Mathematical Modeling and Analysis of COVID-19 and Tuberculosis Coinfection with Optimal Control

Abstract

Coronavirus disease (COVID-19) continues to claim the lives of many people globally and controlling the disease has become the most challenging part of the modern health care system. Tuberculosis (TB) is also a major global health threat affecting millions of people every year. Both are infectious diseases that transmit mainly through close contact, and both primarily attack the respiratory system with similar symptoms such as cough, fever, and difficulty breathing. In this dissertation, we studied the dynamics of COVID-19, and the coinfection of TB and COVID-19 to forecast the possible features and establish mitiga tion strategies for their spread. We developed an extended SIR type compartmental model for the COVID-19 pandemic and analyzed the model with analytical methods and numeri cal simulations. The biological meaningfulness of the model was proved and the stability analysis was performed by obtaining the basic reproduction number. The parameter values of the proposed model are estimated with a modified combination of the Bayesian and least square estimation technique. The sensitivity analysis of the model parameters has also been performed using the normalized forward analysis. We have then proposed a deterministic mathematical model to give insight into the coinfection of COVID-19 and TB. The coinfec tion model is qualitatively studied by proving the properties such as the existence, bounded ness, and positivity of the solutions. In each sub-model, the disease-free equilibrium points are proved to be both locally and globally stable if the reproduction numbers are less than unity. We computed the basic reproduction number of the coinfection, and the disease-free equilibrium point was proved to be conditionally stable. The analytical findings reveal that an increase in infected individuals with TB has a positive impact on the spread of COVID-19 while under some conditions, an increase in the number of COVID-19 cases has a positive impact on the spread of TB disease. Furthermore, the coinfection model is extended into an optimal control problem by adding four control measures as the prevention effort against TB, prevention techniques against COVID-19, treatments for TB infections, and medical care for COVID-19 infection to optimally manage the diseases. The optimal control problem was analyzed with Pontryagin’s Minimum Principle. Different simulation cases were performed to supplement the analytical results and to identify the most appropriate control intervention strategies. The simulation results show that the prevalence of the coinfection was reduced when all the four control measures were concurrently implemented.