Transmission of Tuberculosis and Reproduction Number Using Lyapunov Function of SIR Model

Mohd Rahimie Bin Md Noor; Mohamed Zuheel Bin Zaid; Rozaimie Bin Rosli; Mohamad Danial Bin Hairudin; Sharifah Athirah Bt Syed Adnan; Mohd Azry Bin Abdul Malek; Muhammad Firdaus B. Mustapha; Ahmad Bukhari Bin Mohd Yasin; Omar B. Kairan

doi:10.47405/mjssh.v7i11.1873

Mohd Rahimie Bin Md Noor Department of Mathematics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia.
Mohamed Zuheel Bin Zaid Department of Mathematics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia.
Rozaimie Bin Rosli Department of Mathematics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia.
Mohamad Danial Bin Hairudin Department of Mathematics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia.
Sharifah Athirah Bt Syed Adnan Medical Department, Klinik Kesihatan Padang Luas, 22000 Jertih, Terengganu, Malaysia.
Mohd Azry Bin Abdul Malek Department of Statistics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia.
Muhammad Firdaus B. Mustapha Department of Computer Science, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 18500 Machang, Kelantan, Malaysia.
Ahmad Bukhari Bin Mohd Yasin Department of Business, Faculty Business Management Universiti Teknologi Mara Kelantan, 15050 Kota Bharu, Kelantan, Malaysia.
Omar B. Kairan Department of Statistics, Faculty Mathematic and Computer Sciences Universiti Teknologi Mara Kelantan, 15050 Kota Bharu, Kelantan, Malaysia.

DOI: https://doi.org/10.47405/mjssh.v7i11.1873

Keywords: Nonlinear System Dynamic, Basic Reproduction Number, SIR Model

Abstract

This paper studies the mechanism of transmission of Tuberculosis (TB) caused by the mycobacterium tuberculosis bacteria. TB is an airborne disease that endangering human population globally. Mathematical modelling of tuberculosis was developed using the Susceptible-Infected-Recovered (SIR) model in this analysis. The model generated by the four-dimensional, nonlinear dynamic system is reduced to three dimensions, with some assumptions. Then, the model will be analyzed by creating a mathematical theorem that will prove the existence of a TB event, the disease-free equilibrium phase, and the TB endemic disease stage. By using the Lyapunov function method, the three theorems can be proved. The basic reproduction number R0 also can be obtained from the model. If the basic reproduction number R0 ≤ 1, then the disease-free equilibrium is global asymptotically stable and if R0 > 1, then the endemic equilibrium is asymptotically stable globally. Carrying out a simulation using data in a selected region, the result of the model successfully describes and forecast the number of TB cases and it may also be used for assessing the TB disease status.

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