Power-law population heterogeneity governs epidemic waves

Abstract

We generalize the Susceptible-Infected-Removed (SIR) model for epidemics to take into account generic effects of heterogeneity in the degree of susceptibility to infection in the population. We introduce a single new parameter corresponding to a power-law exponent of the susceptibility distribution at small susceptibilities. We find that for this class of distributions the gamma distribution is the attractor of the dynamics. This allows us to identify generic effects of population heterogeneity in a model as simple as the original SIR model which is contained as a limiting case. Because of this simplicity, numerical solutions can be generated easily and key properties of the epidemic wave can still be obtained exactly. In particular, we present exact expressions for the herd immunity level, the final size of the epidemic, as well as for the shape of the wave and for observables that can be quantified during an epidemic. In strongly heterogeneous populations, the herd immunity level can be much lower than in models with homogeneous populations as commonly used for example to discuss effects of mitigation. Using our model to analyze data for the SARS-CoV-2 epidemic in Germany shows that the reported time course is consistent with several scenarios characterized by different levels of immunity. These scenarios differ in population heterogeneity and in the time course of the infection rate, for example due to mitigation efforts or seasonality. Our analysis reveals that quantifying the effects of mitigation requires knowledge on the degree of heterogeneity in the population. Our work shows that key effects of population heterogeneity can be captured without increasing the complexity of the model. We show that information about population heterogeneity will be key to understand how far an epidemic has progressed and what can be expected for its future course.

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

“…The same result has also been recently obtained in Ref. (47). Note however that the full quasi-homogeneous description of the epidemic dynamics requires both R e ( S ) and S e ( S ), that in a general case a characterized by different scaling exponents.…”

Lockdown Effects on Sars-CoV-2 Transmission – The evidence from Northern Jutland

“…Early computer modelling indicated that mandated lockdown would be highly effective and prevent hundreds of thousands of deaths. [4] However, the homogenous models used to produce counterfactual curves have been criticized in many papers, since heterogeneity in susceptibility, activity, infectivity, and compliance all tend to flatten curves by themselves relative to counterfactual homogeneous populations [6]- [8], [10], [12]. To the extent that test-driven infection control or voluntary compliance do not fully explain our data, our results are consistent with this criticism and with country-comparison studies showing smaller direct effects of full lockdown itself [16].…”

Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus and an endemic state

“…We assume persistent levels of social activity α i to also follow a Gamma distribution with dispersion parameter κ . In several recent studies of epidemic dynamics in populations with persistent heterogeneity [15, 17, 52] it has been demonstrated that dispersion parameter κ determines the herd immunity threshold. Multiple studies [10, 48, 53] of real-world contact networks report an approximately exponential distribution of α , which corresponds to κ 1.…”