Optimal Control of a Tuberculosis Model with State and Control Delays

TL;DR

This paper models tuberculosis dynamics incorporating delays in diagnosis and treatment, analyzes stability, and formulates optimal control strategies considering these delays, which is a novel approach in TB modeling.

Contribution

It introduces the first TB model with delays in both state and control variables and studies their impact on stability and optimal control.

Findings

Stability of disease-free and endemic states depends on delays.

Optimal control strategies are derived considering delays.

Delays significantly affect the effectiveness of TB interventions.

Abstract

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.