A PROBABILISTIC-AUTOMATA NETWORK EPIDEMIC MODEL WITH BIRTHS AND DEATHS EXHIBITING CYCLIC BEHAVIOR

A probabilistic automata network model for the spread of an infectious disease in a population of moving individuals is studied. The local r ule consists of two subrules. The first one, applied synchronously, mo dels infection, birth and death processes. It is a probabilistic cellu lar automaton rule. The second, applied sequentially, describes the mo tion of the individuals. The model contains six parameters: the probab ilities p for a susceptible to become infected by contact with an infe ctive; the respective birth rates b(s) and b(i) of the susceptibles fr om either a susceptible or an infective parent; the respective death r ates d(s) and d(i) of susceptibles and infectives; and a parameter in characterizing the motion of the individuals. The model has three fixe d points. The first is trivial, it describes a stationary state with n o living individuals. The second corresponds to a disease-free state w ith no infectives. The third and last one characterizes an endemic sta te with non-zero densities of susceptibles and infectives. Moreover, t he model may exhibit oscillatory behaviour of the susceptible and infe ctive densities as functions of time through a Hopf-type bifurcation. The influence of the different parameters on the stability of all thes e states is studied with a particular emphasis on the influence of mot ion which has been found to be a stabilizing factor of the cyclic beha viour.