DETERMINISTIC SIRB CHOLERA MODEL WITH LOGISTIC APPROACH IN DORMAA-AHENKRO, GHANA

CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
Cholera has long been, and continues to be, a world health problem. It is a water-borne disease caused by the pathogenic bacterium, Vibrio cholerae. It causes a significant suffering and death. Cholera has great effects on public health, social and economic development. Cholera outbreaks cause loss of lives and hinder social and economic development of a country. As a result of this it is studied by many people.

Cholera was first identified in ancient Greece around 470-400 B.C. by Hippocrates and in the Sushrute Samhita, India around 500-400 B.C. Since then seven cholera outbreaks have occured during the 19th and early 20th centuries worldwide (Hays, 2005). The seventh cholera outbreak worldwide began in 1961 and it affected most of the southern hemisphere (WHO, 2001). It continues affecting countries in the southern hemisphere today (Hays, 2005).

John Snow’s investigations on cholera outbreaks in London in the 1800s was significant in establishing the germ theory of disease (Yamai et al., 1997; Faruque et al., 1998; Ramamurthy et at., 1993; Felsenfeld, 1996; Glass et al., 1985). Still, severe cholera outbreaks continue to occur. In recent years, there has been a significant trend in cholera outbreaks especially in developing countries, for example cholera outbreak in Haiti in 2010 (WHO, 2010). A large number of people die from cholera annually in the world. More than 3.5 million people are reported from cholera worldwide every year (WHO, 2010). More than 90% of the world’s reported cholera cases occur in Africa (Goodgame and Greenough, 1975). Since 1998, cholera outbreaks have been reported consistently from Benin, Ghana, Guinea, and Togo (http://www.cdc.gov_global/cholera). In 1999, the southern African bloc countries accounted for 51% of all cholera cases in Africa and 40% of the resulting deaths (http://en.who.int/cholera/cases).

In 2013, 43% of cases were reported from Africa whereas between 2001-09, 93% to 98% of total cases worldwide were reported from the continent. This proportion changed in 2010 due to the outbreak in the Haiti and Dominican Republic (WHO, 2010). Africa has seen a decrease in the number of reported cases of cholera worldwide since 2012 (en.who.int/cholera/cases).

Cholera reached West Africa and Ghana during the seventh pandemic (WHO, 2001). Cholera has been endemic in Ghana since its introduction in the 1970’s (Goodgame and Greenough, 1975). The prevalence of cholera in Ghana dates back to September 1, 1970 when a Togolese in transit at the Kotoka International Airport from Conakry, Guinea, collapsed and was found to have cholera. However, it was not until much later when cholera took root in Ghana. Some Ghanaians went for fishing in waters of Togo, Liberia and Guinea, and one of the fishermen died. His corpse was smuggled to his home town for burial. Since then cholera began to spread along the shores of Ghana and swept through many coastal villages in epidemic proportions. The disease kept on spreading and by July 1971, the Ashanti Region began to report cases indicting that cholera were spreading across the country. Cholera then became endemic in most parts of Ghana over the years with the coastal regions of Ghana: Greater Accra, Central and Western regions boasting of high cases (WHO, 2011).

There have been several major cholera outbreaks over the past three decades (WHO, 2011). In 1982, as many as 15,032 cases were recorded (www.ghanaweb.com/health/artikel.php). Since 1998, cholera outbreaks have been reported consistently in Ghana (Griffith, Kelly-Hope and Miller, 2005).

All regions in Ghana except the Northern Region have recorded cases of cholera outbreaks (www.afro.who.int/en/ghana/press). There have been cases of cholera outbreaks in Ghana in recent times.

The current ongoing outbreak which started in early June 2014 in the Accra Metropolis with 6 cases and no death has rapidly escalated reaching alarming magnitude of 16,527 cases including 128 deaths (Case fatality rate of 0.8%) as of 14 September 2014 (www.afro.who.int/en/ghana/press). The epidemic has spread to 91 districts in 8 out of the 10 regions in the country. No cases reported from the Upper West and Northern Regions of Ghana (www.afro.who.int/en/ghana/press). The Greater Accra is the most affected region of the current cholera outbreak in Ghana (Oteng-Ababio, 2014).

