The aim of the present paper is to show a way in which the problem of predicting the evolution of an epidemic may be tackled by describing it in the framework of Boltzmann’s kinetic theory, as it has been developed and applied in the last years to complex systems by a suitable modification of the Boltzmann equation, a suitable re-interpretation of state variables and the introduction of the notion of «functional subsystems». Accordingly, in this paper we model an arbitrary (national) population S as a complex system, split in two functional subsystems, the first containing all single individuals of S and the second containing the «care tools», that are to be meant as available places in hospitals with a sufficient number of physicians and of equipments for intensive cares. The state variable on the first subsystem will be the «health state», and the state variable on the other will be the «effectiveness». We shall then write a system of non-linear ordinary differential equations which gives the evolution of the probability distribution on the set of possible values of the health states. By assigning data partly on the basis of plausibility assumption and partly as estimated from those furnished by institutions of Campania region, the system takes a form allowing the numerical simulation of such evolution, which will be performed and presented in a forthcoming paper.