Bifurcation and stability analysis of an SEITFR conjunctivitis model with standard incidence and treatment failure

This study formulates and analyzes an susceptible–exposed–infectious–treated–failed treatment–recovered (SEITFR) model for conjunctivitis transmission. The compartments represent susceptible individuals, exposed individuals, infectious cases, treated cases, detected treatment failure cases, and recovered individuals. The model separates two clinically relevant effects: Treated individuals may still transmit infection at a reduced rate, while detected treatment-failure cases are assumed to be isolated or referred for further care and therefore do not contribute directly to community transmission. We prove the positivity, boundedness, and uniqueness of solutions; derive the basic reproduction number $ \mathcal{R}_0 $; and establish the local stability of the disease-free equilibrium for $ \mathcal{R}_0\leq 1 $ and its global stability for $ \mathcal{R}_0 < 1 $. The endemic equilibrium is globally asymptotically stable for $ \mathcal R_0>1 $ under a constant-population assumption and in the absence of immunity loss. Eigenvalue computations using the baseline loss-of-immunity rate further support the local stability of the endemic equilibrium over the tested supercritical range. A center manifold calculation for the full system confirms a forward transcritical bifurcation at $ \mathcal{R}_0 = 1 $. The sensitivity analysis shows that reducing contacts and residual infectiousness during treatment is essential and that increasing the treatment rate is beneficial only when treatment is accompanied by sufficient isolation or rapid clinical resolution. Numerical simulations illustrate the threshold behavior, the convergence to disease-free or endemic states, and the parameter regions where eradication is expected.