Dynamically consistent discrete-time SI and SIS epidemic models
Dynamically consistent discrete-time SI and SIS epidemic models
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动态一致的离散时间 SI 和 SIS 流行病模型
DOI：
10.3934/proc.2013.2013.653
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
L. Roeger
中科院分区：
文献类型：
--
作者：
L. Roeger
离散时间 $SI$ 和
$SIS$ 流行病模型是通过将非标准有限差分 (NSFD) 格式应用于微分方程模型来构建的。差分方程系统与其模拟连续时间模型动态一致。
考虑了无出生和死亡、有出生和死亡以及有移民的基本标准发生率 $SI$ 和 $SIS$ 模型。连续模型要么满足总人口为常数的守恒定律，要么总人口$N$满足$N'(t)=\lambda-\mu N$，因此当$t$接近无穷大时，$N$接近常数$\lambda/\mu$。通过 NSFD 方案的差分方程系统保留了所有属性，包括解的正性、守恒定律以及平衡的局部和部分全局稳定性。据说它们在这些属性方面与连续模型动态一致。
我们证明，选择某种 NSFD 方案以保留正解的简单标准通常是适当 NSFD 方案的指示。
Discrete-time $SI$ and
$SIS$ epidemic models are constructed by applying the nonstandard finite difference (NSFD) schemes to the differential equation models. The difference equation systems are dynamically consistent with their analog continuous-time models.
The basic standard incidence $SI$ and $SIS$ models without births and deaths, with births and deaths, and with immigrations, are considered. The continuous models possess either the conservation law that the total population is a constant or the total population $N$ satisfies $N'(t)=\lambda-\mu N$ and so that $N$ approaches a constant $\lambda/\mu$ as $t$ approaches infinity. The difference equation systems via NSFD schemes preserve all properties including the positivity of solutions, the conservation law, and the local and some of the global stability of the equilibria. They are said to be dynamically consistent with the continuous models with respect to these properties.
We show that a simple criterion for choosing a certain NSFD scheme such that the positivity solutions are preserved is usually an indication of an appropriate NSFD scheme.