In this paper, the interest is in the use of time-discretized models as approximations to the continuous-time birth–death (BD) process \(\mathcal{I}=\{I(t): t\ge 0\}\) describing the number I(t) of infective hosts at time t in the stochastic \(susceptible \rightarrow infective \rightarrow susceptible\) (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 \(\mathcal{I}\), and we show the similarities and differences between the discrete-time BD process \(\widetilde{\mathcal{I}}\) of Allen and Burgin (Math Biosci 163:1–33, 2000), which is inspired from the infinitesimal transition probabilities of \(\mathcal{I}\), and an alternative discrete-time Markov chain \(\bar{\mathcal{I}}\), which is defined in terms of the number \(I(\tau _n)\) of infective hosts at a sequence \(\{\tau _n: n\in \mathbb {N}_0\}\) of inspection times. Processes \(\widetilde{\mathcal{I}}\) and \(\bar{\mathcal{I}}\) can be thought of as a uniformized version and the discrete skeleton of process \(\mathcal{I}\), 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 \(\mathcal{I}\) and its discrete-time counterparts \(\widetilde{\mathcal{I}}\) and \(\bar{\mathcal{I}}\) behave asymptotically the same in the limit of large time index, while the processes \(\widetilde{\mathcal{I}}\) and \(\bar{\mathcal{I}}\) differ from the continuous-time BD process \(\mathcal{I}\) 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 \(\mathcal{I}\) with those of the discrete-time processes \(\widetilde{\mathcal{I}}\) and \(\bar{\mathcal{I}}\) 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 \(\mathcal{I}\) and \(\widetilde{\mathcal{I}}\), or \(\mathcal{I}\) and \(\bar{\mathcal{I}}\).

在本文中，兴趣在于使用时间离散模型作为连续时间生死 (BD) 过程的近似值\(\mathcal{I}=\{I(t): t\ge 0\} \)描述了随机\(易感 \右箭头感染 \右箭头易感\) (SIS) 流行模型中在时间t时感染宿主的数量I ( t )，该模型假设来自环境的额外感染源。我们说明了一些用于分析连续时间 BD 过程\(\mathcal{I}\) 的离散时间版本的简单技术，并展示了离散时间 BD 过程\(\widetilde{\mathcal{一世}}\）Allen 和 Burgin (Math Biosci 163:1–33, 2000) 的灵感来自\(\mathcal{I}\)的无穷小转移概率，以及另一种离散时间马尔可夫链\(\bar{\mathcal {I}} \） ，其在数量方面定义\（I（\ tau蛋白_n）\）在序列感染主机的\（\ {\ tau蛋白_n：N \在\ mathbb {N} _0 \} \)的检查次数。进程\(\widetilde{\mathcal{I}}\)和\(\bar{\mathcal{I}}\)可以被认为是一个统一的版本和进程的离散骨架\(\mathcal{I}\ )分别，并且通常用于在更一般的马尔可夫链设置中，通过应用离散时间马尔可夫链的已知定理来推导出关于连续时间马尔可夫链的定理。我们将在这里证明连续时间 BD 过程\(\mathcal{I}\)及其离散时间对应物\(\widetilde{\mathcal{I}}\)和\(\bar{\mathcal{I} }\)在大时间索引的极限上表现渐近相同，而过程\(\widetilde{\mathcal{I}}\)和\(\bar{\mathcal{I}}\)不同于连续的 -时间 BD 过程\(\mathcal{I}\)就爆发的随机长度而言，或者在考虑它们在预定时间间隔内的动态时\([0,t']\)。比较过程\(\mathcal{I}\)与离散时间过程\(\widetilde{\mathcal{I}}\)和\(\bar{\mathcal{I}}\)的动力学在\([0,t']\) 期间，我们考虑了这三个过程中的极值（即在\([0,t']\)期间同时观察到的最大和最小感染数）。最后，我们通过一些数值例子来说明我们的分析结果，其中我们使用两个概率分布之间的海灵格距离来量化\(\mathcal{I}\)和\( \widetilde{\mathcal{I}}\)，或\(\mathcal{I}\)和\(\bar{\mathcal{I}}\)。