Oscillating epidemics: a discrete-time model

We study a model of an epidemic where the individuals which are cured from the infection are not permanently immunised but have a finite probability of becoming reinfected. We show that the epidemic does not follow the usual pattern of growth and decay but rather oscillates towards an “endemic” fixed point, where a certain number of infectives may persist. We present a discrete-time version of the model which may also serve as an integrator of the continuous-time one. An appropriate choice of the discrete system allows the derivation of a cellular-automaton version, which incorporates the essential dynamical behaviour of the model.