Act Now or Forever Hold Your Peace: Slowing Contagion with Unknown Spreaders, Constrained Cleaning Capacities and Costless Measures

What can be done to slow contagion when unidentified healthy carriers are contagious, total isolation is impossible, cleaning capacities are constrained, contamination parameters and even contamination channels are uncertain? Short answer: reduce variance. I study mathematical properties of contagion when many people share successively a limited number of devices (e.g. restrooms, which have been identified as a potential contamination channel for COVID19) and may get contaminated if the device has been used by an unidentified already contaminated person. I show that the number of exposures is a convex function of the number n of people using the device between two cleanings. As a direct application of Jensen inequality, contamination can be reduced at no cost by limiting the variance of n. These results are qualitatively robust to large changes in parameters, which is relevant in contexts of high uncertainty. The gains from an optimal use and cleaning of the devices can be substantial in this baseline framework: with a 1% proportion of (unknown) contaminated people, cleaning one device after 5 uses and the other after 15 uses increases contamination by 26 % with respect to the optimal organization (cleaning each device after 10 uses). The relative gains decrease when the proportion of spreaders increases. The absolute gains reach a peak for a low proportion of contaminated people, especially when the cleaning capacities are highly constrained. Thus, optimal organization is more beneficial at the beginning of an epidemic, providing additional reason for early action during an epidemic (the traditional reason, which is first-order, is that contamination is approximately exponential over the expansion phase of an epidemic). These convexity results extend only partially to simultaneous use situations, since the exposure function becomes concave above a threshold which decreases with the proportion of spreaders. Still, reducing convexity is beneficial overall. If the number of spreaders affects the probability of contamination, relative effects of better organization can be much larger than in the baseline framework. Eventually, reducing variance makes it possible to slow contamination during an existing epidemic, not to reduce the probability of an outbreak when only a handful cases exist.