Dynamical Behaviors Of Two Classes Of Impulsive Biomathematical Models

Impulsive differential equations are widely applied in life sciences.In recent years,the basic theory and methods of impulsive differential equations have been improved.This paper studies two classes of impulsive differential equations: one is a COVID-19 model with periodic impulse vaccination,and the other is a tumor immune model with state-dependent impulse.This article mainly consists of two parts.In the first part,the COVID-19 model of continuous vaccination is established.The conditions are given for the stability of the disease-free equilibrium.Based on the continuous model,the new model with periodic impulse is given.Then the relevant qualitative theory and methods of impulsive differential equations are used to study the local and global asymptotic stability of the disease-free periodic solution.The numerical results are given to illustrate the main results.The relations of the critical parameters are also explored.Based on the model established by Kuznetsov and Taylor in 1994,the second part studies a tumor model with state-dependent impulse.The dynamical behaviors,including the existence and stabilities of the tumor-free periodic solution,are studied by using the Floquet theories.With the help of Poincare mapping of the system,the bifurcations near the tumor-free periodic solution are analysed.The bifurcations include transcritical bifurcation and supercritical pitchfork bifurcation et al..