### Mathematical Model of Infectious Disease with Multistage Vaccine

Assefa Erba Bikila

#### 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 R0, which is a threshold quantity for the stability of equilibrium point is calculated. If R0 < 1 then the disease-free equilibrium point is globally asymptotically stable and it is the only equilibrium point. On the contrary, if R0 > 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 31st 2019

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