Mathematical Model of Infectious Disease with Multistage Vaccine

Adugna Mosissa Bikila

doi:10.7176/mtm/9-10-04

Abstract

Many diseases, such as the seasonal influenza, tetanus, and smallpox, can be vaccinated against with a single dose of a vaccine. However, some diseases require multiple doses of a vaccine for immunity. Diseases requiring a multistage vaccine such as Hepatitis B can have extra complications with its vaccination program, as some who start the doses may forget to complete the program or could become infected before completing the program. This thesis concerns the setup and analysis of a model for developing a mathematical model to describe the dynamics of an infectious disease with a multistage vaccine. In this thesis, we considered Susceptible-Infected-Removed (SIR) epidemic models and discussed the mathematical analysis and simulation study is conducted. We discuss an epidemic model which represents the direct transmission of infectious disease. The model assumes that individuals are equally likely to be infected by the infectious individuals in a case of contact except those who are immune. We formulated SIR epidemiological model to determine the transmission disease by using compartmental model approach to using a system of nonlinear differential equations. We study about basic reproduction number and equilibrium point for compartmental mathematical models of infectious disease transmission. The basic reproduction number R 0 , which is a threshold quantity for the stability of equilibrium point is calculated. If R 0 < 1 then the disease-free equilibrium point is globally asymptotically stable and it is the only equilibrium point. On the contrary, if R 0 > 1 then an endemic equilibrium point appears which is locally asymptotically stable. Keywords : Equilibrium Stability, SIR, Multistage Vaccine and Basic Reproduction Number DOI : 10.7176/MTM/9-10-04 Publication date : October 31 st 2019

Highlights

Disease has played an important role in the history of mankind
Facts indicate that hepatitis B virus is liable for almost 80 percent of the primary liver cancer, which is one of the most common tumours in man
We develop a mathematical model which describes the dynamics of infectious disease and its transmission using a system of non-linear ordinary differential equations

Summary

Introduction

Disease has played an important role in the history of mankind. The spread of communicable diseases has influenced the growth of populations and economies of countries. In 2009, a novel strain of the H1N1 influenza virus, which was referred to as the “Swine Flu” in the mainstream media, caused a worldwide pandemic, eventually causing about 18,500 laboratory-confirmed deaths worldwide (the estimated number of deaths worldwide was around 285,000) (Dawood 2012). In the summer of 2009, a vaccine against the virus was eventually developed. Studies indicated that children between the ages of 6 months and 9 years would need to receive two doses of the vaccine, spread about 28 days apart, in order to be fully immunized (Centres for Disease Control and Prevention 2009)

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