On time-discretized versions of the stochastic SIS epidemic model: a comparative analysis.

Abstract

In this paper, the interest is in the use of time-discretized models as approximations to the continuous-time birth-death (BD) process [Formula: see text] describing the number I(t) of infective hosts at time t in the stochastic [Formula: see text] (SIS) epidemic model under the assumption of an additional source of infection from the environment. We illustrate some simple techniques for analyzing discrete-time versions of the continuous-time BD process [Formula: see text], and we show the similarities and differences between the discrete-time BD process [Formula: see text] of Allen and Burgin (Math Biosci 163:1-33, 2000), which is inspired from the infinitesimal transition probabilities of [Formula: see text], and an alternative discrete-time Markov chain [Formula: see text], which is defined in terms of the number [Formula: see text] of infective hosts at a sequence [Formula: see text] of inspection times. Processes [Formula: see text] and [Formula: see text] can be thought of as a uniformized version and the discrete skeleton of process [Formula: see text], respectively, and are commonly used to derive, in the more general setting of Markov chains, theorems about a continuous-time Markov chain by applying known theorems for discrete-time Markov chains. We shall demonstrate here that the continuous-time BD process [Formula: see text] and its discrete-time counterparts [Formula: see text] and [Formula: see text] behave asymptotically the same in the limit of large time index, while the processes [Formula: see text] and [Formula: see text] differ from the continuous-time BD process [Formula: see text] in terms of the random length of an outbreak, or when considering their dynamics during a predetermined time interval [Formula: see text]. To compare the dynamics of process [Formula: see text] with those of the discrete-time processes [Formula: see text] and [Formula: see text] during [Formula: see text], we consider extreme values (i.e., maximum and minimum number of infectives simultaneously observed during [Formula: see text]) in these three processes. Finally, we illustrate our analytical results by means of a number of numerical examples, where we use the Hellinger distance between two probability distributions to quantify the similarity between the resulting extreme value distributions of either [Formula: see text] and [Formula: see text], or [Formula: see text] and [Formula: see text].