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

摘要

In this paper, the interest is in the use of time-discretized models as approximations to the continuous-time birth-death (BD) process I = {I (t) : t >= 0} describing the number I (t) of infective hosts at time t in the stochastic susceptible -> infective -> susceptible (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 I, and we show the similarities and differences between the discrete-time BD process (I) over tilde of Allen and Burgin (Math Biosci 163:1-33, 2000), which is inspired from the infinitesimal transition probabilities of I, and an alternative discrete-time Markov chain <()over bar>, which is defined in terms of the number I (tau(n)) of infective hosts at a sequence {tau(n) : n is an element of N-0} of inspection times. Processes (I) over tilde and (I) over bar can be thought of as a uniformized version and the discrete skeleton of process I, 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 I and its discrete-time counterparts (I) over tilde and (I) over bar behave asymptotically the same in the limit of large time index, while the processes (I) over tilde and (I) over bar differ from the continuous-time BD process I in terms of the random length of an outbreak, or when considering their dynamics during a predetermined time interval [0, t ']. To compare the dynamics of process I with those of the discrete-time processes (I) over tilde and (I) over bar during [0, t '], we consider extreme values (i.e., maximum and minimum number of infectives simultaneously observed during [0, t ']) 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 I and (I) over tilde, or I and (I) over bar.