Nonlinear state-dependent feedback control strategy in the SIR epidemic model with resource limitation

In present work, in order to avoid the spread of disease, the impulse control strategy is implemented to keep the density of infections at a low level. The SIR epidemic model with resource limitation including a nonlinear impulsive function and a state-dependent feedback control scheme is proposed and analyzed. Based on the qualitative properties of the corresponding continuous system, the existence and stability of positive order-k ( $k\in\mathbf{Z}^{+}$ ) periodic solution are investigated. By using the Poincare map and the geometric method, some sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution are obtained. Moreover, the sufficient conditions which guarantee the nonexistence of order-k ( $k\geq3$ ) periodic solution are given. Some numerical simulations are carried out to illustrate the feasibility of our main results.