Fig 1.
A framework for counterfactual analysis, strategy evaluation, and feedback control of epidemics using reproduction number estimates.
The framework consists of three pieces and three methods. The first piece is to leverage real-world spreading data to estimate the effective reproduction number, where we propose a method to quantify the impact of the isolation rate on the basic reproduction number. The second step involves performing counterfactual analysis by introducing a method to reverse engineer the effective reproduction number, enabling simulation of hypothetical spreading scenarios without the implemented intervention strategy or with an alternative strength of intervention. The third component introduces a closed-loop control algorithm that uses the effective reproduction number as feedback to adjust the isolation rate, which in turn influences the effective reproduction number to manage the spread.
Fig 2.
A) The infection profile v of COVID-19 w/o testing-for-isolation strategies. The infection profile v is captured by Eq (5). We leverage the infection profile to capture the spreading process across UIUC and Purdue. B) The infection profile v(αP) of COVID-19 w/ testing-for-isolation strategies. The infection profile v(αP) reflects the impact of the overall isolation rate αP at Purdue, which reduces the average daily number of infected cases generated by a single infected individual under Purdue’s testing-for-isolation strategy.
Fig 3.
A) Daily confirmed cases at UIUC during Fall 2020 and Spring 2021. We mark five shaded areas for five important events across the two semesters, corresponding to the entry screening of Fall 2020 (I), the start of the Big Ten football season (II), Christmas Break (III), the entry screening of Spring 2021 (IV), and Spring Break (V). The simulated spreading process (dotted solid line) accurately captures the spreading trend observed on the UIUC campus during Fall 2020 and Spring 2021, including spikes and weekly confirmed pattern. We simulate the spreading process over Fall 2020 and Spring 2021 separately, since at the beginning of each semester, the entry-screening resets the spreading process. B) Estimated effective reproduction number for UIUC from Fall 2020 to Spring 2021. The estimated effective reproduction number (95% confidence interval) is greater than one during multiple periods, particularly at the beginning of Fall 2020, around the middle of October 2020, and the middle of Spring 2021, corresponding to the three events marked by Shaded Areas I, II, and V in Fig 3A. The effective reproduction number aligns with the confirmed cases at UIUC during Fall 2020 and Spring 2021, as shown in Fig 3A, where several mild spikes were observed. There is no estimated effective reproduction number from 2020-08-18 to 2020-08-24 because we use data from the seven-day window between 2020-08-18 and 2020-08-25 to estimate the effective reproduction number for 2020-08-25. We do not have sufficient data to estimate the effective reproduction number prior to 2020-08-25.
Fig 4.
A) Daily confirmed cases at Purdue during Fall 2020 and Spring 2021. We mark five shaded areas for five major events across the two semesters at Purdue, corresponding to the entry screening of Fall 2020 (I), the start of the Big Ten football season (II), Thanksgiving Break (III), the entry screening of Spring 2021 (IV), and Spring Break (V). Purdue allowed students to stay home after Thanksgiving break, leading to a significant decrease in confirmed cases, as shown in Shaded Area III. The reconstructed spreading process (dotted solid line) matches the confirmed cases observed on the Purdue campus during Fall 2020 and Spring 2021. Unlike UIUC, where we reset the initial condition at the start of Spring 2021, not resetting it for Purdue results in overestimating daily cases during Spring 2021. B) Estimated effective reproduction number for Purdue from Fall 2020 to Spring 2021. The estimated effective reproduction number (95% confidence interval) was around one for most of the time, reflecting that Purdue’s testing-for-isolation strategy avoided potential large outbreaks. Two major spikes were observed in the estimated effective reproduction number (95% confidence interval) at Purdue: one around the beginning of August (I) and the other around the beginning of January (IV). As shown in Fig 4A, these two spikes correspond to the infection process during the Summer and Christmas breaks. Unlike UIUC, Purdue provided sufficient data prior to 2020-08-18, allowing us to estimate the effective reproduction number from 2020-08-18 to 2020-08-24, as shown in Fig 4B. To match the dates in both figures, we did not include data prior to 2020-08-18 in Fig 4A.
Fig 5.
A) Daily confirmed positive cases from hypothetical spreading scenario at UIUC during Fall 2020. The solid dark blue line represents the hypothetical spreading scenario without employing any testing-for-isolation strategies. The solid red line represents the simulated spreading process under UIUC’s testing-for-isolation strategy. The hypothetical spreading scenario captures the worst-case scenario where every individual on campus does not take actions against the pandemic. B) Reverse engineered effective reproduction number at UIUC. The dashed dark blue line illustrates the reverse engineered effective reproduction number of the hypothetical spreading scenario without the implementation of testing-for-isolation strategies. The dashed red line represents the estimated effective reproduction number obtained from the data at UIUC featuring the implemented testing-for-isolation strategy. C) Daily confirmed positive cases from hypothetical spreading scenario at Purdue during Fall 2020 and Spring 2021. Compared to the hypothetical spreading scenario at UIUC, the simulated outbreak at Purdue is less severe since, by assumption, all symptomatic cases are caught and isolated through the voluntary testing-for-isolation strategy. D) Reverse engineered effective reproduction number at Purdue. During the first two months of the Fall 2020 semester, the effective reproduction number of the hypothetical spreading scenario without isolation under surveillance testing consistently exceeds the estimated effective reproduction number obtained from the real spreading data on campus. After October 2020, the reduction in the susceptible population leads to the effective reproduction number of the hypothetical spreading scenario becoming smaller than the estimated effective reproduction number of the COVID-19 data from Purdue.
Fig 6.
A) Daily confirmed cases over the UIUC campus with different isolation rates. The fixed weekly isolation rates of the hypothetical spreading scenario are drawn from {0%, 10%, 20%, …, 90%} and {100%, 120%, …, 180%, 200%}. Higher isolation rates will result in relatively smoother and flatter curves in terms of outbreak. The shape of the brighter area also indicates that a higher isolation rate will lead to lower spikes, while these lower spikes will also be further delayed. B) Daily confirmed cases over Purdue campus with different isolation rates. The weekly isolation rates for asymptomatic cases are drawn from {0%, 10%, 20%, …, 90%} and {100%, 120%, …, 180%, 200%}. The daily confirmed cases of the hypothetical outbreak at Purdue are significantly lower than those on the UIUC campus, due to the existence of a voluntary testing-for-isolation strategy. Higher isolation rates generate relatively smoother and flatter curves in terms of confirmed cases.
Fig 7.
Closed-loop feedback control for the outbreak at UIUC.
When controlling the effective reproduction number at 0.95, the closed-loop feedback control algorithm we propose aligns with the testing-for-isolation policy implemented by UIUC in terms of daily and total confirmed cases, which are around 5000. Under the condition that the isolation rate is proportional to the number of tests, the implemented testing-for-isolation strategy by UIUC will result in a total of 32 tests per individual, whereas our proposed feedback control strategy will require 28 tests per individual.