Abstract
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 (kinmathbf{Z}^{+}) periodic solution are investigated. By using the Poincaré 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 (kgeq3) periodic solution are given. Some numerical simulations are carried out to illustrate the feasibility of our main results.
Highlights
1 Introduction Infectious diseases have had a tremendous influence on human life and have brought huge panic and disaster to mankind once out of control from antiquity to the present
Measles, dengue, tuberculosis, cholera, Ebola and avian influenza have had a tremendous influence on human health during the last few years
Long-term clinical data show that the vaccination strategies lead to infectious disease eradication if the proportion of the successfully vaccinated individuals is larger than a certain critical value, for example, approximately equal to % for measles
Summary
Infectious diseases have had a tremendous influence on human life and have brought huge panic and disaster to mankind once out of control from antiquity to the present. In Section , we discuss the SIR model with the nonlinear impulsive vaccination as state-dependent feedback control, the existence and stability of positive periodic solution of this model. Depends on the threshold RL as a function P(RL) is a sufficient condition for any point lying in section RL to map to section pmax after once impulsive effect.
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