Analysis of a mathematical model with nonlinear susceptibles-guided interventions

In this paper, we considered a mathematical model describing the nonlinear susceptibles-guided vaccination and isolation strategies, incorporating the continuously saturated treatment. In this strategy, we find that the disease-free periodic solution can always exist, and consequently the control reproduction number can be defined through analyzing the stability of the disease-free periodic solution. Also, we discussed the existence and stability of the positive order-1 periodic solution from two points of view. Initially, we investigated the transcritical and pitchfork bifurcation of the Poincaré map with respect to key parameters, and proved the existence of a stable or an unstable positive order-1 periodic solution near the disease-free periodic solution. For another aspect, by studying the properties of the Poincaré map, we verified the existence of the positive order-1 periodic solution in a large range of the control parameters, especially, we verified the co-existence of finite or infinite countable different positive order-1 periodic solutions. Furthermore, numerical simulations show that the unstable order-1 periodic solution can co-exist with the stable order-1, or order-2, or order-3 periodic solution. The finding implies that the nonlinear susceptibles-triggered feedback control strategy can induce much rich dynamics, which suggests us to carefully choose key parameters to ensure the stability of the disease-free periodic solution, indicating that infectious diseases die out.