Mathematical analysis of tuberculosis vaccine models and control stategies.
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The epidemiological study of tuberculosis (TB) has been ongoing for several decades, but the most effective control strategy is yet to be completely understood. The basic reproduction number, R₀, has been found to be plausible indicator for TB control rate. The R₀ value is the average number of secondary TB cases produced by a typical infective individual in a completely susceptible population during its entire infectious period. In this study we develop two SEIR models for TB transmission; one involving treatment of active TB only, with the second incorporating both active TB treatment and post-exposure prophylaxis (PEP) treatment for latent TB. Using the next generation matrix method we obtain R₀. We determine the disease free equilibrium (DFE) point and the endemic equilibrium (EE) point. Global stability conditions of DFE are determined using the Castillo-Chavez theorem. Through model analysis of the reproduction number, R₀, we find that for R₀ < 1, the infection will die out. The value of R₀ > 1 implies that the disease will spread within the population. Through stability analysis, we show that the model exhibits backward bifurcation, a phenomenon allowing multiple stable states for fixed model parameter value. MATLAB ode45 solver was used to simulate the model numerically. Using the Latin Hypercube Sampling technique the model is sensitive to treatment and disease transmission parameters, suggesting that to control the disease, more emphasis should be placed on treatment and on reducing TB transmission. For the second model, which incorporated treatment with post-exposure prophylaxis for latently infected individuals, by means of simulations, we found that treatment of latently infected individuals may reduce R₀. Numerical simulations on the latter model also showed that it may be better to introduce a hybrid of active treatment and post-exposure treatment of the latent class. The force of infection was found to reduce when this hybrid control strategy is present. Contour plots and PRCC values highlighted the important parameters that influence the size of the Infective class. The implications of these findings are that TB control measures should emphasise on treatment. Our simplified models assume that there is homogeneous mixing. The model used have not been validated against empirical data.