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 people may be contaminated by using successively devices, such as restrooms, which have been identified as a potential contamination channel for COVID19. The expected number of exposures (at least one previous user was already contaminated and is thus a “spreader”) and new contaminations (which may increase with the number of spreaders among previous users and may also decrease with time) are always convex functions of the number n of users. As a direct application of Jensen inequality, contamination can be reduced at no cost by limiting the variance of n. The gains from 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, which is cleaning each device after 10 uses. The relative gains decrease when the proportion of spreaders increases. 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: once again, this calls for early action. Simultaneous use is the framework most often analyzed in the network literature, which may explain why the above convexity results have been overlooked. When multiple spreaders increase the probability of contamination, the degree of convexity depend on the precise effects of each additional spreader. With linear probabilities, the expected contamination curves are semi-parabolas, both for successive and simultaneous use. For other inverse link functions, convexity is always ensured in the successive use case but must be determined case by case for simultaneous use.