We consider a stochastic susceptible-infected-recovered (SIR) model with contact tracing on random trees and on the configuration model. On a rooted tree, where initially all individuals are susceptible apart from the root which is infected, we are able to find exact formulas for the distribution of the infectious period. Thereto, we show how to extend the existing theory for contact tracing in homogeneously mixing populations to trees. Based on these formulas, we discuss the influence of randomness in the tree and the basic reproduction number. We find the well known results for the homogeneously mixing case as a limit of the present model (tree-shaped contact graph). Furthermore, we develop approximate mean field equations for the dynamics on trees, and - using the message passing method - also for the configuration model. The interpretation and implications of the results are discussed.

中文翻译：

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在随机树上进行接触追踪的精确和近似公式。

我们考虑随机易感者感染者恢复（SIR）模型，在随机树和配置模型上进行接触追踪。在一棵有根的树上，除了被感染的根之外，最初所有个体都易感，我们能够找到感染期分布的精确公式。为此，我们展示了如何将现有的均匀混合种群接触追踪理论扩展到树木。基于这些公式，我们讨论了树中随机性和基本再生数的影响。我们发现均匀混合情况的众所周知的结果作为当前模型的限制（树形接触图）。此外，我们还为树上的动力学开发了近似平均场方程，并且 - 使用消息传递方法 - 也为配置模型开发了近似平均场方程。讨论了结果的解释和含义。