A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents’ decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.

中文翻译：

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线性链条技巧多快？行为流行病学背景下的严格分析

根据其地方平衡的稳定性分析了出生时接种疫苗的原型SIR模型。有关该疾病的可用信息会影响父母是否接种疫苗的决定。此信息会根据Erlang分布进行延迟建模。后者包括衰落存储器的退化情况以及集中存储器的极限情况。线性链条技巧是用于调查一般情况的基本工具。除了对其新颖性的分析和对集中案例的分析之外，研究还表明，通过线性链技巧，分布延迟以线性速率接近离散延迟。根据相关线性化问题的特征值给出了严格的证明，并且还提供了对通用模型的扩展。