Mathematical Modeling Of Tuberculosis

Malaria and tuberculosis are two major co-endemic diseases in most parts of the world,particularly in the tropics.In recent years,a lot of researches has been done on these two diseases using mathematical models.In this thesis,we discuss two mathe-matical models of tuberculosis and one model of tuberculosis and malaria co-infection.In Chapter 2,we present and analyze a deterministic model of tuberculosis with exogenous re-infection and treatment.Our model includes classes for the undiagnosed infective as well as the treatment for diagnosed infective.We assume that individuals in the latently infected class develop active TB due to exogenous re-infection and relapse of recovered individuals.We evaluate the basic reproduction number and present the existence of the local stability of the disease-free and endemic equilibria.We also proved that when the model is analyzed with mass balance incidence with the exogenous re-infection set to zero,it has no positive equilibrium whenever the reproduction number is less than unity;therefore the disease-free equilibrium is G.A.S.We show that the model exhibits a backward bifurcation phenomenon.The exogenous re-infection play a key role on the existence of backward bifurcation.We performed numerical simulations to confirm our analytic results.Chapter 3 outlines a tuberculosis transmission model with optimal control s-trategies.We present the mathematical model and calculate the basic reproduction number R0.The mathematical analysis for the stability is performed.We showed that the disease-free equilibrium is asymptotically stable whenever R0<1.Also,we proved that the model has a unique endemic equilibrium whenever R0>1.We performed the bifurcation analysis.To control the disease from spreading,we included three in-tervention strategies called controls into the model.These controls are case finding,prevention,and case holding.We aim to reduce the number of infectious and increase the individuals who are receiving treatment.Finally,we presented a numerical simula-tion and gave some recommendations on the control of the tuberculosis transmission.Lastly,in Chapter 4,we propose a mathematical model which considers the epi-demiological features of tuberculosis and malaria and their co-infections.We analyzed the malaria-only model,TB-only model,dual malaria-TB only model,and the full Malaria-TB model.We performed the bifurcation analysis of the malaria-only and the tuberculosis-only models.The study of the dual malaria-TB model shows that it has no positive endemic equilibrium when the reproduction number is less than unity;thus,it does not exhibit the phenomenon of backward bifurcation.Therefore,we conclude that the disease-free equilibrium of the dual malaria-TB model is globally asymptotically stable whenever the reproduction number is less than unity.Also,the analysis of the full malaria-TB model shows that when RM>1 and RT>1,irrespective of which one is higher,malaria and TB co-exist in the population.Secondly,we found out that there is no competitive exclusion.We perform the numerical simulations of the various sub-models.Finally,we perform the sensitivity analysis to determine the parameters that significantly affect the transmission of the disease.