Modelling And Simulation Of The Spread Of HBV Disease With Infectious Latent
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 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
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 modeling 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 modeling 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
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.
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.
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