# An Analysis of CTMC Stochastic Models with Quarantine on the Spread of Tuberculosis Diseases

## DOI:

https://doi.org/10.5614/j.math.fund.sci.2021.53.1.3## Keywords:

CTMC Approach, disease outbreak probability, expected time until disease extinction, quarantine, SIQRS model## Abstract

The SIQRS epidemic model developed in this study is intended to analyze the spread characteristics of the infectious disease tuberculosis. It is a modification of the SIQR model developed by Cao et al., using a stochastic model called the Continuous Time Markov Chains (CTMC) approach. Further analysis of the SIQRS model was done to determine the transitional probability, the outbreak probability, the expected time until disease extinction and to simulate the effect of quarantine treatment on the expected time until disease extinction. Based on the simulation it can be concluded that a decrease of the healing rate together with an increase of the transmission rate changes the basic reproduction number (*R _{0}*), the expected number of infected individuals (

*m*), the time until disease extinction, and the outbreak probability. A disease outbreak will occur if both

*R*>1 and

_{0}*m*>1 hold. Also, based on the simulation it was concluded that the decrease of the healing rate and the increase of the transmission rate cause increases of

*R*and

_{0 }*m*. An increase of the quarantine rate reduces the expected time to disease extinction,

*R*and

_{0 }*m*. As a consequence, the disease will gradually disappear from the system.

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