Application of Optimal Control to the SEITRS Mathematical Model of Tuberculosis Transmission with Control Variables Socialization and Therapy

Muhammad Rafiq; Sri Wigantono; Indriasri Raming

doi:10.29303/emj.v8i2.322

Authors

Muhammad Rafiq Study Program of Mathematics, Universitas Mulawarman
Sri Wigantono Laboratory of Mathematical Modeling and Machine Learning, Universitas Mulawarman
Indriasri Raming Laboratory of Mathematical Modeling and Machine Learning, Universitas Mulawarman

DOI:

https://doi.org/10.29303/emj.v8i2.322

Keywords:

Numerical simulation, optimal control, stability analysis, SEITRS mathematical model, tuberculosis

Abstract

Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis, which remains a serious public health concern. The objective of this study is to develop, analyze, and propose an optimal control strategy for the transmission dynamics of TB using an SEITRS mathematical model. The model consists of five population compartments: Susceptible (S), Exposed (E), Infected (I), Treatment (T), and Recovered (R). The methodology involves constructing the SEITRS model, determining the equilibrium points, and analyzing their stability under different conditions of the basic reproduction number. The model has two equilibrium points, namely the non-endemic and endemic equilibrium. If the basic reproduction number is less than one and certain conditions are satisfied, the non-endemic equilibrium is locally asymptotically stable. Conversely, if the basic reproduction number is greater than one and specific conditions are met, the endemic equilibrium becomes locally asymptotically stable. Furthermore, this study provides optimal control strategies in the SEITRS model. We use two control variables in this model, namely socialization and therapy, to reduce the number of infected individuals. The sufficient conditions for the existence of optimal controls are derived using Pontryagin’s Maximum Principle. Numerical simulations are then conducted to examine the impact of applying these controls on the system. The simulation results indicate that the simultaneous implementation of socialization and therapy controls is effective in reducing the number of TB-infected individuals.

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