On limits of contact tracing in epidemic control

Contact tracing and quarantine are well established non-pharmaceutical epidemic control tools. The paper aims to clarify the impact of these measures in evolution of epidemic. The proposed deterministic model defines a simple rule on the reproduction number R in terms of ratio of diagnosed cases and, quarantine and transmission parameters. The model is applied to the early stage of Covid19 crisis in Poland. We investigate 3 scenarios corresponding to different ratios of diagnosed cases. Our results show that, depending on the scenario, contact tracing prevented from 50% to over 90% of cases. The effects of quarantine are limited by fraction of undiagnosed cases. The key conclusion is that under realistic assumptions the epidemic can not be controlled without any social distancing measures.


A.1 Optimization algorithm and initial data
In order to fit the values β 1 , β 2 , β 3 we use a standard gradient descent algorithm. Namely, we define error function as At each step we approximate the gradient of the error function with respect to β 1 , β 2 , β 3 by differential quotients and move in the direction opposite to the gradient. The algorithm reveals a good performance provided we start sufficiently close to the minimum, which is not difficult to ensure in our case.
It remains to choose the initial data. A closer look on results of simulations shows that the choice of initial data mostly influence the fitting in the beginning of period under consideration and hence the value of β 1 , while for analysis of future scenarios β 3 is the most important. Taking all this into account we do not struggle for sharp optimization of data fitting with respect to initial data and restrict to the following heuristic choice. It is natural to assume I u (0) = 1−κ κ I d (0). Concerning the choice of E(0) we assume it in a form E(0) = m(I d (0) + I u (0)). We set initial values I d (0) ∈ {10, 20, 30} and for each value we set I u (0) according to the above formula and three values of E(0) corresponding to m ∈ {2, 3, 4}. For each of these 9 combinations we run the optimization algorithm looking for the best fit of β i , i = 1 . . . 3. We have repeated this approach for κ ∈ {0.2, 0.5, 0.8}. It turns in that for all values of κ the best fit was obtained for I d (0) = 20 and m = 2. More careful analysis around I d (0) = 20 did not improve the quality of fitting, therefore: is our final choice. We obtain the following fitting error defined by (A.1): To close the set of initial data we put K(s) = 0 for −T ≤ s ≤ 0, which is a requirement of ODE with delay T .
A.2 Choice of fixed parameters 1. The parameter σ describes the rate of transition from non-infectious incubation state E into the infectious states I d or I u . The median incubation time from exposure till the onset of symptoms was estimated at 4 to 5 days [5,6,7]. However, there exists evidence that typically infectivity preceeds symptoms, by 1 to 3 days [8,1,2].
A modelling study identified the rate of transition between the non-infectious and infectious states at 1 3.69 [10], which corresponds to an average time lag of 3.69 days untill the case becomes infectious.
2. The parameter γ u represents the period of infectivity during the natural course of disease. We discuss the period of infectivity, especially as applied to mild cases. The median duration of viral shedding was estimated among 113 Chinese hospitalized patients. Overall it was 17 days, but it was shorter among cases with milder clinical course [14]. A study among 23 patients in Hong Kong confirmed viral shedding longer than 20 days among a third of patients, although the peak level of shedding was noted during the first week of infection [15]. In the mission report from China WHO reports viral shedding in mild and moderate cases to last 7 -12 days from symptom onset. Among younger and asymptomatic or mild cases the shedding may be shorter: in a study among 24 initially asymptomatic youngsters the median duration was 9.5 days [9].
3. The value of κ generally depends on the testing policy. However, recommended testing policies often rely on the presence of respiratory symptoms. This is also the case in Poland. It was observed that some infected people never develop symptoms, although the precised rate of such truly asymptomatic infections is still under investigation. Some studies may be biased by a too short follow-up time. A small study among residents of a long-term care skilled nursing facility found that even though more than half of individuals with confirmed infection were asymptomatic at the time of test, majority of them subsequently developed symptoms. The proportion of people who remain asymptomatic may be higher among younger individuals [9]. A study among Japanese nationals repatriated from Wuhan suggests the proportion of asymptomatic infections is about 30% [11]. An analysis among the passengers of Diamond Princess ship, where a COVID-19 outbreak occurred, taking into account this delayed onset of symptoms estimated the proportion of asymptomatic infections to be about 18%, even though almost 50% were asymptomatic on initial test. In addition, large scale screening implemented in Italian village Vo'Euganeo indicated that 50% to 75% of infected individuals did not report symptoms [3]. Similarly, in population screening in Iceland 50% were asymptomatic at the time of screening [12]. It may be stipulated that some of the people diagnosed through screening developed symptoms latter, consistently with the findings from the Diamond Princess study.
On the other hand a sizable proportion of infected people, especially at younger ages, experience only mild symptoms, for which they may not seek medical attention. In the study of Li [10], the proportion of undocumented cases was estimated as 86%.
4. The parameter γ d was estimated basing on a sample of case-based data available in routine surveillance, by fitting gamma distribution to the time from onset to diagnosis, for cases who were not in quarantine before diagnosis. Time from onset to diagnosis was estimated based on surveillance data available in the Epidemiological Reports Registration System for COVID-19, as of 28.04.2020.The system collects epidemiological data on cases diagnosed in Poland and is operated by local public health departments. All cases eventually are entered into the database. However, substantial reporting delays are noted. There were altogether 4976 cases registered in the system, including 1995 (40.1%), who did not have symptoms at the time of diagnosis. Plausible onset date and plausible diagnosis date were available for 2884 cases ( 96.7% of 2981 cases that were not asymptomatic) Gamma distribution was fitted by maximum likelihood to cases who were not diagnosed in quarantine. The observed and fitted distributions are shown below (figure A.1).
We next fitted gamma-regression model with week of diagnosis as an explanatory variable. We found no significant trend in time. We therefore adopted the average time from onset to diagnosis to be 4.6 days, and taking into account the probability of asymptomatic spread we assumed the parameter γ d to be 1/5.5.
5. Next we base θ on available data. We calculate prevalence of infection among the quarantined individuals, according to data published by the Chief Sanitary Inspectorate on the number of cases diagnosed among quarantined people and the total number of quarantined. We used a series of data 8.04 -20.04 to estimate a likely value of θ. We chose this time period due to data availability. Data are shown on the figure below. During this time period there was an increasing trend in the proportion of diagnosed from 0.5% to 0.8% A.2. We presume that this parameter could change with changing procedures of contact tracing and testing. However, since no detailed data were available, for the modelling purposes we chose a simplifying assumption that θ is stable (i.e. we always take a similar group of contacts under quarantine) selecting an average value of 0.6%. This proportion could be also viewed as attack rate among the contacts of cases. The proportion in Poland is in line with what was observed in Korea, where an estimated attack rate was 0.55% overall and 7.56% among household contacts [13], although household attack rate was higher (> 19%) in other studies [16]. 6. Furthermore we fix the parameter α. Here we make another simplification assuming this parameter to be constant. The main difficulty is a lack of precise data concerning the number of newly quarantined people per day, distinguishing between reasons of quarantine (travel related or contact tracing relate). At the beginning of epidemic in Poland the average amount of quarantined following one diagnosed case was definitely higher. Moreover, people coming back from abroad were subject to obligatory quarantine starting from March 16 and constituted a considerable part of quarantined in the second half of March and beginning of April. In particular, around 54 000 Polish citizens staying abroad came back within a special program of charter flights operated by Polish Airlines which ended on April 5. We can assume that after this date the ratio of people coming from abroad among all people subject to quarantine was negligible. As our model does not take migration into account, we have to take into account only quarantine from contact. For above reasons, for fitting α we restrict our analysis only to a period of 2 weeks of April. Assuming already θ = 0.006 we then choose α minimizing the square error between the number of quarantined from the data and computed K(t). This way we obtain α = 75.

