Dynamical Analysis And Control Strategies Of An SIVS Epidemic Model With Imperfect Vaccination On Scale-Free Networks

Infectious diseases have always threatened human life and health,and vaccination is one of the most effective methods to prevent and control infectious diseases.In this paper,based on the actual transmission process of infectious diseases,we establish an SIVS epidemic model with imperfect vaccination on scale-free networks by means of complex networks,epidemic dynamics and optimal control,and then we study its dynamical behaviors and control measures.The main research results are as follows:Firstly,three types of people with network characteristics were investigated:sus-ceptible people,infectious people and vaccinated people,and then we establish an SIVS epidemic model with imperfect vaccination on scale-free networks.Secondly,we obtain two threshold parameters????₀and₀,and derive the conditions for the ex-istence of disease-free equilibrium and endemic equilibrium.Then,we prove that the disease-free equilibrium is globally asymptotically stable by using comparison theo-rem of differential equations when????₀is not greater than one,and we study the at-tractivity of the endemic equilibrium by constructing iterative sequences when₀is greater than one.In terms of control measures,we propose immunization schemes such as uniform immunity,target immunity and acquaintance immunity.Taking the vaccinated rate function in the epidemic system as the control variable,the epidemic model as the main constraint,and the weighted sum of the number of infectious pop-ulation and relative costs of intervention as the objective function,an optimal control problem is proposed,and its necessary conditions are derived by using the Pontrya-gin's maximum principle.Finally,based on a scale-free network with 5000 nodes,Forward-Backward sweep method and the fourth-order Runge-Kutta method are used to numerically verify the previous theoretical results.In particular,the numerical lim-its of the weight ratio in the objective function of the optimal control problem are discussed.The problem discussed in this dissertation is the study of Applied Mathematic-s driven by practical problems,involving the fields of propagation dynamics,com-plex networks,ordinary differential equations,optimal control,etc.,and it is a cross-disciplinary subject.The work in this paper can not only enrich the theory of opti-mal control on complex networks,but also apply it to practical problems such as the prediction and control of infectious diseases with imperfect vaccination,and provide qualitative and quantitative references for the effective prevention and control of such diseases.