## 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 CONTENTS

Title page i

Certificate of Approval ii

Declaration iii

Dedication iv

Acknowledgement v

Abstract vi

Table of content vii

**INTRODUCTION**

1.1 The Epidemiology of Cholera 1

1.2 Definition of terms 3

**LITERATURE REVIEW**

2.1 Review of Mathematical Models 6

2.2 SIBR Model 8

**CONTROL MODELS FOR CHOLERA EPIDEMICS**

3.1 Vaccination model 19

3.2 Provision of clean water model 24

3.3 Treatment model 27

3.4 Combined control measure model 35

3.5 Global Stability of the Combined Control Model 36

**NUMERICAL SIMULATIONS**

4.1 Explanation of Simulation 38

4.2 Sensitivity analysis 39

4.3 Numerical Simulation graph 39

**DISCUSSION AND RECOMMENDATIONS**

5.1 Discussion of results 41

5.2 Observations 41

5.3 Recommendation 42

5.4 Suggestion for further work 42

References 43

## 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 10^{2} – 10^{3} 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.

**Figure 1.1:** The life cycle of *V. Cholerae* **[7].**

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].

Cholera can either be transmitted through interaction between human (i.e., fecal – oral), or through interaction between humans and their environment (i.e., ingestion of contaminated water and food from the environment). Some of the recommended controlling mechanisms (by WHO) are providing safe and clean-drinking water (chlorination), intensified promotion to improve the population awareness and practice (like washing hands after defection and before handling food), proper disposal of human excreta and sanitation practices especially in highly populated areas. Once cholera emerges in naive or endemic regions, it requires integrated action from multi-sector to limit its spread. The success of treatment is highly impacted by the speed and the method of treatment. If treatment is quickly and properly administered in cholera dynamics, it will keep the mortality rate below 1%, while untreated cases rise the mortality rate from 50% – 60% [10].

In Nigeria, out breaks of the disease have been occurring with increasing frequency since the first outbreak in modern times in 1970 [19, 21]. The UN humanitarian unit reveals that out of a total of 5,600 cholera cases in Nigeria, 340 cholera deaths were reported. Of this number, 102 cases were known to have drunk street vended water [23] Similar cases were also reported in Gwer West and Apa, LGA of Benue State. According to the UN humanitarian Unit (2005), a source attributed the outbreaks to the fact that many residents in Makurdi depend on the River Benue (which flows through the city) for drinking water [22]. In February, 2005, at least 46 people died of cholera in Kusa village in Oyo State [21]. In July 16, 2008, it was reported that cholera claims six lives leaving 30 others hospitalized in Zaria LGA of Kaduna State [20].

In summary UN humanitarian unit (2006), reports: “despite Nigeria’s oil wealth, more than 70% of the country’s 126 million people live below the poverty line and cholera outbreak are common in poor urban areas which lack proper sanitation and clean drinking water” [22]. Evidently, cholera is endemic in Nigeria. It is against this background that this mathematical model study of the causative agent of this disease with a bias on its control measures is undertaken.

1.2 **Definition of Terms**

Here we present the definition of some keywords in the study

**Equilibrium point.** Consider the following autonomous initial value problem (IVP)

Where A point is said to be an equilibrium point (or stationary point or critical point) of (1.3) if

**Dynamical System**: A dynamical system may be regarded as a process which is changing in time [46]. Examples of dynamical system includes the spread of an epidemic, the motion of the stars in heavens, chemical reactions, variations in the stock market, population growth etc. Mathematically, the study of dynamical system involves deriving mathematical models of such time – dependent processes and then using the models to predict future behaviour [46]. These models often take the form of a differential equation or a difference equation. A dynamical system is made up of two parts: a state vector which exactly describes the state of the system at any given time, and a function which maps the state at one instant of time to the state at a later time. A more precise definition of a dynamical system is given as follows [46]:

Let X represents some state space and Let A function

that satisfies the two properties

is called a dynamical system on *X*.

**Epidemic: **This is a disease temporarily prevalent in a community or throughout a large area. That is, affecting many in a community at once.

**Endemic: **This is the constant presence of a disease or infectious agent within a given geographical area. It may also refer to the usual prevalence of a given disease in the population without the need for external inputs. That is, each person who becomes infected with the disease must pass it on to one other person on average.

**Pandemic: **An epidemic of infectious disease that has spread through human population across a large region is called pandemic; for instance, multiple continents, or even worldwide.

**Susceptible: **A person or animal not possessing sufficient resistance against a particular pathogenic agent to prevent contracting infection or disease when exposed to the agent.

**Infected: **A person or animal that harbors an infectious agent and who has either manifest disease or in apparent infection (i.e., carrier).

**Reproductive Number: **This is the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population. It is denoted as ** R_{0}**. The metric is useful because it helps determine whether or not an infectious disease can spread through a population. When

**, the infection will die out in the long run. But if**

*R*_{0}<1**, the infection will be able to spread in a population.**

*R*_{0}>1**Spectral Radius: **In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by that is the spectral radius of a real matrix A is defined by

In other words, the spectral radius measures the largest magnitude attained by any eigenvalue.

**Mathematical Modeling**

Mathematical model is a simplified representative of certain aspects of a real system, created using mathematical concept such as functions, graphs, diagrams and equations to solve problems in the real world [42]. Other definitions of mathematical modeling have also been given by various authors. For instance, [43] defined mathematical modeling as the art of translating physical problems into tractable mathematical formulations whose theoretical and numerical analysis provides understanding of the real life phenomenon and solution to the problem. Modeling involves identifying and selecting relevant features symbolically, analyzing and reasoning about the model and characteristics of the situation, and considering the accuracy and limitations of the model [44].

A mathematical model can be formulated either through intuitive reasoning about the Phenomenon or from physical law based on evidence from experiment. It is usually constructed in the language of mathematics, logic, and computer following the algebraic rules of syntax. A mathematical model often takes the form of differential equation or system of differential equations. Since our goal is to use the equation to solve specific problems, we are interested in specific rather than general solutions. We obtain this specific solutions by imposing on the equation some initial and boundary conditions [43]. Generally, the success of a model depends on how easily it can be used and how accurate are its predictions [42]. The use of computers has extended modeling by allowing combination of data, interaction, repetition, sound graphics, and other displays for various types of mathematical modeling. However, models usually have a limited range of validity and should not be applied outside this range [45]. According to [44] a mathematical model is a set of equations, which are the mathematical translation of hypothesis (or assumptions) when interpreting model prediction. It is thus important to bear in mind the underlying assumptions. An assumption, by our definition is an unverified proposition, tentatively accepted to explain certain facts or to provide a basis for further investigation.

Modeling has been applied in virtually every sphere of man’s existence and it is as wide as nature itself [43]. The list is not exhaustible. Modeling has been used to solve problems of robotics in the area of Artificial intelligence; detection of planetary systems in Astronomy; population dynamics and spread of infectious disease in Biology; planning of production units in Chemical Engineering; stability of electric circuits, microchip analysis and power supply network optimization in Electric Engineering; prediction of oil or ore deposits and earthquake in Geosciences; stability of structures and structural optimization in Civil Engineering; and so on. There is hardly any problem that cannot be modeled mathematically if one is versed in modeling [43]. Thus, our goal/aim is to use mathematical modeling in studying dynamics and control spread of cholera epidemics, in order to understand the cholera dynamics better and to predict the occurrence of some hazards so that effective preventive measures can be taken.

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