Global analysis of tuberculosis dynamical model and optimal control strategies based on case data in the United States

Abstract

Tuberculosis is a chronic infectious disease with a high death rate, which has attracted worldwide attention. In this paper, we focus on the annual data of tuberculosis in the United States from 1988 to 2019. Firstly, we propose an SVEITR dynamical model with vaccination, fast and slow progression, incomplete treatment, and relapse. Mathematical analyses show that the disease-free equilibrium is globally asymptotically stable (unstable) if R0≤1 (if R0>1). There exists a unique endemic equilibrium that is globally asymptotically stable when R0>1. Next, we use the Markov-chain Monte-Carlo (MCMC) method to acquire the model’s optimal parameters. From R0=1.3763, tuberculosis will not be eliminated without control measures taken in the United States. Through partial rank correlation analysis, the most sensitive parameters to the basic reproduction number are found. Based on this, we take the actual epidemic situation of tuberculosis in the United States into considerations to get the required conditions for optimal control using Pontryagin’s maximum principle and cost-effectiveness analysis. Implementation of three public health measures: tuberculosis prevention and control education, timely treatment, and enhanced efficacy can reduce the prevalence of tuberculosis in the United States. But at present, it is challenging for us to achieve the goal of eliminating tuberculosis by 2030 as called for by WHO.