Department of Plant Sciences, University of Cambridge, Cambridge, UK.
J R Soc Interface. 2011 Feb 6;8(55):201-9. doi: 10.1098/rsif.2010.0325. Epub 2010 Jul 14.
The percolation paradigm is widely used in spatially explicit epidemic models where disease spreads between neighbouring hosts. It has been successful in identifying epidemic thresholds for invasion, separating non-invasive regimes, where the disease never invades the system, from invasive regimes where the probability of invasion is positive. However, its power is mainly limited to homogeneous systems. When heterogeneity (environmental stochasticity) is introduced, the value of the epidemic threshold is, in general, not predictable without numerical simulations. Here, we analyse the role of heterogeneity in a stochastic susceptible-infected-removed epidemic model on a two-dimensional lattice. In the homogeneous case, equivalent to bond percolation, the probability of invasion is controlled by a single parameter, the transmissibility of the pathogen between neighbouring hosts. In the heterogeneous model, the transmissibility becomes a random variable drawn from a probability distribution. We investigate how heterogeneity in transmissibility influences the value of the invasion threshold, and find that the resilience of the system to invasion can be suitably described by two control parameters, the mean and variance of the transmissibility. We analyse a two-dimensional phase diagram, where the threshold is represented by a phase boundary separating an invasive regime in the high-mean, low-variance region from a non-invasive regime in the low-mean, high-variance region of the parameter space. We thus show that the percolation paradigm can be extended to the heterogeneous case. Our results have practical implications for the analysis of disease control strategies in realistic heterogeneous epidemic systems.
渗滤范式广泛应用于空间明确的传染病模型中，在这些模型中，疾病在相邻宿主之间传播。它成功地确定了入侵的流行病阈值，将非入侵状态（疾病从未入侵系统）与入侵状态（入侵的概率为正）区分开来。然而，它的功能主要限于均匀系统。当引入异质性（环境随机性）时，通常需要通过数值模拟才能预测流行病阈值的值。在这里，我们分析了在二维格点上的随机易感感染消除传染病模型中的异质性的作用。在均匀的情况下，相当于键渗滤，入侵的概率由单个参数控制，即相邻宿主之间病原体的传播性。在非均匀模型中，传播性成为从概率分布中抽取的随机变量。我们研究了传播性的异质性如何影响入侵阈值的值，发现系统对入侵的恢复能力可以用两个控制参数来很好地描述，即传播性的平均值和方差。我们分析了一个二维相图，其中阈值由一个相界表示，将高均值、低方差区域的入侵状态与低均值、高方差区域的非入侵状态分隔开来。因此，我们表明渗滤范式可以扩展到非均匀的情况。我们的结果对分析现实中异质流行病系统中的疾病控制策略具有实际意义。
