Mathematics

# Mathematical Model for the Spread and Control of Tuberculosis Disease

Contents

## 1.1 Background of the Study

Tuberculosis or TB (short for Tubercles Bacillus) is an air borne and highly infectious disease caused by infection with the bacteria mycobacterium tuberculosis. An individual is infected with the disease when he or she inhales the TB germs which are released into the air when infected individuals cough, sneeze, spit or talk.

The first period of infection is the period of Latency when individual exhibits no symptoms of the disease and is not infectious to others. Such an individual is said to have Latent TB infection. ( A.U. Kalu and S.C. Inyama 2012), The Latent period can be extremely variable as a great majority ( 90 %) may live with the disease as long as possible without it progressing to Active TB whereas a small proportion ( 10 %) will progress to Active TB infection, falling ill within months or several years after infection.

The second stage is the period of Active TB infection when the individual start to exhibit some or all the symptoms of TB.

The highest risk group to acquire TB when exposed to it are children under five years of age, persons who are immuno compromised (i.e. have weakened immunity), especially those who are HIV-Positive, persons who have diabetes or kidney failure, people that take excessive alcohol and drugs, those with poor nutrition and lack of food, those suffering from stress and those living in poorly ventilated rooms.

Tuberculosis usually attacks the lungs but can also attack other parts of the body like the kidney, Spine, brain, bones, joints etc. The classic symptoms of TB of the lungs are a chronic cough which may result in blood-tinged sputum, fever, loss of appetite, weight loss and fatigue, Infection of other organs causes a wide range of symptoms. Pneumonia, lung collapse and enlarged lymph nodes may also occur.

Two forms of tuberculosis that become life- threatening are:

1. Miliary TB, which means the bacteria have spread throughout the lungs and into the bloodstream.

2. TB meningitis (infection of the covering of the spinal cord and /or brain by TB bacteria).

Diagnosis relies on radiology (commonly chest X- ray), a tuberculin skin test, blood tests, as well as microscopic examination and microbiological culture of bodily fluids (such as sputum).

The infectiousness of a TB patient is directly related to the number of droplet nuclei carrying M. tuberculosis (tubercle bacilli) that are expelled into the air. Depending on the environment, these tiny particles can remain suspended in the air for several hours. M. tuberculosis is transmitted through the air, notby surface contact. Infection usually occur when a person inhales droplet nuclei containing M. tuberculosis, and the droplet nuclei traverse the mouth or nasal passages, upper respiratory tract, and bronchi to reach the alveoli of the lungs.

Environmental factors that enhance the probability that m. tuberculosis will be transmitted are:

Concentration of infectious bacilli suggest that the more baccili in the air, the more probable that M.tuberculosis will be transmitted.

Space; This is an exposure in small, enclosed space.

Ventilation is inadequate local or general ventilation that results in insufficient dilution or removal of infectious droplet nuclei.

Air circulation is the recirculation of air containing infectious droplet nuclei.

Specimen handling, improper specimen handling procedures generate infectious droplet nuclei.

Air pressure ; this is a positive air pressure in infectious patient’s room that causes M.tuberculosis organisms to flow to other areas.

Figure 1.0: Life cycle of Mycobacterium tuberculosis.

Prevention relies on screening programs and vaccination, usually with Bacillus calmette – Guérin (BCG) vaccine given to infants.

The Directly Observed Treatment Short Course (DOTS) is the internationally recommended strategy for the control and cure for TB. Treatment for tuberculosis uses antibiotics to kill the bacteria. Effective TB treatment is difficult, due to the unusual structure and chemical composition of the mycobacterium cell wall, which makes many antibiotics ineffective and hinders the entry of drugs.

The two antibiotics most commonly used are Rifampicin and Isoniazid.

However, instead of the short course of antibiotics typically used to cure other bacterial infections, TB requires much longer periods of treatment (around 6 to 24 months) to entirely eliminate mycobacterium from the body (Center for Disease Control and Prevention 2000).

TB has remain a global problem despite many decades of study, the wide spread availability of vaccines, an arsenal of anti-microbial drugs as well as a highly visible World Health Organization (WHO) effort to promote a unified global strategy.

The World Health Organization (WHO) declared Tuberculosis (TB) a global emergency in 1993 and it remains one of the world’s major causes of illness and death. One third of the world’s population, two billion people, carries the TB bacteria. More than nine million of these become sick each year with active TB that can be spread to others. TB poses significant challenges to developing economy as it primarily affects people during their most productive years. More than 90% of new TB cases and deaths occur in developing countries. ( Health protection Agency 2000)

Nigeria ranks 10th among the 22 high-burden TB countries in the world, WHO estimates that 210,000 new cases of all forms of TB occurred in the country in 2010. There Are an estimated 320,000 prevalent cases of TB in 2010, equivalent to 199/100,000 cases. The TB burden is compounded by a high prevalence of HIV in the country which stands at about 4.1% in general population; the age groups commonly affected by TB are the most productive age groups, with the 25 – 34 age group accounting for 33.6% (15,303) of the smear positive cases registered in 2010.

This project is a formulation of mathematical model to study the dynamics of TB, One of the principal attribute of these model is that the force of infection (the rate at which susceptible leave the susceptible class and move into the infected category i.e. become infected) is a function of the number of infectious hosts in the population at any time t and is thus a non-linear term. Other transitions such as the recovery of infectious individuals and death are modelled as linear terms with constant coefficients. The model is a deterministic or compartmental, MSLIR- type model where the population is partitioned into 5 components or classes based on the epidemiological state of individuals, and it is assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic.

## 1.2 Aim and Objectives

The aim is to formulate Mathematical model for the spread and control of Tuberculosis Disease.

The objective are;

1. To formulate Mathematical Model the spread and control of Tuberculosis Disease.

2. To establish the existence of Equilibrium State, Disease free Equilibrium and Endemic Equilibrium.

3. To carry out the stability analysis for the disease free equilibrium state.

4. To draw conclusion on the nature of the disease so as to help international institution and policy makers know the most appropriate control measures to be used.

## 1.3 Significance of the Study

The results of this study aim to assist the different agencies and institutions in the development of pragmatic policies and programs on TB. Specifically, the results should serve the following purpose:

1. Help the Department of Health in shaping policies that will increase Case Detection Rates (CDR) among vulnerable population.

2. Guide local government officials in crafting localized policies and programs on TB control and prevention.

3. Help in deciding whether there is need to streamline or enhance existing health promotion efforts designed for vulnerable segments, as the study results would identify key variables that need to be addressed and invested on.

## 1.4 Scope and Limitation of the Study

This project is restricted only to the solutions of spread and control of tuberculosis using mathematical modelling and the limitations include learning and familiarization with modelling in solving physical problems, rapport with the experimenter, and interest in modelling as a tool for solving problems among others.