# Modeling The Effects Of Carriers On Transmission Dynamics Of Infectious Diseases

Modeling The Effects Of Carriers On Transmission Dynamics Of Infectious Diseases

## Abstract

An S-Ic-I-R epidemic model is investigated for infectious diseases that can be transmitted through carriers, infected individuals who are conta-gious but do not show any disease symptoms. Mathematical analysis is carried out that completely determines the global dynamics of the model. The impacts of disease carriers on the transmission dynamics are discussed through the basic reproduction number and through numerical simulations.

## Chapter One

### 1.0 Introduction

#### 1.1 Background of Study

For certain infectious diseases, there are individuals who are able to transmit their illness but do not exhibit any symptoms. These individuals are called “carriers” and they play an important role in the transmission of the disease. There are two types of carriers. Genetic carriers carry the illness on their recessive genes. They can only pass on their disease to their children and are not contagious. The focus of our study is on infectious disease carriers. These individuals are asymptomatic and are likely unaware of their conditions, and therefore are more likely to infect others. An infectious disease that produces long-term asymptotic carriers is the Typhoid fever caused by the bacteria Salmonella Typhi. Typhoid fever reached public notoriety at the beginning of the 20th century with the cases of “Mr. N the milker” in England and Typhoid Mary in the US. These individuals infected hundreds of people over the decades while they worked in the food production industry and private homes. Even today, Typhoid fever infects 21 million people and kills 200,000 worldwide every year. Asymptomatic carriers are believed to play an essential role in the evolution and global transmission of Typhi, and their presence greatly hinders the eradication of Typhoid fever using treatment and vaccination.

Another major infectious disease that causes long-term asymptomatic carriage is hepatitis B, a liver disease caused by the HBV virus of the Hepadnavirus family. Most people infected with HBV recover completely and develop a lifelong immunity to the virus. However, about 5-10% of adults will develop chronic HBV infection, and 15-25% of these will develop liver disease. Hepatitis B’s symptoms include jaundice, abdominal pain, nausea, fatigue and joint pain. About 30% of people with the disease do not show any of these symptoms. A major public-health challenge in the control of hepatitis B infection in many countries is the existence of a large pool of chronic carriers who are responsible for transmitting most of the new infections. Infections of other pathogens are also know to produce asymptomatic carriers. The Epstein-Barr Virus (EBV) of the herpes family is one of the most common viruses in humans. EBV infection commonly causes infectious mononucleosis, also known as glandular fever. Most people infected with EBV are asymptomatic, as it remains dormant in those who have had it for the rest of their lives in the cells of the throat and the immune system. Clostridium difficile is a bacterium that causes Clostridium difficile-associated diseases (CDAD). CDAD remains the most common cause of acute hospital-acquired diarrhea, responsible for more than 300,000 cases of diarrhea annually in acute-care facilities in the United States. Asymptomatic carriage rates of up to 30% have been reported in long-term care facilities. It is believed that carriers are responsible for transmission and large outbreaks of CDAD in Europe and North America.

Despite their public health significance, the effects of carriers on the transmission dynamics of the disease have not received adequate research attention in the mathematical modeling literature. One of the earlier attempts was Kempe, in which a general mathematical model that incorporates disease carriers was devel- oped and analyzed. Medley et al. used a mathematical model for hepatitis B with carriers to discuss the effects of HBV vaccination. Several other studies using large-scale computational models with carriers are specifically aimed at hepatitis B and other diseases

#### 1.2 Statement of Problem

What really instigated the study was the prevalence of highly infectious diseases in Nigeria. the growth of diseases in Nigeria and most African countries has increased overtime. It is evident that researchers have put in so much effort to come out with model on the effect of carriers on transmission dynamics of infectious diseases but some prove abortive.

#### 1.3 Aims and Objectives of Study

The main aim of the research work is to examine the modeling of the effect of carriers on transmission dynamics of infectious diseases. Other specific aims of the research work are:

1. To investigate the effects of carriers on the transmission dynamics
2. To determine the basic reproduction number and through numerical simulations
3. To investigate on the factors affecting the model
4. To proffer solution to the above model

#### 1.4 Research Questions

The study came up with research questions so as to be able to ascertain the objectives of the study. The research questions are stated below as follows:

1. What are the effects of carriers on the transmission dynamics?
2. How to determine the basic reproduction number and through numerical simulations?
3. What are the factors affecting the model?

#### 1.5 Significance of Study

The study on the modeling the effects of carriers on transmission dynamics of infectious diseases will be of immense benefit the ministry of health, hospitals, the state government and other researcher that wishes to carryout similar research as the study will be able to discuss the effects of carriers on the transmission dynamics and the factors affecting the model

#### 1.6 Scope of Study

The study on modeling the effects of carriers on transmission dynamics of infectious diseases is limited to the modeling of selected infectious diseases like typhoid; how they are transmitted and the model for their transmission.

#### 1.7 Definition of Terms

##### Diseases:

Is a particular abnormal condition, a disorder of a structure or function that affects part or all of an organism. The study of disease is called pathology which includes the causal study of etiology

##### Carriers:

Individual harboring specific pathogenic organisms who, though often immune to the agent harbored, may transmit the disease to others

##### Infectious:

Liable to be transmitted to people, organisms, etc. through the environment.

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