Stability of epidemic model with time delays influenced by stochastic perturbations

Many processes in automatic regulation, physics, mechanics, biology, economy, ecology etc. can be modelled by hereditary equations (see, e.g. [1±6]). One of the main problems for the theory of stochastic hereditary equations and their applications is connected with stability. Many stability results were obtained by the construction of appropriate Lyapunov functionals. In [7± 11], the procedure is proposed, allowing, in some sense, to formalize the algorithm of the corresponding Lyapunov functionals construction for stochastic functional differential equations, for stochastic difference equations. In this paper, stability conditions are obtained by using this procedure for the mathematical model of the spread of infections diseases with delays influenced by stochastic perturbations. # 1998 IMACS/Elsevier Science B.V.