Structured epidemic models can be formulated as first-order hyperbolic PDEs, where the "spatial" variables represent individual traits, called structures. For models with two structures, we propose a numerical technique to approximate $R_{0}$, which measures the transmissibility of an infectious disease and, rigorously, is defined as the dominant eigenvalue of a next-generation operator. Via bivariate collocation and cubature on tensor grids, the latter is approximated with a finite-dimensional matrix, so that its dominant eigenvalue can easily be computed with standard techniques. We use test examples to investigate experimentally the behavior of the approximation: the convergence order appears to be infinite when the corresponding eigenfunction is smooth, and finite for less regular eigenfunctions. To demonstrate the effectiveness of the technique for more realistic applications, we present a new epidemic model structured by demographic age and immunity, and study the approximation of $R_{0}$ in some particular cases of interest.
翻译:结构性的流行病模型可以作为一阶双曲式PDE, “ 空间” 变量代表个体特性, 称为结构。 对于有两种结构的模型, 我们提出一个数字技术, 大约为R ⁇ 0}美元, 用来测量传染病的传染性, 严格地说, 被定义为下一代操作者的主导性电子元值。 在高温网格上, 双倍地合用和幼稚可与一个有限维基矩阵相近, 这样它的主要电子元值可以很容易地用标准技术来计算。 我们用试验示例来实验近似行为: 当相应的电子元功能平稳时, 趋同顺序看起来是无限的, 并且限制不那么普通的电子元。 为了证明这种技术在更现实的应用上的有效性, 我们提出了一个按人口年龄和免疫度构建的新的流行病模型, 并研究在某些特定感兴趣的情况下, 近似值为 $R ⁇ 0 美元 。