Modelling on Effects of Carriers on Transmission Dynamic of Infectious Disease on Hepatitis B

Modelling on Effects of Carriers on Transmission Dynamic of Infectious Disease on Hepatitis B

CHAPTER ONE

Objective

The general objective: assess the knowledge, attitude and practices concerning hepatitis B (HBV) among rural and urban adolescents in Nigeria using SVEIRS model .

The specific objectives of the study are:

To formulate a modified constant population SVEIRS model for hepatitis B in Nigeria
To determine the spread of the disease in Nigeria
To perform the stability analysis of the equilibrium states of the SVEIRS model
To identify the behavior of the
To compare uncontrolled and controlled state strategies on the dynamics of the
Perform simulations to illustrate analytical
Using pplane 8 we can analyze the phase portrait of the differential
Determine the general behavior of the disease using pplane8

CHAPTER TWO

LITERATURE REVIEW

This chapter reviews the work of other researchers related to the objectives of this thesis. Some of the related works are as follows;

Grob and Esteban (1995) stated that HBV may be transmitted horizontally and vertically. Horizontal transmission occurs during adolescence or childhood, throughout sexual exposure, needle stick (both accidental or through intravenous drug use), and blood transfusion. Therefore, any person with a bad history of sexually transmitted diseases (STDs), multiple sexual partners or an injecting drug user stands a higher chance of being infected with HBV (CDC, 2002). Exposure to blood is also means of open wounds in households and other close contacts and multiple transfusions in hemophiliacs. This view of exposure to risk was also shared who argued that most of the infections occur among adolescents and young adults due to exposure to high risk activities they engage in at this stage of life.

A vertical transmission occurs when an infected mother transmits the virus directly to the neonatal during child birth. Such transmissions are usually possible when the expectant mother suffers an acute infection of hepatitis B during pregnancy or if she is a chronic carrier during that period. The mode of this vertical transmission is not clear cut, but indications are that, infection might occur through a placenta cutting during childbirth. Majority of countries in Southeast Asia, the Western Pacific and Africa have high endemicity of HBV. In these settings the major mode of HBV transmission has been identified as vertical, where by mothers directly trans- mit virus to their infants during prenatal periods or where infected siblings, play- mates,other members of different households transmit the virus to their younger ones. A cross-sectional study clarified that without prophylaxis, an estimated num- ber of 6000 infants born to carrier mothers each year in the USA would develop chronic HBV infection as a consequence of prenatal transmission.

Unsafe blood transfusion has been a major force in the transmission of HBV glob- ally (Wang & Wong, 1960). The enactment of a law for the donation and management of blood in blood banks across the world has aggressively fought this channel of HBV transmission. This notwithstanding, current researches have showed that blood transfusion is regaining its position as one of the major risk factors for HBV transmission globally. This finding is attributed to the presence of occult HBV in- fection (OHBVI) among blood donors (Shang et al, 2007). It is also worth mention- ing that the global acceptance of the auto-disposable syringes (ADS) has consider- ably reduced the incidence of HBV infections that occur due to unsafe injections. Also, as a result of the extensive use of invasive medical procedures, iatrogenic HBV infections are no longer frequent. There have also been speculations that den- tal care operations which are capable of causing oral mucous membrane injuries is becoming a major route to HBV transmission if steps are not taken to prevent it.

Gashu Afework from Abuja University,(March, 2015) stated that Occupa- tional exposure occurs frequently among health care workers (HCWs). The most serious occupational health hazard faced by HCWs worldwide is exposure to hep- atitis B virus (HBV). Having enough Knowledge and proper attitudes toward this infection is crucial in prevention of occupational hepatitis infection. Therefore this study was conducted to assess knowledge, atti-tude and practice of HBV and its vaccination among health care professionals (HCPs) in selected public hospitals of Abuja, Nigeria. Both qualitative and quantitative methods were used. Qual- itative study involved key informants interviews whereas quantitative method was carried out by using structured self-administered questionnaire. This study revealed that HCPs’ knowledge regarding occupational exposure of HBV is found to be good as majority of our par-ticipants correctly answered most of the question pertaining to it. Relatively larger proportion, 103 (44.8%) of the respondents were found to have neutral attitude score. One hundred seventy eight (77.7%) of them had been screened for and received hepatitis B Vaccine, and one hundred sixty (94.1%) of them had received full course of the vaccine. Knowledge score was significantly associated with: education level, type of profession and area of practice; whereas profession type and history of training were the only predictors of vaccination status. In conclusion, majority of the survey participants had moderate to high knowledge score, however, larger percentage of HCPs had negative or neutral attitudes on HBV and its vaccination. In spite of this, vaccination status was encouraging. The gaps identified in some areas of knowledge, attitude and practice of HBV call for concern among all stakeholders since HCPs have a high risk of being infected with HBV owing to their high frequency of exposure to infectious fluids coupled with the high infectivity of HBV.

CHAPTER THREE

THE MODEL

Introduction

Hepatitis B models are either deterministic or stochastic. Deterministic models have a finite number of states, and specify rules by which individuals move from one state to another through a series of differential equations.

In this chapter, we are going to formulate our SVEIRS model based on the deterministic approach and develop expressions for the equilibrium points. Expressions which will be used to test for the stability of these steady states will also be devel- oped, as well as the formula that will be used to calculate for the basic reproductive number [4].

