Drug-sensitivity and passive immunity mathematical epidemiological model for tuberculosis

A.A Danhausa,E.E Daniel,A.M Nuhu,C.J Shawulu,L Philemon

doi:10.4314/jasem.v25i9.18

Abstract

Regardless of many decades of research, the widespread availability of a vaccine and more recently highly visible WHO efforts to promote a unified global control strategy, Tuberculosis remains a leading cause of infectious mortality. In this paper, a Mathematical Model for Tuberculosis Epidemic with Passive Immunity and Drug-Sensitivity is presented. We carried out analytical studies of the model where the population comprises of eight compartments: passively immune infants, susceptible, latently infected with DS-TB. The Disease Free Equilibrium (DFE) and the Endemic Equilibrium (EE) points were established. The next generation matrix method was used to obtain the reproduction number for drug sensitive (𝑅𝑜𝑠) Tuberculosis. We obtained the disease-free equilibrium for drug sensitive TB which is locally asymptotically stable when 𝑅𝑜𝑠 < 1 indicating that tuberculosis eradication is possible within the population. We also obtained the global stability of the disease-free equilibrium and results showed that the disease-free equilibrium point is globally asymptotically stable when 𝑅𝑜𝑠 ≤ 1 which indicates that tuberculosis naturally dies out.

Highlights

(DFE) and the Endemic Equilibrium (EE) points were established
Completing a vicious circle, HIV were male, 10% were people living with HIV (74% in itself has been the single most important factor in the Africa) and 56% were in five countries: India, rising incidence of TB in Africa since 1990
Method of Model Analysis: various methods adopted in carrying out this study were discussed: Equilibrium State: The equilibrium state for the model is obtained by setting the model equations to be zero. i.e dM = dS = dEs = dIs = dRs = 0 dt dt dt dt dt

Summary

Introduction

(DFE) and the Endemic Equilibrium (EE) points were established. The generation matrix method was used to obtain the reproduction number for drug sensitive (Ros) Tuberculosis. We obtained the disease-free equilibrium for drug sensitive TB which is locally asymptotically stable when Ros < 1 indicating that tuberculosis eradication is possible within the population. According to the World Health Organization (WHO), one-third of the world's population is infected, either latently or active, with tuberculosis (WHO, 2016) It is an ancient disease with evidence of its existence being found in relics from ancient Egypt, India and China. An estimated 10.4 million people fell ill with TB in 2016: 90% were adults, 65% were male, 10% were people living with HIV (74% in Africa) and 56% were in five countries: India, Indonesia, China, the Philippines and Pakistan (WHO, 2017). Tuberculosis (TB) is a preventable disease linked to poverty, was declared an emergency in Africa in 2005 Each year it claims the lives of half a million Africans, many young and in their most productive years. HIV/AIDS die from TB every year, most of them in people fell ill with TB in 2016: 90% were adults, 65%

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