Analysis of the Asymptotic Behavior Solutions of Some Models of Epidemic Processes
Pertsev N.V., Pichugin B.Yu., Pichugina A.N.
Omsk Branch of Sobolev Institute of Mathematics SB RAS
Omsk F.M. Dostoevsky State University
Abstract. A collection of mathematical models of epidemic processes in the form of nonlinear systems delay differential equations, integro-differential equations and high-dimensioned ordinary differential equations is built. The results of analysis of asymptotic stability of trivial equilibrium of models are presented. The sufficient conditions for asymptotic stability of such equilibrium are got. The problem of stability of the models solutions for permanent disturbances is studied. Sufficient conditions are formulated in terms of smallness of the groups of susceptible individuals. Provides recommendations for the implementation of activities aimed at the containment of the epidemic process and reduce the incidence rate for tuberculosis and HIV-infection.
Key words: SIRS model, mathematical models of the spread of tuberculosis and HIV-infection, asymptotic stability of the equilibrium, stability by the first approximation, linear delay differential equation, quasi-nonnegative matrix, M-matrix.