Complex Dynamics Of A Discrete-Time SIR Model With Nonlinear Incidence And Recovery Rates

Infectious diseases have been paid much attention to all over the world for a long time.The struggle between human beings and infectious diseases runs through the whole history of human development.For example,the severe infectious diseases in our country in recent years include SARS in 2003,Influenza A(H1N1)in 2009,and Covid-19 pneumonia,which has not ended since the end of 2019.Every outbreak of the spread of disease will bring great hidden dangers to our lives.In order to fight against infectious diseases,human beings should understand the law and route of transmission of infectious diseases in the crowd,and determine whether the disease will continue to spread.Recently,researchers have constructed different mathematical models according to the transmission rules and routes of different infectious diseases,which provide a theoretical basis for making better strategies to prevent and control infectious diseases.In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler method.Firstly,the existence and stability of fixed points in the model are analyzed.Using the central manifold theorem and bifurcation theory,the effects of the intervention level and the integration step on the dynamic behavior of the number of susceptible individuals and infected individuals in the model were investigated.Through theoretical analysis and numerical simulation,three kinds of codimension-one bifurcations(transcritical bifurcation,period-doubling bifurcation and Neimark-Sacker bifurcation)and four kinds of codimension-two bifurcations(fold-flip bifurcation,1:2 resonance,1:3 resonance and1:4 resonance)are given in detail.In addition,the phase diagrams,bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of the theoretical analysis.Particularly,the diagrams of local attraction basins and periods distributions of two-parameter bifurcation points are discovered.By comparing the discrete-time model with the corresponding continuous system and the local domain of attraction of the discrete-time model under different integration time steps,we find that when the integration time step is small,the dynamic behavior of the discrete-time model is consistent with that of the corresponding continuous system.However,when the integration step is large,the discrete-time model can show more complex dynamic behavior.That is,a discrete-time model can capture the underlying dynamics that the corresponding continuous model does not exhibit.Such a discrete-time model can not only be widely used to detect the pathogenesis of infectious diseases,but also make contributions to the prevention and control of infectious diseases.