A.3 Confidence intervals
Bootstrap. To estimate confidence intervals we use a method of parametric bootstrap. We generate M = 200 sequences of perturbed data assuming that for each time t ∈ {1, 54} the increment of R (i.e. daily number of new diagnoses) is a random number from Poisson distribution with mean value equal to increment of observed data.
Model parameters are also perturbed, see below. For each series of perturbed data we estimate the values of β i and take estimated confidence intervals as appropriate quantiles of obtained sets.
In order to estimate confidence intervals for R d (t), we proceed as follows. For each sequence of perturbed data we compute fitted R d (t). This way we obtain a set of curves  Distribution of parameters. Following other Authors [17] as well as experimental data, for the uncertainty analysis we used the following distributions of the parameters.
We take N = 200 (approximate average of daily number of diagnosed cases from the data). We approximate the mean value of N samples from Gamma distribution using Central Limit Theorem. Namely, we generate

A.4 Stability analysis -computation of R
Based on the classical approach to epidemiological models we address the basic question concerning the propagation of the disease. Namely, how many persons are infected by one infectious individual, a quantity which is usually called reproductive number, R. In order to compute this quantity we use the approach from [4]. Recall, the system readṡ We look then at the system assuming S ∼ N and E, I d , I u are close to zero, then we consider the following linearizationĖ Then one deduces (see [4]) that if we define R = max{eigenvalues of T Σ −1 } then the system is stable for R < 1 and it is unstable for R > 1.
Stability of system (A.6) means that the whole vector (E, I d , I u ) is going to zero, it follows that the main system (A.2) also tends to the zero solution for (E, I d , I u ). Instability implies that for "almost all" small data, the vector (E, I d , I u ) is growing in time (exponentially fast), causing the nonlinear system also evolves far away from the trivial state, i.e. E, I d , I u rapidly grow. By (A.5) we have Hence the stability of our system is determined by the following factor: To make a final comment, let us note that in case of spread of pandemia, as R d , R u grow, the above analysis become less reliable. Recall that β is normalized by N , so as S/N is not close to one and the analysis of stability becomes more complex.