Mathematical model of tuberculosis with seasonality, detection, and treatment

Abstract

In this study, we use a modified SEIR compartmental model to investigate the transmission dynamics of tuberculosis as well as its detection and treatment. The construction of this model is based on the hypothesis that the total population can be divided into seven compartments: susceptible, vaccinated, latently infected, diagnosed infected, undiagnosed infected, treated individuals, delayed treated individuals, and recovered individuals with a nonautonomous system. The effective reproduction number shows that the amplitude of the effective reproduction number decreases with the increasing vaccination rate and increases with the increase in the degree of seasonality of TB. The stability analyses of the model show that the value of the basic reproduction number R0 acts as a threshold between disease-free and endemic equilibrium. The model is found to be locally and globally asymptotically stable at the disease-free equilibrium when R01. The sensitivity of the parameters of the corresponding autonomous system is examined using the partial rank correlation coefficients (PRCC) analysis, which demonstrates that identification has a positive index and treatment has a negative index. The model is simulated using the RK-45 numerical method, and the parameter values for the model are taken from the available literature. Finally, the model outcome was compared with the real field data and found to be consistent.

Keywords

• Tuberculosis
• Delay treatment
• Effective reproduction number
• Sensitivity
• Seasonality
• Bifurcation