Mathematical modelling of TB–COVID-19 co-infection with time delay and optimal control analysis

Abstract

Tuberculosis (TB) and COVID–19 remains a major global public health threat, particularly in low- and middle-income countries where healthcare systems face persistent structural constraints. The co-existence of these two diseases introduces complex clinical and epidemiological challenges, such as delayed diagnosis, delayed treatment initiation and overlapping symptoms, which generally increase the disease burden and mortality risk. In this study, we formulate and analyse a deterministic compartmental model for TB–COVID–19 co-infection incorporating time delays representing diagnostic and treatment lags. The model integrates three public health interventions i.e. physical distancing, vaccination and treatment, formulated as time-dependent control functions. Qualitative analysis establishes positivity and boundedness of the solutions, existence of invariant regions and threshold dynamics governed by reproduction numbers. An optimal control framework is then introduced in our work to reduce the number of infections and also the implementation costs. Using Pontryagin’s Maximum Principle, the corresponding Hamiltonian system and adjoint equations are derived, together with the necessary conditions for optimal controls. To further explore the impact of these interventions, numerical simulations are carried out using delay differential equation techniques combined with a forward–backward sweep method. The results suggests that applying multiple interventions simultaneously can significantly lower the prevalence of co-infection and related fatalities. However, the presence of delays reduces the overall effectiveness of these strategies. The model offers quantitative proof that effective management of TB and COVID-19 co-infection in nations with limited resources requires early diagnosis, quick treatment initiation, and coordinated intervention methods.