Persistent heterogeneity not short-term overdispersion determines herd immunity to COVID-19

Abstract

It has become increasingly clear that the COVID-19 epidemic is characterized by overdispersion whereby the majority of the transmission is driven by a minority of infected individuals. Such a strong departure from the homogeneity assumptions of traditional well-mixed compartment model is usually hypothesized to be the result of short-term super-spreader events, such as individual's extreme rate of virus shedding at the peak of infectivity while attending a large gathering without appropriate mitigation. However, heterogeneity can also arise through long-term, or persistent variations in individual susceptibility or infectivity. Here, we show how to incorporate persistent heterogeneity into a wide class of epidemiological models, and derive a non-linear dependence of the effective reproduction number R_e on the susceptible population fraction S. Persistent heterogeneity has three important consequences compared to the effects of overdispersion: (1) It results in a major modification of the early epidemic dynamics; (2) It significantly suppresses the herd immunity threshold; (3) It significantly reduces the final size of the epidemic. We estimate social and biological contributions to persistent heterogeneity using data on real-life face-to-face contact networks and age variation of the incidence rate during the COVID-19 epidemic, and show that empirical data from the COVID-19 epidemic in New York City (NYC) and Chicago and all 50 US states provide a consistent characterization of the level of persistent heterogeneity. Our estimates suggest that the hardest-hit areas, such as NYC, are close to the persistent heterogeneity herd immunity threshold following the first wave of the epidemic, thereby limiting the spread of infection to other regions during a potential second wave of the epidemic. Our work implies that general considerations of persistent heterogeneity in addition to overdispersion act to limit the scale of pandemics.

Correlation of the rise and fall in COVID-19 cases with the social isolation index and early outpatient treatment with hydroxychloroquine and chloroquine in the state of Santa Catarina, southern Brazil: A retrospective analysis

“…This situation leads to the reduction of disease incidence, but this effect would vane easily over time as soon as social interactions within a cluster or between the different clusters of the population increases. This concept is better defined as “transient collective immunity” [ 28 ].…”

A Novel Model for Simulating COVID-19 Dynamics Through Layered Infection States that Integrate Concepts from Epidemiology, Biophysics and Medicine: SEI3R2S-Nrec

“…This leads to theoretical elegance, and such models have been useful as targets of the past such sexually-transmitted diseases. One nice simplification has involved combining two λ's in series, with one for societal and one for biological variance, each representing a "heterogeneity" distribution and assumed to be independent of each other (15). Such models tend to be best equipped for evaluating steady-state behavior, for formulating various alternatives for reproduction ratios, and for HIT estimates.…”

“…SEIRS-type models, often in simple forms, have been used extensively to illustrate epidemiological principles. In most cases, the flux rates have been assumed to be constants, often with one exception for the S→E flux: a modulated distribution "force of infection" or "immunity factor" called λ (14)(15)(16) or multiplicative flux modulator called β in the literature. More sophisticated models use statistics and covariates to obtain λ or β (e.g., 11) or include multi-SEIR models for partitioning age, spatial location, etc.…”

“…(9,12,17,18). Principles that have emerged from such modeling efforts include: formalizing the Reproduction Ratio (Ro, secondary cases/case) as a function of rates for various model formulations that can include additions such as birth rates (5)(6)(7)(8), illustrating advantageous properties of successful pathogen mutation (5)(6)(7), identifying conditions for stability and steadystate behavior (5)(6)(7)(8), identifying conditions for a "herd immunity threshold" (HIT) (14)(15)(16), and accommodating seasonal fluctuations (11,13). As subpopulations such as S change, often an "effective" or "instantaneous" reproductive ratio Re (12)(13)(14)(15)(16)17), Ri (19) are calculated over time for a deterministic or stochastic model.…”

“…Still another body of work is based on combining complex connection networks and/or geolocation "spacetime" network-type models to SEIRS-type compartmental models, which can be challenging since the two UoD's tend not to mesh well. Such approaches involve making assumptions on networks, using SEIR networks with partitioned subpopulations such as by age (12,17,18) or make heterogeneous connectivity distribution networks usually assumed to persistent over time (13,15), and have developed a strong theoretical foundation (12)(13)(14)(15). A key feature is capturing population heterogeneity assuming a unimodal distribution with a single mean and then a spread, for instance, with examples of assumed variational distribution including normal (Gaussian), binomal, Poisson, a power law, or centered exponential decay.…”