Mathematical analysis of the transmission dynamics of tuberculosis–diabetes co-infection with a maturation delay

Abstract

Tuberculosis (TB) and diabetes mellitus (DM) form a bidirectional syndemic: diabetes roughly triples the risk of progression to active TB, while active TB worsens glycaemic control. Because the transition from infection to infectious disease is not instantaneous, we formulate a seven-compartment deterministic model of TB–DM co-infection that incorporates a discrete maturation delay \(\tau\) representing the time required for a newly infected latent individual to become infectious, with a survival factor \(e^{-\mu\tau}\). We establish the well-posedness of the resulting system of delay differential equations (positivity, boundedness and existence of a positively invariant compact set), derive the basic reproduction number \(\mathcal{R}_0\) through the next-generation operator, and prove the local and global asymptotic stability of the disease-free equilibrium. Global stability of the disease-free equilibrium is obtained by a matrix-theoretic Lyapunov function built from the left Perron eigenvector of the next-generation matrix, while global stability of the endemic equilibrium is established by the graph-theoretic method of Li and Shuai combined with a Lyapunov–Krasovskii functional that absorbs the delay. A centre-manifold analysis shows that the model undergoes a forward (supercritical) transcritical bifurcation at \(\mathcal{R}_0=1\), so that no backward bifurcation or bistability occurs and reducing \(\mathcal{R}_0\) below unity is sufficient for elimination. We characterise a time-dependent optimal control strategy combining case-finding/treatment, latent-TB preventive therapy and transmission reduction using Pontryagin's maximum principle for delayed systems. Numerical experiments confirm every analytical result, quantify the effect of the delay, and show that an integrated control programme averts a large fraction of the infectious burden. The findings translate into concrete programmatic guidance: bidirectional TB–DM screening, glycaemic control and preventive therapy of latent infection act synergistically and are the levers to which \(\mathcal{R}_0\) is most sensitive.