A mathematical study of tuberculosis infections using a deterministic model in comparison with continuous Markov chain model

Abstract

Mathematical model of the transmission of tuberculosis infection was widely studied to capture the transient behavior of the disease transmission. In this study we model the dynamic of the disease by considering an epidemiological model called SEIIR model to capture the deterministic behavior of the disease. We also applied a continuous-time Markov chain model to take into consideration the randomness of the system. In the deterministic model, a disease-free equilibrium point and a basic reproduction number of the model are found which are mainly influenced by the contact rate of susceptible individuals with infective individuals. Other parameters such as progression rate of the latent individuals to be infectious individuals and the treatment rate of latent individuals also influence not only the deterministic model but also the stochastic sample paths. For a certain critical value of the treatment rate of latent class, deterministic and stochastic solutions show different behavior at the final time of observation. A high degree of randomness is observed in the latent and infected class (hospitalize or not-hospitalized). While in the susceptible class, effect of randomness is almost not observed. This suggests the robustness of deterministic model of susceptible class to the stochastic perturbations.