

CHAPTER FIVE
5.         DISCUSSION AND RECOMMENDATIONS
5.1       Discussion of results
5.2       Observations
5.3       Recommendation
5.4      Suggestion for further work

References

CHAPTER ONE
INTRODUCTION
1.1       The Epidemiology of Cholera
Cholera is an acute intestinal infection caused by the ingestion of contaminated food and water with Vibrio cholerae bacterium. Among the 200 Serogroups of V. cholera, it is only V. cholerae 01 and 0139 that are known to be the cause of cholera disease [4]. Vibrio Cholerae is a motile gram negative curved-rod bacterium with polar flagellum that causes cholera in human [1, 2]. V. cholerae was first identified by the Italian anatomist Filippo Pacini (1854), though his discovery was not known until Robert Koch thirty years later [3]. Cholera is characterized, in its most severe form, by the sudden onset of acute watery diarrhea that can lead to death by severe dehydration. The etiological agent, Vibrio cholerae 01, (and more recently V. cholerae 0139), passes through and survives the gastric acid barrier of the stomach and then penetrates the mucus lining that coats the intestinal epithelial [5]. Once they colonize the intestinal gut, they produce enterotoxin (which stimulates water and electrolyte secretion by the endothelia cells of the small intestine) that leads to watery diarrhea. If left untreated, it leads to death within hours. In human volunteer studies, the infection dose was determined to be  102 – 103 cells [6].  V. cholerae can stay in faeces without losing its infections ability for 7-14 days and shed back to the environment.

A pictorial representation of the life cycle of V. cholerae is depicted in the diagram below.

Historical records tracing back to 200 years, in both Greek and Sanskrit, describe diseases similar to cholera [8]. Thus, from the literature, it is clear that there were cases of cholera before the first pandemic began. The first cholera pandemic (1816-1826) was fairly limited in scope and related to the Oman war and the war between Persia and Turkey. About 10,000 British troops and a huge number of Indians died during this period. The second pandemic (1829-1851) is believed to have begun in Russia and Hungary (about 100, 000 deaths), the United Kingdom (more than 55,000 deaths) and 100,000 deaths in France. The epidemic reached New York in the same year and the Pacific coast of North America by 1834. The third Pandemic (1985-1860), mainly hit Russia with over a million deaths. In 1852, the outbreak in Chicago took the lives of 5.5% of its population. The fourth pandemic (1863-1873) spread mostly in Europe and Africa and was followed by the fifth (1881-1896), and the sixth pandemic (1899-1923), which had little effect in Europe because of advances in public health, but had hit major cities in Russia and caused the death of more than half a million people [9]. The seventh pandemic began in 1961 and continued to the present on the six continents [3]. A new V. cholerae strain, or biotype called V. cholerae EI Tor, emerged in Indonesia in 1961 and was responsible for the seventh pandemic (and more recently V. cholera 0139).
According to the World Health Organization (WHO), 45, 159 cases and 3, 488 deaths in ten African nations were reported up to July 1991. Since 2005, the reoccurrence of cholera is linked with the ever-increasing size of the population living in unsanitary conditions. For instance, from August 2008 to February 2009, more than 79, 000 cases and 3,700 deaths were reported from a single country Zimbabwe [3]. Regardless of the advancement of medical science and health care service, cholera remains a global threat to public health and one of the key indicators of social development. While the disease is not an issue in the developed nations where minimum hygiene standards are met, it still remains a threat in developing countries. In 2006, 236,896 cases were reported from 52 nations, including 6, 311 deaths, which is 79% greater than the reported cases in 2005. It is estimated that only a small proportion of cases less than 10% - were reported to WHO. The burden of the disease was therefore highly under estimated because of poor surveillance and under-reporting [6]......

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