| Abstract | We study two different approaches to the Susceptible-Infective-Susceptible (SIS) epidemiological model. The first one consists of a single differential equation, while the second one is given by a discrete time Markov chain (DTMC) model. The large time behaviour of the dynamics of these two models is known to differ whenever the basic reproductible number, R0, is larger than one. We show, however, that their behaviour is similar for finite time, and that the maximum time (for a given maximum admissible difference between the solution of both models) diverge when the population goes to infinity. We introduce a new model, based on a partial differential equation of drift-diffusion type. The corresponding diffusion is degenerated at one of the boundaries, and we show that this model approximates the evolution of the DTMC in all time scales. We also show that the solution of the SIS ordinary differential equation model gives the most probable state of the DTMC, assuming that the DTMC is not absorbed. In addition, we study the effect of a finite population comparing the DTCM and the PDE model with the classical ODE model. We find that, for initial conditions far from the absorbing state, the ODE is a very good approximation for an exponentially long time, even if the population is not very large. For the initial conditions close to the absorbing state, such as the ones used for the study of the onset of a disease, we find that both the discrete and PDE models differ considerably from the ODE model. In particular, for R0 > 1, we obtain numerically that, with a probability 1=R0, the disease extinguishes itself without becoming endemic |
