Persistence and extinction are key processes in infectious disease dynamics that, due to incomplete reporting, are seldom directly observable. For fully-immunizing diseases, reporting probabilities can be readily estimated from demographic records and case reports. Yet reporting probabilities are not sufficient to unambiguously reconstruct disease incidence from case reports. Here, we focus on disease presence (i.e., marginal probability of non-zero incidence), which provides an upper bound on the marginal probability of disease extinction. We examine measles and pertussis in pre-vaccine era U.S. cities, and describe a conserved scaling relationship between population size, reporting probability, and observed presence (i.e., non-zero case reports). We use this relationship to estimate disease presence given perfect reporting, and define cryptic presence as the difference between estimated and observed presence. We estimate that, in early 20 th century U.S. cities, pertussis presence was higher than measles presence across a range of population sizes, and that cryptic presence was common in small cities with imperfect reporting.While the methods employed here are specific to fully-immunizing diseases, our results suggest that cryptic incidence deserves careful attention, particularly in diseases with low case counts, poor reporting, and longer infectious periods.Recurrent epidemics are a common feature of these diseases, driven by long-term host demographics and periodic forcing of disease transmission via changes in host density, such as school terms [2][3][4] or economic migration [5,6]. At high incidence, susceptible hosts are rapidly depleted, leading to subsequent inter-epidemic troughs of low incidence, where stochastic extinction can occur. When infection is low or absent from a population, susceptible replenishment proceeds via the host demographic processes of birth and migration. These forces combine to yield characteristic yearly and multi-annual epidemic cycles in a range of diseases and human populations [7][8][9][10][11][12][13][14].The life histories of measles and pertussis differ significantly in pace: measles has a shorter life cycle, is more "invasive", and experiences more pronounced epidemics, while pertussis is the superior "colonizer". The slower life history of pertussis is expected to dampen the effects of isolation relative to measles, and is predicted to enhance dynamical stochasticity [15,16]. In pertussis, the contribution of waning immunity to observed dynamics has been a subject of extensive debate, both in infection-derived and vaccine-derived immunity [17][18][19]. In the pre-vaccine era, however, pertussis dynamics are consistent with the dynamics of infections that confer relatively long-lasting immunity, irrespective of whether the mechanism is long-term protection or natural immune boosting [20,21].
Reporting probabilities vary widely between these diseases, as well as between locations [3, 22-24]. Measlesinfection causes characteristic symptoms of fever, rash, and pathognomoni...