Stability Analysis of Mathematical Model of a Relapse Tuberculosis Incorporating Vaccination

O Odetunde,M.O Ibrahim,Department Of Mathematics, University Of Ilorin, Kwara State, Nigeria

doi:10.52968/28301280

Abstract

A general SIQRM epidemic model with vaccination and relapse possibility is proposed for analysis in this work. The idea behind the proposed model is to check the effect of immunity obtained from vaccine or treatment, quarantine effect as well as waning effect of immunity on the transmission rate of Tuberculosis within a population that is subjected to proper education without restricted access. Some other infectious diseases in this category include measles and Ebola. Two equilibrium states of the proposed model are obtained as well as the effective reproduction number(Reff). Stability analysis of the model at the Infection Free Equilibrium(I.F.E) state is established on the condition that Reff<1. Numerical simulation for the general SIQRM model was done using specific data for Tuberculosis disease and the result shows that proper education, vaccination and early diagnosis of an infectious individual for quarantining is an efficient way by which the spread of Tuberculosis can be reduced in the population while adequate medical attention yield better result for detected cases

Highlights

General interest in infectious diseases and their control are fueled by discovery of new infectious disease, frequent re-appearance of plague and emergence of both preventive and therapeutic antibiotics to curb, manage and possibly eradicate the threat of such an infection [2]
A generalized S − I − Q − R − M model is presented for analysis where S, I, R retain their meaning and Q denotes Quarantine and M denotes Immunity
The generalized S − I − Q − R − M model is scarcely used in literature, it can be of immense importance in predicting the course of an infection as well as proffering effective and efficient ways by which an infection in this category can be treated and eliminated

Summary

Introduction

General interest in infectious diseases and their control are fueled by discovery of new infectious disease, frequent re-appearance of plague and emergence of both preventive and therapeutic antibiotics to curb, manage and possibly eradicate the threat of such an infection [2]. Tuberculosis shall be used as a case study to show the effect of early detection, quarantine, vaccination and treatment of an infected individual on the overall epidemic of the disease within the host population. A non-linear deterministic model for the transmission dynamics of infectious diseases in the presence of immunity loss and relapse is built by dividing the total human population at time t, denoted by N(t) into five disjoint epidemiological sub-populations, which are, Susceptible (S), Infected (I), Quarantine (Q), Recovered (R) and Partial Immunity (M).

Results

Conclusion