In Accra, cholera cases come from overcrowded peri-urban ghettos with poor infrastructures and hygiene conditions, inadequate environmental management and poverty (like Agbogbloshie, James Town and Glife slums) (www.afro.who.int/en/ghana/press).

A total of 304 new cholera cases have been reported in the first 16 days of 2015 in six districts in the Greater Accra and Volta regions of the country (www.afro.who.int/en/ghana/press).The situation is more alarming in the capital, Accra, which has a high infection rate and has claimed many lives (www.theafricareport.com/West-Africa/ghana-cholera-kills-hundreds.html).

The main market centres and slums in Accra are close to refuse dumps, filthy and open gutters with stinking stagnant water which are very dirty but many people queue to buy food sold at these areas. Cases of cholera outbreaks will be persistent because of poor sanitary conditions currently in Accra.

1.1.1 Causes of Cholera
Cholera is a living testimony of poor sanitary conditions. It is a severe water-borne infectious disease caused by the bacterium Vibrio cholerae (Ryan, 2004; WHO, 2010). Vibrio cholerae is a large and very diverse species. It is divided into about 200 sero-groups, of which only serotypes denoted O1 and O139 contain pathogenic members (Alexander, 2008). Cholera has short incubation period, from less than one day to five days. It is characterized by severe watery diarrhoea caused by the production of cholera toxin (an enterotoxin) by Vibrio cholerae bacteria in the small intestine. It is caused by eating food or drinking water contaminated with Vibrio cholerae (Kaper et al., 1995). The infection occurs in two ways: first, through the introduction of a small, but not very small, number of infected individuals into the population; second, through small, but not very small, fluctuations in the pathogen density in a reservoir (water source).

When an individual takes in food or water containing the bacterium, it releases a toxin in the intestines which causes severe diarrhoea. It has been shown that pathogenic Vibrio cholerae can survive refrigeration and freezing in food supplies (Desmarchelier, 1997; Greenough, 1999).

Vibrio Cholerae bacteria live in, and are transmitted by contact with contaminated water or food. Environmental and climatic conditions, such as water and temperature have:

* direct impact on the abundance and/or toxicity of Vibrio cholerae.

* indirect impact on other aquatic organisms such as zooplankton, phytoplankton and macrophytes to which we find Vibrio cholerae attached. Phytoplankton blooms have a strong effect on the development of zooplankton blooms; they both have impact on the life cycle of Vibrio cholerae (Codeço, 2001)

The ability of the bacteria to connect to and live inside aquatic organisms enables them to survive in harsher environments (Codeço, 2001). War also contributes to the disease’s ability to invade communities (Faruque et al., 1998).

1.1.2 Symptoms of Cholera
Most people do not get infected with cholera when exposed to Vibrio cholerae, yet they can still infect other susceptible individuals via contaminated water because they shed the pathogen in their stool for 7 to14 days. People begin to show symptoms as soon as few hours or as long as five days after infection. Most people exposed often show mild symptoms or are asymptomatic, but sometimes symptoms are grave (Akor, 2007).

About one in every 20 infected individuals develops severe diarrhoea accompanied by vomiting, which can quickly result in dehydration.

Severe diarrhoea. Cholera causes dangerous fluid loss-as much as 950 cm3 an hour. The diarrhoea is pale, and milky in appearance which is similar to rice-
water stool.

Nausea and vomiting. Take place in the early and later stages. Vomiting may persist for a number of hours.

Dehydration. The following are the signs and symptoms of dehydration: irritability, lethargy, sunken eyes, dry mucous membranes, especially inside of the mouth, throat, nose, and eyelids, extreme thirst, loss of skin elasticity, little or no urine output, low blood pressure, rapid heartbeat and muscle cramps.

