A mathematical model for virus infection in a system of interacting computers

Abstract

Abstract. We introduce a simplified theoretical model to describe a virtual virus propagation process in a set of interacting computers. The propagation mechanisms considered here are those related to the reception of messages through internet as well as the ones concerning the simple exchange of files using recording devices as compact disks or the commonly used floppy disks.In spite of its inherent simplicity, this model provides a good idea of the infection process and trends. From the mathematical point of view, the nonlinear delay integral equation that we obtain here presents certain interesting features which are explored and enlightened in this paper.Mathematical subject classification: 45G10, 49K25, 92D99.

Dynamics of a virtual virus infection process involving a spatial distribution of interacting computers

“…Notice that it reduces to the simplest case of homogeneously mixed computers if pðx, yÞ ¼ 1, ðt, xÞ ¼ 1, is independent of x, and ðx, yÞ ¼ t 0 . This particular case is just the one already studied in [4].…”

“…From now on, the reasoning is the same followed in [4] to get the general evolution of infection processes in the framework of SIS models, namely…”

“…As in the simpler homogeneous mixing case considered in [4], stationary solutions can be easily and explicitly obtained in certain particular cases. For instance, in the case that T ðxÞ ¼ T and pðx, yÞ ¼ p 0 are constant functions, then the condition (2) implies that p 0 ¼ 1 and it is easily seen that…”

“…Although a lot of successful effort has been devoted to the development of virtual vaccines during the last years, progress in understanding the propagation process itself, quantitatively, has been rather slow. Recently [4], a mathematical model for a virus infection propagation process in a system of interacting computers has been reported by the authors, but only the simplest homogeneous mixing case was considered there. Although in relatively small communities some form of homogeneous mixing can be assumed, such an approach has no sense when dealing with widely dispersed populations.…”

“…In order to describe the space-time evolution of the virtual infection process, we propose a generalization of the model developed by us in [4], which consists of an integral equation involving densities of susceptible and infected computers.…”