Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.
J Theor Biol. 2018 Jul 14;449:35-52. doi: 10.1016/j.jtbi.2018.04.023. Epub 2018 Apr 16.
While deterministic metapopulation models for the spread of epidemics between populations have been well-studied in the literature, variability in disease transmission rates and interaction rates between individual agents or populations suggests the need to consider stochastic fluctuations in model parameters in order to more fully represent realistic epidemics. In the present paper, we have extended a stochastic SIS epidemic model - which introduces stochastic perturbations in the form of white noise to the force of infection (the rate of disease transmission from classes of infected to susceptible populations) - to spatial networks, thereby obtaining a stochastic epidemic metapopulation model. We solved the stochastic model numerically and found that white noise terms do not drastically change the overall long-term dynamics of the system (for sufficiently small variance of the noise) relative to the dynamics of a corresponding deterministic system. The primary difference between the stochastic and deterministic metapopulation models is that for large time, solutions tend to quasi-stationary distributions in the stochastic setting, rather than to constant steady states in the deterministic setting. We then considered different approaches to controlling the spread of a stochastic SIS epidemic over spatial networks, comparing results for a spectrum of controls utilizing local to global information about the state of the epidemic. Variation in white noise was shown to be able to counteract the treatment rate (treated curing rate) of the epidemic, requiring greater treatment rates on the part of the control and suggesting that in real-life epidemics one should be mindful of such random variations in order for a treatment to be effective. Additionally, we point out some problems using white noise perturbations as a model, but show that a truncated noise process gives qualitatively comparable behaviors without these issues.
虽然在文献中已经对人群之间传染病传播的确定性元胞自动机模型进行了很好的研究，但疾病传播率和个体或群体之间相互作用率的可变性表明，需要考虑模型参数中的随机波动，以便更全面地表示现实中的传染病。在本文中，我们扩展了一个随机 SIS 传染病模型 - 在感染力（从感染人群到易感人群的疾病传播率）中引入白噪声形式的随机扰动 - 到空间网络中，从而获得了一个随机传染病元胞自动机模型。我们通过数值方法求解了随机模型，并发现白噪声项不会相对于相应的确定性系统动力学，使系统的整体长期动力学（对于噪声的方差足够小）发生巨大变化。随机和确定性元胞自动机模型的主要区别在于，对于大时间，解在随机环境中趋于准静态分布，而在确定性环境中趋于常数稳态。然后，我们考虑了控制随机 SIS 传染病在空间网络上传播的不同方法，比较了利用传染病状态的局部到全局信息的一系列控制方法的结果。白噪声的变化被证明能够抵消传染病的治疗率（治疗治愈率），需要控制部分更大的治疗率，并表明在现实生活中的传染病中，人们应该注意这种随机变化，以便治疗有效。此外，我们指出了使用白噪声扰动作为模型的一些问题，但表明截断噪声过程可以在没有这些问题的情况下给出定性可比的行为。
