具有跳Lévy过程的随机SICA流行病模型
Stochastic SICA Epidemic Model with Jump Lévy Processes
摘要：
本文提出并研究了一个移位的SICA传染病模型，将Silva和Torres（2017）的模型推广到由布朗运动过程和跳跃Lévy噪声共同驱动的随机环境.现有的流行病学数学建模工作通常忽略了Lévy噪声扰动，但将其纳入SICA流行病模型是值得考虑的，因为艾滋病毒/艾滋病动态存在强烈波动，往往导致研究过程中出现一些不连续性。我们的工作组织如下：（i）我们首先介绍我们的模型，明确证明其使用的形式，即与Lévy噪声相关的分量;（ii）通过构造合适的停时，我们证明了全局正解的存在性和唯一性;（iii）在某些假设下，我们证明了艾滋病毒/艾滋病的灭绝;（iv）我们获得了确保艾滋病毒/艾滋病持续存在的充分条件;（v）通过数值模拟说明了我们的数学结果。
英文摘要：
We propose and study a shifted SICA epidemic model, extending the one of Silva and Torres (2017) to the stochastic setting driven by both Brownian motion processes and jump Lévy noise. Lévy noise perturbations are usually ignored by existing works of mathematical modelling in epidemiology, but its incorporation into the SICA epidemic model is worth to consider because of the presence of strong fluctuations in HIV/AIDS dynamics, often leading to the emergence of a number of discontinuities in the processes under investigation. Our work is organised as follows: (i) we begin by presenting our model, by clearly justifying its used form, namely the component related to the Lévy noise; (ii) we prove existence and uniqueness of a global positive solution by constructing a suitable stopping time; (iii) under some assumptions, we show extinction of HIV/AIDS; (iv) we obtain sufficient conditions assuring persistence of HIV/AIDS; (v) we illustrate our mathematical results through numerical simulations.
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