Model Formulation Preliminaries

Differential equations have been developed as mathematical models to study the dynamics of disease transmission for many communicable diseases.

When a community identifies people infected with hepatitis B, new individuals get infected by coming in contact with members of the infected population. In epidemics, it is of high interest to know how the disease will spread. Thus, what we really want to know in many cases is how many infected individuals there will be in the next period. In this chapter, a mathematical model will be developed to study the epidemiology of hepatitis in the central region of Nigeria. The specific model to be developed is the Susceptible-Vaccinated-Exposed-Infected-Recovered (SVEIRS) model.

CHAPTER FOUR

OPTIMAL CONTROL

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal con- trol is a set of differential equations describing the paths of the control variables that minimize the cost function. The optimal control can be derived using Pon- tryagin’s maximum principle (a necessary condition also known as Pontryagin’s minimum principle or simply Pontryagin’s Principle) or by solving the Hamilton- Jacobi-Bellman equation (a sufficient condition) [25].

Introduction to Optimal Control

Optimization is the process in which the best feasible solution for a problem is found. This usually entails finding either a maximum or minimum, which are called extrema, of the possible solutions. This can be done in various ways, though the most common involves using some version of the derivative of the function numerical solution of optimal control was difficult using by hand computation. Numerical mathematics is study of quantitative approximations to the solutions of mathematical problems including consideration of and bounds to the errors involved. Optimal control theory is no exception to this rule. The purpose here is to implement three different numerical algorithms in MATLAB to approximate the solution to an optimal control problem. Once the methods are developed, the concept of convergence for each method will be discussed as well as any flaws or problems with each specific method.

CHAPTER FIVE

MODEL APPLICATION

Introduction

In this chapter, we are going to estimate parameter values that will be used for the analysis. The values will be substituted into equation obtained in chapter three to get the exact value for the basic reproductive number. Sensitivity analysis will also be carried out on the parameter values to investigate their impact on study results.

CHAPTER SIX

CONCLUSION AND RECOMMENDATIONS

Conclusions

This model focused on HBV transmission through the deterministic approach. The basic reproductive number, R₀ of our SVEIRS model was calculated to be 0.6463, which implies that on average, each infectious individual transmits virus to 0.6463 people. The value of R₀ indicted the model is unstable or the disease die out. The stability analysis of the endemic equilibrium has been found to be stable; hence, the transmission rate of HBV is high and will persist.

Routh-hurwitz stability criterion the best way to identify the stability of higher power Λ formulating Hurwitz matrix.

The sensitivity analysis that was carried out both R₀ and simulation showed that the initial infection rate, β has a very great influence on the spread of HBV in the region than all the other parameters. We conclude that HBV transmission is primarily as a result of the effective interaction between active HBV patients and the susceptible group.

Recommendations

It is highly recommended that besides the attempts being made by Minister of Health Service to fight HBV in the Nigeria, there are other effective measures that can be implemented to reduce drastically the burden of HBV on the health of the people in the region.

All children should get their first dose of hepatitis B vaccine at birth and should have completed the vaccine series by 6 through 18 months of
Children and adolescents through 18 years of age who did not get the vaccine when they were younger should also be
Minimize the amount of drinking
take drug when HBV infected recommend by doctors

References

Tahir Khana, Gul Zamana and M.,Ikhlaq Chohanb, The transmission dynamic and optimal control of acute and chronic hepatitis B.Journal of Biological Dy- namics, 11:1, 172-189,
Caroline Dickens., Occult Hepatitis B Virus (HBV) Infection In The Chacma Baboon, PhD University of the Witwatersrand, Johannesburg, 2011
Afiba Manza-Azele Agovi, Hepatitis B, C Virus And HIV CO-Infection Among Reported Female Cases In South Carolina, 2004 – 2011: An Epidemiological Analysis Of Pregnancy Outcomes. Master University of South Carolina ,Columbia,2016
Sarkodie Eric Modeling The Spread Of Tuberculosis In Central Region Using The Susceptible-Exposed-Infected-Susceptible (SEIS) Mathematical Model. Master Thesis. Kwame Nkrumah University Of Science And Technol- ogy,Ghana, 2014
Sacrifice Nana-Kyere, Joseph Ackora-Prah, Eric Okyere, Seth Marmah, Tuah,Afram, Hepatitis B Optimal Control Model with Vertical Transmission. Applied Mathematics, 7(1): 5-13,2017.
LenkaBubniakov, The Mathematics Of Infectious Diseases. Society for Indus- trial and Applied Mathematics, 42(4): 599-653, 2000
Tigistu Demisse, Sero-Prevalence and Risk Factors Of Hepatitis B and C Viruses Among Healthy Adult Blood Donors in Wolita Zone, Nigeria.BSc The- Abuja University College of Health Science, Nigeria, 2015
Dongmei Chen, Bernard Moulin, Jianhong Wu, Introduction to Analyzing, Mod-eling Spatial and Temporal Dynamics of Infectious Diseases. Department of Geography Faculty of Arts and Science, Department of Computer Science and Software Engineering, Faculty of Science and Engineering, Department of Mathematics and Statistics Faculty of Science, 8:17 , 2014

Other Topics