Barnard Rosanna C, Berthouze Luc, Simon Péter L, Kiss István Z
Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
Centre for Computational Neuroscience and Robotics, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
J Math Biol. 2019 Aug;79(3):823-860. doi: 10.1007/s00285-019-01380-1. Epub 2019 May 11.
The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.
在网络上的流行病建模中，流行阈值可能是研究最多的量。对于一大类网络和动力学，它已经得到了充分的研究和理解。然而，对于聚类网络来说情况并非如此，其理论结果大多局限于理想化网络。在本文中，我们关注一类被称为成对模型的模型，据我们所知，这类模型不存在关于流行阈值的解析结果。我们表明，通过利用快速变量的存在并运用微扰理论中的一些标准技术，我们能够解析地得到流行阈值。我们将这个新阈值与基于完整系统数值解的阈值进行比较，从而验证它。结果发现在聚类系数、传播率和网络平均度的广泛取值范围内，二者吻合得非常好。有趣的是，我们发现阈值的解析形式取决于封闭条件的选择，这凸显了在处理实际流行病时模型选择的重要性。尽管如此，我们预计我们的方法将推广到其他存在快速变量的系统。
