Attempt to Model Tuberculosis with the Effects of Both Treatment and Case Detection

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Samson W. Wanyonyi
Wafula Mike Erick
John M. Hunja
Collins Ochieng Onyaga


Tuberculosis (TB) is an infectious disease that usually affect the lungs. It is caused by bacteria and spread through the air. Due to the high rate of spread and later leads to death due to TB, this led to formulation of a model in order to enlighten the public and health department at large of its effects. A non-linear mathematical model (NLMM) of TB with the effect of case discovery and treatment was formulated and analyzed. The population under study was divided into four compartments namely susceptible ( ), exposed ( ), infected  and recovered  individuals move to  class or once they come into contact with an infected person and this is usually based on immunity level of an individual. This is incorporated through progression rate which could be quick or slow. The basic reproduction number (Rₒ) and an equilibrium of the model were computed. It was found out that the disease-free equilibrium of the model is locally asymptotically stable when . The model exhibits backward bifurcation (BB) under certain constraints on parameters, which results to existence of multiple endemic equilibrium for . This tells that an accurate estimation of parameters and the level of curb measures are required to lower the infection prevalence of TB regions where it is common and just not enough to get rid of the disease from the population.  needs to be highly reduced to confirm the global stability of the disease free-equilibrium. Numerical simulations (NS) was done using MATLAB software to graphically illustrate the effect of case discovery, detection and treatment of TB infection. It was found that the rise in the rate of case detection shifts the BB diagram towards right which led in the rise of the threshold value of Rₒ. It was also shown that the equilibrium level of infective population reduces when infective population is subjected to treatment. NS were carried out to support the analytic results. This project intends to help the heath sector and Kenyans at large to seek tests and treatment earlier so as to minimize the death rate.

Mycobacterium tuberculosis, basic reproduction number, mathematical model, case detection, stability and simulation.

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How to Cite
W. Wanyonyi, S., Erick, W., M. Hunja, J., & Onyaga, C. (2019). Attempt to Model Tuberculosis with the Effects of Both Treatment and Case Detection. Asian Research Journal of Mathematics, 12(1), 1-12.
Original Research Article