ArticleJournal of mathematical biology2021
On time-discretized versions of the stochastic SIS epidemic model: a comparative analysis.
Article in Journal of mathematical biology, 2021. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Not yet cited in PubMed.
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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].
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