Backward bifurcation in a seasonal conjunctivitis model

This study develops a non-autonomous compartmental model for conjunctivitis transmission with seasonal forcing, treatment failure, temporary immunity, modified infectiousness during treatment, and limited treatment capacity. The model is shown to be mathematically well posed, and the disease threshold is characterized using the periodic next-generation approach. The analysis shows that the disease-free periodic solution is locally stable below the seasonal threshold and unstable above it, while uniform persistence and the existence of at least one positive periodic solution are established when the threshold is exceeded. Although saturated treatment does not affect the local invasion threshold when the baseline treatment rate is fixed, it changes the nonlinear dynamics and can generate backward bifurcation under specified parameter conditions. In the autonomous limiting case, this mechanism is confirmed through an explicit endemic branch calculation, numerical equilibrium-stability checks, and continuation methods; backward branching can also occur when treated individuals are noninfectious under sufficient saturation. A comparison argument establishes global elimination under a sufficient condition. In the seasonal model, simulations from multiple initial conditions provide evidence of subthreshold bistability, where disease extinction and persistent seasonal infection may coexist below the invasion threshold depending on the initial state. The simulation-estimated transition is not an exact seasonal fold. These results show that reducing transmission below the invasion threshold alone may not guarantee global elimination when treatment capacity is limited. The numerical findings describe assumed scenarios rather than outbreak-calibrated predictions.