Marchuk’s Models of Infection Diseases: New Developments

Abstract

We consider mathematical models of infectious diseases built by G. I. Marchuk in his well-known book on immunology. These models are in the form of systems of ordinary delay differential equations. We add a distributed control in one of the equations describing the dynamics of the antibody concentration rate. Distributed control looks here naturally since the change of this concentration rather depends on the corresponding average value of the difference of the current and normal antibody concentrations on the time interval than on their difference at the point t only. Choosing this control in a corresponding form, we propose some ideas of the stabilization in the cases, where other methods do not work. The main idea is to reduce the stability analysis of a given integro-differential system of the order n, to one of the auxiliary systems of the order \(n+m\), where m is a natural number, which is “easy” for this analysis in a corresponding sense. Results for these auxiliary systems allow us to make conclusions for the given integro-differential system of the order n. We concentrate our attempts in the analysis of the distributed control in an integral form. An idea of reducing integro-differential systems to systems of ordinary differential equations is developed. We present results about the exponential stability of stationary points of integro-differential systems using the method based on the presentation of solution with the help of the Cauchy matrix. Various properties of integro-differential systems are studied by this way. Methods of the general theory of functional differential equations developed by N. V. Azbelev and his followers are used. One of them is the Azbelev W-transform. We propose ideas allowing to achieve faster convergence to stationary point using a distributed control. We obtain estimates of solutions using estimates of the Cauchy matrices.