We adopt a maximum-likelihood framework to estimate parameters of a stochastic susceptible-infected-recovered (SIR) model with contact tracing on a rooted random tree. Given the number of detectees per index case, our estimator allows to determine the degree distribution of the random tree as well as the tracing probability. Since we do not discover all infectees via contact tracing, this estimation is non-trivial. To keep things simple and stable, we develop an approximation suited for realistic situations (contract tracing probability small, or the probability for the detection of index cases small). In this approximation, the only epidemiological parameter entering the estimator is $R_0$. The estimator is tested in a simulation study and is furthermore applied to covid-19 contact tracing data from India. The simulation study underlines the efficiency of the method. For the empirical covid-19 data, we compare different degree distributions and perform a sensitivity analysis. We find that particularly a power-law and a negative binomial degree distribution fit the data well and that the tracing probability is rather large. The sensitivity analysis shows no strong dependency of the estimates on the reproduction number. Finally, we discuss the relevance of our findings.

翻译：我们采用最大似然框架来估计具有接触追踪的随机易感-感染-恢复（SIR）模型在根随机树上的参数。给定每个指示病例的检测人数，我们的估计量能够确定随机树的度数分布以及追踪概率。由于我们未能通过接触追踪发现所有感染者，这一估计并非易事。为保持简洁稳定，我们开发了一种适用于现实情况的近似方法（接触追踪概率较小，或指示病例检测概率较小）。在该近似中，唯一进入估计量的流行病学参数是$R_0$。该估计量通过模拟研究进行了测试，并进一步应用于来自印度的新冠肺炎接触追踪数据。模拟研究强调了该方法的效率。对于实证新冠肺炎数据，我们比较了不同的度数分布并进行了敏感性分析。我们发现幂律分布和负二项式度数分布尤其能很好地拟合数据，且追踪概率相当大。敏感性分析显示，估计值对再生数的依赖性不强。最后，我们讨论了研究结果的相关性。