Numerical Solution and Analysis for Acute and Chronic Hepatitis B

Muhammad Farman ,Zafar Iqbal ,Aqeel Ahmad ,Ali Raza ,Ehsan Ul Haq

doi:10.28924/2291-8639-16-2018-842

Muhammad Farman , Zafar Iqbal + Show 3 more

Open Access

https://doi.org/10.28924/2291-8639-16-2018-842

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Abstract

In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem to control the spread of hepatitis B in a community. In order to do this, first we present sensitivity analysis of the basic reproduction number R0. We develop a unconditionally convergent nonstandard finite difference scheme by applying Mickens approach φ(h) = h + O(h^2) instead of h to control the spread of this infection, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. The stability analysis of the scheme has been developed by theorems which shows the both stable locally and globally. Comparison is also made with standard nonstandard finite difference scheme. Finally numerical simulations are also established to investigate the influence of the system parameter on the spread of the disease.

Highlights

The scope of mathematics includes mathematical modeling and esoteric mathematics
Materials and Method we used a mathematical model for HBV transmission by extending the work presented in [28].We divide the host population denoted by T(t) into four compartments: susceptible individuals S(t), who are not infective but have the chance to catch the disease; infected I1(t) represents those individuals who are infective with acute hepatitis; I2(t) are those individuals, who are infected with chronic hepatitis and R(t) represents those individuals who have recovered after the infection with a life-time immunity
We have considered a mathematical system of equation which describes the hepatitis B disease

Summary

Introduction

The scope of mathematics includes mathematical modeling and esoteric mathematics. The flow of work, process, predictions and outcomes can be measured with the help of mathematical concepts and theory. The virus reproduces and releases large numbers of new viruses into the blood stream [8] This infection has two possible phases: (1) acute and (2) chronic. Mathematical models have been used to help understand the dynamics of viral infections, such as human immunodeficiency virus and hepatitis C infection [11,12]. 2. Materials and Method we used a mathematical model for HBV transmission by extending the work presented in [28].We divide the host population denoted by T(t) into four compartments: susceptible individuals S(t), who are not infective but have the chance to catch the disease; infected I1(t) represents those individuals who are infective with acute hepatitis; I2(t) are those individuals, who are infected with chronic hepatitis and R(t) represents those individuals who have recovered after the infection with a life-time immunity. With initial conditions S(0) ≥ 0, I1(0) ≥ 0, I2(0) ≥, R(0) ≥ 0, Here b represents the birth rate, α is the moving rate from susceptible to infected with acute hepatitis B, β is the moving rate from acute stage to infected with chronic hepatitis, γ1 is the recovery rate from acute stage to recovered, γ2 is the recovery rate from chronic stage to recovered compartment, μ0 is the death rate occurring naturally, which is called natural mortality rate, μ1 is the death rate occurring due to hepatitis B and ν represents hepatitis B vaccination rate

Qualitative Analysis

Sensitivity Analysis of R0

Conclusion