Optimal control analysis of tuberculosis dynamics with dual quarantine strategies using real data in Indonesia

Abstract

Tuberculosis is a serious health problem worldwide, with high transmission rates specifically in developing countries. Indonesia lies on the second rank with an estimated more than a million of new cases per years and about hundred thousand of death. This study aimed to explore an optimal control analysis of tuberculosis involved two kind of quarantines strategies in Indonesia. First, models were constructed by considering several subhuman populations, such as healthy, infected, and diagnosed individuals. Some parameters such as transmission rate, recovery rate, and medical interventions are involved. Dynamical analysis has been conducted through several algorithms in mathematical modelling analysis, which consists of 1) critical point analysis, 2) stability analysis, and 3) threshold value analysis. Optimal control has been applied in the constructed model for the final study to observe the long-term behavior of the disease and the impact of interventions on reducing infected individual. This research resulted in two critical points, which is a critical point of disease-free equilibrium and endemic equilibrium. Regarding the next generation matrix and the disease-free equilibrium, basic reproduction number was generated, and it had been used to explore local and global analysis stability. The first condition, basic reproduction number will be less than one means that disease-free equilibrium is stable which the infection will die out of the human population while in the second, basic reproduction number will be more than one and it means that disease-free equilibrium is unstable or endemic equilibrium will arise and the infection will persist in the human population. To enhance control, Pontryagin Maximum Principle is applied to characterize the necessary condition in reducing the prevalence of TB. This study highlights that the implementation of two quarantines strategies, combined with medication treatment, significantly lower the number of infected individuals and minimizes both infection burden and interventions cost.