The Galton–Watson branching process for TB dynamics: The potential for disease persistence or extinction

Abstract

Tuberculosis is a critical problem particularly in developing countries of Africa that has resulted in a high mortality rate. The disease is transmitted through air when a susceptible individual inhales Mycobacterium tuberculosis bacteria that have been released by an infected person during coughing, sneezing, singing or speaking. In this work, we formulate and analyze a deterministic model and its corresponding continuous time Markov chain (CTMC) stochastic model to study the dynamics of tuberculosis in humans. The Galton–Watson multitype branching process is adopted to compute the stochastic threshold ρ(M) and determine the possibility of tuberculosis persistence or extinction. The next generation method is employed to find the basic reproduction number R0 and the partial rank correlation coefficient method is used to determine the most sensitive parameters in the spread of tuberculosis. Sensitivity analysis shows that the probability of TB infection, the fraction of individuals who progress to pulmonary TB, pulmonary TB induced death and human recruitment rate influence the transmission of TB disease. If the basic reproduction number R0<1 and the stochastic threshold ρ(M)<1 then tuberculosis clears in human population. If R0>1 then tuberculosis persists in the population. However, there is a possibility of tuberculosis to persist or clear if ρ(M)>1 depending on the number of infectious agents at the beginning of the tuberculosis outbreak. Results reveal that the probability of tuberculosis extinction is small if TB emerges from individuals with pulmonary TB and it is large if TB emerges from individuals with latent TB. If TB emerges from individuals with extra-pulmonary TB, then the disease clears in the population. These results suggest that it is essential to treat humans with pulmonary tuberculosis particularly at the beginning of the disease outbreak so as to control human TB.