We rigorously investigate the convergence of a new numerical method, recently proposed by the authors, to approximate the reproduction numbers of a large class of age-structured population models with finite age span. The method consists in reformulating the problem on a space of absolutely continuous functions via an integral mapping. For any chosen splitting into birth and transition processes, we first define an operator that maps a generation to the next one (corresponding to the Next Generation Operator in the case of R0). Then, we approximate the infinite-dimensional operator with a matrix using pseudospectral discretization. In this paper, we prove that the spectral radius of the resulting matrix converges to the true reproduction number, and the (interpolation of the) corresponding eigenvector converges to the associated eigenfunction, with convergence order that depends on the regularity of the model coefficients. Results are confirmed experimentally and applications to epidemiology are discussed.

翻译：本文严格研究了一种新数值方法的收敛性，该方法由作者近期提出，用于逼近一大类具有有限年龄跨度的年龄结构种群模型的再生数。该方法通过积分映射将问题重新表述在绝对连续函数空间上。对于任意选定的出生与转移过程划分，我们首先定义一个将一代映射到下一代的算子（对应于R0情况下的下一代算子）。然后，我们使用伪谱离散化方法，用矩阵逼近该无穷维算子。本文证明了所得矩阵的谱半径收敛于真实再生数，且对应特征向量（的插值）收敛于关联的特征函数，其收敛阶取决于模型系数的正则性。实验结果验证了理论结论，并讨论了该方法在流行病学中的应用。