Estimate On Convergence Rate Of A SIRk Epidemic Model In Non-Autonomous System

SIR epidemic model is a very important class of mathematical models in the study of epidemic dynamics. Because under normal circumstances, the epidemic model in the non-autonomous system can’t obtain the exact solution. So, in order to analyze the SIR epidemic model in the non-autonomous system, we usually used classical theory of ODE for qualitative analysis.In this paper, first, we consider the classical SIR epidemic model as follows. Assuming that the total amount of people in the area is constant, the proportion of Susceptible people is S, I is the proportion of infective people, the proportion of Removed people is R, the. Daily contact rate of Susceptible people is λ, and the daily cure rate is μ:On the basis of the SIR model, we consider the more general case. During the epidemic, if we suppose the population birth rate is k, and assuming that the newborns are health and susceptible. So, the SIR model can be modified for SIRk model:, and k will be recorded as contact number, σ=μ/λ is Contact number. It is obvious that, the system has a unique equilibrium point which is stable. If considering the daily contact rate λ and the daily cure rate μ will change with time, then we will add into two influence factors which are the increase speed of the knowledge of infectious diseases prevention and control means called a and social medical progress speed b. a and b are constant. If the daily contact rate isA(t)=A(l+e), daily cure rate isμ(t)=μ(1-e-bt), easy to know that when tâ†'∞, the equilibrium point of the system is Thus, the original epidemic SIRk model of autonomous system turns into the SIRk model of non-autonomous system.This paper is according to the the SIRk model under non-autonomous system, using the classical theory of ordinary differential equations and the method of variation of parameters, inequality zooming method to estimate the vector form solution by using L2norm estimation. Then we prove that the solution of the system will Exponential converge to the equilibrium point by using the nonlinear Gronwall inequality, and last estimate the convergence rate of the exponential convergence to equilibrium. Conclusions are as follows:If λk>4μ2, the differential equation containing two different real eigenvalues of the coefficient matrix, and we will estimate the module of the solution of the vector form is exponential convergence as em1t, which m1<0,and m1is the larger one among two negative real eigenvalues larger. At last, we estimate the convergence rate;If λk=4μ2, the differential equation containing two same real eigenvalues of the coefficient matrix, and we will estimate the module of the solution of the vector form is exponential convergence as which m1<0,and m1is the larger one among two negative real eigenvalues larger. In the end, we estimate the convergence rate;If λk<4μ2, the differential equation containing two different complex eigenvalues of the coefficient matrix, and we will estimate the module of the solution of the vector form is exponential convergence as which m1<0,and m1is the larger one among two negative real eigenvalues larger. And, we estimate the convergence rate;...