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者数量，我们的估计量能够确定随机树的度分布以及追踪概率。由于我们并非通过接触追踪发现所有感染者，因此该估计并非易事。为了保持简单稳定，我们开发了一种适用于现实情况（接触追踪概率小，或指示病例检测概率小）的近似方法。在这种近似中，唯一进入估计量的流行病学参数是$R_0$。该估计量在模拟研究中得到测试，并进一步应用于来自印度的covid-19接触追踪数据。模拟研究突显了该方法的有效性。对于经验性的covid-19数据，我们比较了不同的度分布并进行了敏感性分析。我们发现幂律分布和负二项分布特别能很好地拟合数据，且追踪概率相当大。敏感性分析表明估计值对基本再生数没有强烈依赖性。最后，我们讨论了研究结果的相关性。