Using random testing to manage a safe exit from the COVID-19 lockdown

Markus Mueller; Peter Derlet; Christopher Mudry; Gabriel Aeppli

We introduced in Sec. VI a feedback and control strategy to tune to a marginal state with vanishing growth rate k = 0 after an initial reboot. Interventions were only taken based on the measurement of the growth rate. However, in practice, a more refined strategy will be needed. In case the infection rate drops significantly below i * , one might (depending on netting out political and economic pressures, something which the authors of this paper are not doing here) benefit from a positive growth rate k. We thus assume that if i(t)/i * falls below some threshold i low = 0.2, we intervene by relaxing some measures, that we assume to increase k by an amount uniformly distributed in [0, k 1 ], but without letting k exceed the maximal value of k high = 0.23. Likewise, one should intervene when the fraction i(t) grows too large. We do so when i(t)/i * exceeds i high = 3. In such a situation we impose restrictions resulting in a decrease of k by a quantity uniformly drawn from [k high /2, k high ]. The precise algorithm is given in the Supplementary Information. Figure 3 shows how our algorithm implements policy releases and restrictions in response to test data. The initial infected fraction and growth rate are i(0) = i c /4 = 0.0007 and k 1 = 0.1, respectively, with a sampling interval of one day. We choose α = 3 and a number of r = 15 000 tests per day. Figure 3 (a) displays the infection fraction, U (t)/N , as a function of time, derived using our simple exponential growth model, which is characterized by a single growth rate that changes stochastically at interventions [Eq. (5) without the source term]. In the absence of intervention, the infected population would grow rapidly representing uncontrolled runaway of a second epidemic. At each time step (day) the currently infected fraction of the population is sampled. The result is normally distribution with mean and standard deviation given by Eqs. (6e) and (6f) to obtain i(t). The former are represented by small circles, the latter by vertical error bars in Fig. 3 . If i/i * lies outside the range [i low , i high ], we intervene as described above. Otherwise, on each day k fit (t) and its standard deviation are estimated using the data since the last intervention. With this, at each time step, Eqs. (6m) to (6o) decide whether or not to intervene. In Fig. 3 , each red circle represents an intervention and therefore either a decrease or increase of the growth rate constant of our model. Figure 3 shows the evolution of the fraction of currently infected people. After an initial growth with rate k 1 subsequent interventions reduce the growth rate down to low levels within a few weeks. At the same time the fraction of infected people stabilizes at a scale similar to i * . For the given parameter-set this is a general trend independent of realization. Figure 3 (b) displays the instantaneous value of the model rate constant and also the estimated value together with its fitting uncertainty. The estimate follows the model value reasonably well. One sees that the interventions occur when the uncertainty in k is sufficiently small.