Dehydration can result in shock and death if it remains untreated in a matter of hours. Children develop the following symptoms together with the usual symptoms of cholera: extreme drowsiness or even coma, fever and convulsions (en.wikipaedia.org/wiki/cholera_symptoms).

Diagnosis is by finding the bacteria in the stools of people (WHO, 2010). A rapid dipstick test is used to determine the presence of Vibrio cholerae (Sack and Chaignat, 2006). Effective sanitation practices, if instituted and adhered to in time, are usually sufficient to stop an epidemic.

1.2 Statement of the Problem
Cholera outbreaks have serious negative effects on public health and social and economic development. It is therefore important to understand how the bacterium which causes cholera survives in an aquatic environment, what the transmission dynamics are, and how this affects cholera as a human disease.

Thus, we formulate a deterministic cholera model to study the spread of cholera in Dormaa-Ahenkro. However, our attention will be focused on indirect SIR (SIRB) differential equation models. We make use of some parameter values from Dormaa-Ahenkro to help us to study the propagation of cholera.

1.3 Objectives of the Study
The following are the objectives of the study:
To formulate a deterministic SIR differential equations model for the spread of cholera.

To determine the stability of the equilibrium points of the model.

To perform numerical simulations of scenarios of the model

To use these model scenarios to determine how to control the spread of cholera

1.4 Methodology
Mathematical and computer methods will be used for the study. We will also make use of ordinary differential equations and Matlab software to determine of the solutions and stability analysis of the differential equations.

1.4.1 Parameter Values for the Model
Parameter values such as human death rate and recovery rate are obtained from the Dormaa Presbyterian Hospital, Ghana Health Service and CIA World Fact Book. These (Dormaa Presbyterian Hospital and Ghana Health Service) are the two main health facilities in Dormaa-Ahenkro. We also use parameter values, such as Vibrio cholerae growth and death rates, shedding rate of pathogens back to the reservoir, semi-saturation concentration and contact rate from Cash et al., 1974, Codeço, 2001, Hartley et al., 206 and Grad et al, 2012 for the study. The population size of Dormaa-Ahenkro is used in the study.

1.4.2 Mathematical Methods
Here, we formulate the deterministic SIRB cholera model and then determine the critical points and hence the stability of these points. We then determine its local stability and by handling plane phase diagrams. We fit parameter values such as death rate, infection rate, recovery or removal rate, Vibrio cholerae growth-loss rate and the rate of exposure to contaminated water from the Ghana Health Service, CIA World Factbook (demographic statistics) and other sources into our model.

1.4.3 Computer Methods
Computer methods serve as important tools for the analysis of mathematical models; hence they will be used for our analysis. We will use Matlab software for the analysis of the model for:

* determining the eigenvalues of the equivalent linearized system of the model.

* identifying the type of the stability of the stationary points

* plotting the phase planes and the trajectories of the eigenvalues of the steady states or stationary points.

* the numerical simulations of the deterministic SIRB cholera epidemic model

1.5 Justification of the Study
Cholera outbreaks have tremendous effects on public health and social and economic development. We therefore propose a model to study its transmission dynamism.

We will employ mathematical modelling for the study and predict the dynamics of cholera in order to control it.

1.6 Organization of the Thesis
The thesis is divided into five chapters. Chapter one focuses on the introduction, background to the study, the statement of the problem, methodology, justification of the study and how the thesis is organized. Chapter two reviews the research work that has been carried out on models of cholera. Chapter three is about the methodology used for the study. Here, we review some first order systems of ordinary differential equations and phase portraits of these systems are presented. Also, the deterministic cholera model with the reservoir mediated SIR model called SIRB model is presented. We will derive the basic reproduction number and the critical population size. In chapter four, we will analyze the results by using numerical simulation of the model. All in all, chapter five talks about the conclusion of the thesis and recommendations made for the control of cholera in Dormaa-Ahenkro, Ghana.

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Item Type: Ghanaian Postgraduate Material  |  Attribute: 128 pages  |  Chapters: 1-5
Format: MS Word  |  Price: GH50  |  Delivery: Within 30Mins.
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