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

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˜ 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 I¯, which is defined in terms of the number I(τn) of infective hosts at a sequence {τn:n∈N0} of inspection times. Processes I˜ and I¯ 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˜ and I¯ behave asymptotically the same in the limit of large time index, while the processes I˜ and I¯ 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˜ and I¯ 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˜, or I and I¯.