Modelling and Simulation of the Spread of HBV Disease With Infectious Latent
Chapter One
AIMS AND OBJECTIVES OF STUDY
The main aim of the research work is evaluate the modeling 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
CHAPTER TWO:
REVIEW OF RELATED LITERATURE
INTRODUCTION
This chapter gives an insight into various studies conducted by outstanding researchers, as well as explained terminologies with regards to the modeling and simulating of the spread of HBV disease with infectious latent.
The chapter also gives a resume of the history and present status of the problem delineated by a concise review of previous studies into closely related problems
HERPATITIS
“Hepatitis” means inflammation of the liver. The liver is a vital organ that processes nutrients, filters the blood, and fights infections. When the liver is inflamed or damaged, its function can be affected. Heavy alcohol use, toxins, some medications, and certain medical conditions can cause hepatitis. However, hepatitis is most often caused by a virus. In the United States, the most common types of viral hepatitis are Hepatitis A, Hepatitis B, and Hepatitis C
HERPATITIS B VIRUS
Hepatitis B can be a serious liver disease that results from infection with the Hepatitis B virus. Acute Hepatitis B refers to a short-term infection that occurs within the first 6 months after someone is infected with the virus. The infection can range in severity from a mild illness with few or no symptoms to a serious condition requiring hospitalization. Some people, especially adults, are able to clear, or get rid of, the virus without treatment. People who clear the virus become immune and cannot get infected with the Hepatitis B virus again. Chronic Hepatitis B refers to a lifelong infection with the Hepatitis B virus. The likelihood that a person develops a chronic infection depends on the age at which someone becomes infected. Up to 90% of infants infected with the Hepatitis B virus will develop a chronic infection. In contrast, about 5% of adults will develop chronic Hepatitis B. Over time, chronic Hepatitis B can cause serious health problems, including liver damage, cirrhosis, liver cancer, and even death.
CHAPTER THREE
RESEARCH METHODOLOGY
INTRODUCTION
This chapter is designed to describe the procedures adopted in this research. The procedures involve the following:
Formulation of the model, the existing model, the assumption of the existing model, variables and parameters of the existing model, equation of existing model and the extended model.
FORMULATION OF THE MODEL EQUATIONS
The Existing Model
We begin our model formulation by introducing the model by Zou et al (2009). We, first, present the parameters and assumptions of the existing model
Assumptions of the Existing model
The following are the assumptions of the existing model by Zou et al (2009):
- The population is compartmentalized into the proportions of susceptible individuals, latent individuals L(t) acutely infected individuals (I(t) chronic carriers C(t) vaccinated individuals V(t) and the recovered individuals R(t) all at time
- The population is homogeneous
- Influx into the population is by birth only,
- exit out of the population is by natural and HBV-related mortality only,
- The vaccinated individuals do not acquire permanent immunity,
- the newborns to carrier mothers infected at birth proceed to carrier state immediately
CHAPTER FOUR
SIMULATION RESULTS AND DISCUSSION
In this chapter we study numerically the behaviour of the system. The system of linear ordinary differential Equations (1)-(4) is been solved numerically by using the software package XPPAUTO and using the following parameter set from the literature, (b = 0.015, b = 0.000025,s = 6,g = 4). Also we simulate our system for two different states one if R0 < 1 and the other one when R0 > 1 we found that, disease has a thre-shold level Pc for the reproductive number R0 to be under one in value which the disease to die out. If the vaccination value p is not sufficient then R0 stays above one in value and the disease becomes endemic.
CHAPTER FIVE
SUMMARY AND CONCLUSION
This paper research work investigates the effect of using another way of producing new cases. This way is the fact that latent persons can pass the disease into susceptibles. Also, vaccination of all newborns, at a constant rate, has been considered. It is documented that vaccination strategies are applied worldwide to vaccinate children in the early ages. For example, in China an effective vaccination program has been established for newborn babies since the 1990s, which has reduced chronic HBV infection in children. Unfortunately, the incidence of hepatitis B is still increasing. This means that the vaccinated proportion is large enough to force the reproduction number to be less than one in value. Therefore, to control HBV infection vaccination, strategies need a treatment scheme as another leg to have a better control strategy for the disease. The first result of this paper comes from the stability analysis of the DFE of our model. We find that the DFE is locally asymptotically stable when R0, the basic reproduction number, is less than one. If R0 exceeds one, then the DFE point is unstable. When 0 R > 1 , there exists another equilibrium point which is the endemic point P SEIR ( , ,, ) ∗ ∗ ∗∗ ∗ ≡ . We deduced that if 0 R > 1 , then P SEIR ( , ,, ) ∗ ∗ ∗∗ ∗ ≡ is globally asymptotically stable for the system (1)-(4). We used Liapunov’s direct methods to prove this result. Simulation results of our model have been conducted for HBV parameter set using different vaccination parameter values. From these results, we find that there is a critical ratio Pc = 96% approximately, from which all the newborns must be vaccinated. This value is the sufficient condition to reduce susceptible number to be less than a critical value SC. This forces the basic reproduction number R0 to be less than one in value and the disease dies out.
REFERENCES
- Abdulrahman, S., Akinwande, N.I., Awojoyogbe, O.B. and Abubakar, U.Y. (2013) Sensitivity Analysis of the Parameters of a Mathematical Model of Hepatitis B Virus Transmission. Universal Journal of Applied Mathematics, 1, 230-241. [2]
- Saher, F., Rahman, K., Quresh, J.A., Irshad, M. and Iqbal, H.M. (2012) Investigation of an Inflammatory Viral Disease HBV in Cardiac Patients through Polymerase Chain Reaction. Advances in Bioscience and Biotechnology, 3, 417-422. http://dx.doi.org/10.4236/abb.2012.324059 [3]
- Centers for Disease Control and Prevention (2012) http://www.cdc.gov/hepatitis/HBV [4]
- Moneim, I.A., Al-Ahmed, M. and Mosa, G.A. (2009) Stochastic and Monte Carlo Simulation for the Spread of the Hepatitis B. Australian Journal of Basic and Applied Sciences, 3, 1607-1615. [5]
- Li, G. and Jin, Z. (2005) Global Stability of an SEI Epidemic Model with General Contact Rate. Chaos, Solitons and Fractals, 23, 997-1004.
- Li, G. and Jin, Z. (2005) Global Stability of a SEIR Epidemic Model with Infectious Force in Latent, Infected and Immune Period. Chaos, Solitons and Fractals, 25, 1177-1184. http://dx.doi.org/10.1016/j.chaos.2004.11.062
- [7] Li, G., Wang, W. and Jin, Z. (2006) Global Stability of an SEIR
