Mathematics

# Modelling and Simulation of the Spread of HBV Disease with Infectious Latent

## MODELLING AND SIMULATION OF THE SPREAD OF HBV DISEASE WITH INFECTIOUS LATENT

CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF STUDY

The spread of the HBV in Nigeria has posed a lot of threat to health and well being citizens in Nigeria. It is evident that about a third of the world’s population, approximately 2 billion people gets infected with hepatitis B virus in their life time. About 360 million people remain chronically infected carriers of the disease, most of whom are unaware of their HBV status and about 20% – 30% of whom will eventually die from chronic sequel. The prevalence of HBV infection varies from country to country, depending upon a complex behavioral, environmental and host factors. Chronic HBV can lead to hepatocellular carcinoma after 20 years among persons with chronic HBV infection; the risk for premature death from cirrhosis or hepatocellular carcinoma is 15% – 25%.

Hepatitis B is a disease that is characterized by inflammation of the liver and is caused by infection by the hepatitis B virus. According to (WHO, 2002) stated that hepatitis may be caused by drugs or viral agents; these viral agents include the hepatitis A, B, C, D, E, F, G and H viruses. Hepatitis B is one of the world’s most serious health problems. More than a billion people around the world have serological indicators of past or present infection with hepatitis B virus (HBV).

According to (White and Fenner (1994), Platkov et al (2001), Carriapa et al (2004), Fernandez et al (2006), Onuzulike and Ogueri (2007)) in their research stated that Over 300 million people are chronic carriers of the virus. The fast spread of HBV shows that is very communicable.

It is evident according to (WHO, 2002) that HBV infection can be transmitted from mother to child (vertical), contact with an infected person (horizontal transmission), sexual contact (homosexual and heterosexual transmission) with infected partners, exposure to blood or other infected fluids and contact with HBV contaminated instruments.

HBV control measures include vaccination, education, screening of blood and blood products; and treatment (CDC, 2005).

According to (Anderson and May, 1991) stated that epidemiological models help to capture infection or disease transmission mechanisms in a population in a mathematical frame-work in order to predict the behavior of the disease spread through the population.

1.2 STATEMENT OF PROBLEM

What really instigated the study was the massive spread of HBV in Nigeria and most of the African countries. Several efforts has been put in place by the federal government of Nigeria and world health organization (WHO) through the ministry of health in Nigeria to combat HBV.

Secondly mathematicians all over the world have come with up with several model to help solve the model and simulate the spread of HBV; there have been a lot of failed model.

1.3 AIMS AND OBJECTIVES OF STUDY

The main aim of the research work is evaluate the modelling and simulation of the spread of HBV with infectious latent. Other specific objectives of the study include:

To find the existence and uniqueness of the solution to the model.

To carry out sensitivity analysis on Ro to ascertain which parameter that is most sensitive and that should be targeted by way of intervention.

To examine the local stability of the model equation using the modified implicit function theorem.

1.4 RESEARCH QUESTION

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

How to find the existence and uniqueness of the solution to the model?

How to carry out sensitivity analysis on Ro to ascertain which parameter that is most sensitive and that should be targeted by way of intervention?

How to perform the local stability of the model equation using the modified implicit function theorem?

1.5 SIGNIFICANT OF STUDY

The study on modelling and simulating of the spread of HBV disease with infectious latent will be of immense benefit to the ministry of health of Nigeria, the World health organization (WHO) and other researchers that wishes to carryout similar research on the above topic as it will discuss the local stability of the model equation using the modified implicit function theorem and also sensitivity analysis on Ro to ascertain which parameter that is most sensitive and that should be targeted by way of intervention.

1.6 SCOPE OF STUDY

The study on modelling and simulation of the spread of HBV disease with infectious latent will cover the areas of local stability of the model equation and implicit function theorem

1.7 DEFINITION OF TERMS

HBV: Hepatitis B virus is a viral infection that attacks the liver and can cause both acute and chronic disease. The virus is transmitted through contact with the blood or other body fluids of an infected person.

REFERENCES

Abraham, O. J. (2004). Sero-prevalence of hepatitis B virus infection in South-West Nigeria. M.Sc. Thesis, Department of Virology, University of Ibadan.

Alfonseca, M., Martinez-Brawo, M.T and Torrea, J. L. (2000). Mathematical models for the analysis of hepatitis B and AIDS epidemics, Simulation Council Inc. USA.

Ameh, E.J.(2009). The basic reproduction number: Bifurcation and Stability, PGD project at AIMS.

Anderson, R.M. and May, R.M. (1991). Infectious diseases of humans: Dynamics and Control. Oxford University Press.

Anderson, R.M. and May, R.M.(1992). Directly transmitted infectious diseases: Control by vaccination, Science, Vol. 215, pp 1053-1060.

Carriappa, M.M., Jayaram, B.J., Bhalwar, C.R., Praharaj, A., Mehta, V and Kpur, L. (2004). Epidemiological differentials of hepatitis B carrier state in the army: a community-based sero-epidemiological study MJAFI, vol. 60, no.3. CDC (2005). Centre for prevention and control of diseases. CDC (2006). Centre for prevention and control of diseases.

Castillo-Chavez, C. Feng, C. and Huang, W. (2002). On the computation of R0 and its role on global stability. J. Math. Biol. 35:1-22.

Diekmann, O., Heesterbeek, J. A. P. and Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28: 365-382.

Edmunds, W.J., Medley, G.F., Nokes, D.J., Hall, A.J., and Whittle, H.C. (1993). The influence of age on the development of the hepatitis B carrier state. Proc. R. Soc. London. B 253, 197-201.