Dynamical Behavior And Optimal Control For A Delayed Infectious Model With The Impact Of Information

As the information industry develops rapidly,information dissemination plays an increasingly prominent effect in preventing and controlling the spread of infectious dis-eases.Motivated by this,this paper introduces the influence of information related to disease prevalence on increasing vaccination coverage and reducing disease incidence into the traditional infectious disease model,and proposes a class of nonlinear infectious dis-ease model with disease information factor and time delay.Simultaneously,the dynamical behaviors of the proposed model and optimal strategies for controlling disease prevalence are discussed by using related theories and methods of the differential dynamical system and the optimal control theory.Firstly,the dynamical behaviors of a class of multiple-delayed infectious disease model with disease information are studied.On the one hand,ignoring the effect of time de-lay,some basic properties of the proposed model are given,and the sufficient conditions for global asymptotic stability of the endemic equilibrium Ee are obtained by using the geometric approach when the basic reproduction number R0>1.In addition,it is ana-lytically proved that the system will appear bifurcation periodic solution near Ee when the information degradation rate T reaches a certain critical value,which indicates the oscillatory persistence of the disease among the population.On the other hand,consid-ering the incubation period ?1 and the immune period ?2,the stability of the proposed model system is not affected by the time delay under certain conditions when R0<1 The influence of the above time delays on dynamical behaviors of the system is discussed in three cases of(i)?1>0,T2=0,(ii)?1=0,?2>0 and(iii)?1=?2>0 when R0>1,the conditions which can guarantee the existence of Hopf bifurcation and the critical values of the corresponding time delays are obtained.In particular,it is also proved that the system will appear the phenomena of multiple stability switch in case(ii).Furthermore,the results of theoretical analysis are verified by corresponding numerical simulations,and the simulation results show that the information reporting rate b and information degradation rate T will make dynamical behaviors of the proposed model system more complex,the timely and effective disease information has a significant effect on reducing the severity of the disease.Secondly,on the basis of the above model without time delay,this paper regards information-dependent vaccination and treatment as control variables to establish an op-timal control problem,in order to control the disease process and balance the economic loss caused by the disease and the implementation of intervention strategies.Based on the limited of resources,the weighted combination of the number of infected individuals and the cost of vaccination and treatment within a certain period of time is taken as the optimal performance index.The existence of optimal control pair is proved by using opti-mal control theory,and the corresponding optimal paths of control variables are obtained analytically by using Pontryagin's Maximum Principle.In addition,numerical simula-tions are applied to verify the applicability and effectiveness of information-dependent vaccination and treatment as optimal control pair,and the simulation results show that the comprehensive control effect is optimal when these two interventions are combined.