key: cord-0944983-u6a65x9h
authors: Colombo, M.; Mellor, J.; Colhoun, H. M.; M. Gomes, M. G.; McKeigue, P. M.
title: Trajectory of COVID-19 epidemic in Europe
date: 2020-09-28
journal: nan
DOI: 10.1101/2020.09.26.20202267
sha: d3b0540b63b037a24050e5ceff3a4b2ca496c61e
doc_id: 944983
cord_uid: u6a65x9h

The classic Susceptible-Infected-Recovered model formulated by Kermack and McKendrick assumes that all individuals in the population are equally susceptible to infection. From fitting such a model to the trajectory of mortality from COVID-19 in 11 European countries up to 4 May 2020 Flaxman et al. concluded that "major non-pharmaceutical interventions -- and lockdowns in particular -- have had a large effect on reducing transmission". We show that relaxing the assumption of homogeneity to allow for individual variation in susceptibility or connectivity gives a model that has better fit to the data and more accurate 14-day forward prediction of mortality. Allowing for heterogeneity reduces the estimate of "counterfactual" deaths that would have occurred if there had been no interventions from 3.2 million to 262,000, implying that most of the slowing and reversal of COVID-19 mortality is explained by the build-up of herd immunity. The estimate of the herd immunity threshold depends on the value specified for the infection fatality ratio (IFR): a value of 0.3% for the IFR gives 15% for the average herd immunity threshold.

Kermack and McKendrick were careful to state in the abstract of their 1927 paper [1] : 17 In the present communication discussion will be limited to the case in which 18 all members of the community are initially equally susceptible to the disease 19 On this assumption the reproduction number R t at time t is (1 − p t ) R 0 where p t is 20 the proportion of the population that has been infected and is no longer susceptible. 21 The herd immunity threshold H -the value of p t at which R t = 1 -is thus 1 − 1/R 0 .

For natural infection, however, heterogeneity of susceptibility lowers the value of H [3] . 23 If the distribution of susceptibility has a gamma distribution, in the expression above 24 (1 − p t ) is replaced by (1 − p t ) 1+1/α where α is the shape parameter of the gamma 25 distribution [4, 5] . If connectivity, rather than susceptibility, has a gamma distribution, 26 2020-09-26 1/7 the exponent of (1 − p t ) is (1 + 2/α); models with heterogeneity of connectivity or 27 heterogeneity of susceptibility are thus likelihood-equivalent. The classic model is thus a 28 special case of this more general formulation, in which α = ∞ and the distribution of 29 susceptibility or connectivity is a spike at 1.

We compared a model that allows for heterogeneity of susceptibility with the original 32 model that assumes homogeneity. The only change made to the original model was to 33 replace the expression (1 − p t ) by the exponent form given above, with a half-Cauchy(0, 34 5) prior on the shape parameter α. Flaxman et al specified an average infection fatality 35 ratio (IFR) of 1.1%, which is higher than most recent estimates. We repeated the model 36 fitting with country-specific IFRs scaled by a factor of 0.275 to equate the average IFR 37 to a recent estimate [6] of 0.30%. For sensitivity analyses, we repeated the comparison 38 of homogeneity and heterogeneity models after varying other modelling assumptions: Table) . . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.09.26.20202267 doi: medRxiv preprint

Models that allow for heterogeneity favour build-up of herd immunity rather than 71 non-pharmaceutical interventions as the main factor underlying the early slowing and 72 reversal of the COVID-19 epidemic in Europe. This is consistent with observations that 73 epidemic curves in many countries reached a peak less than two months after the first 74 few severe cases appeared [8, 9] . With this dataset it is not possible to distinguish the 75 relative contributions of heterogeneity of connectivity, heterogeneity of susceptibility, or 76 any other process that could have generated a smooth downward trajectory in R t over 77 about one month in each of the 11 European countries studied.

Because the model is fitted to observed deaths, the estimates of cumulative numbers 79 infected and herd immunity threshold depend on the values pre-specified for infection 80 fatality ratios. Specifying an average infection fatality ratio of 0.3% gives an estimated 81 herd immunity threshold of 15%. Whatever value is specified for the infection fatality 82 ratio, a model that allows for heterogeneity has better fit to the data than the 83 homogeneity model and supports herd immunity as the main factor underlying the 84 reversal of the epidemic.

One objection that has been raised to estimates that herd immunity thresholds for 86 COVID-19 are less than 20% is that far higher infection rates have been reached in local 87 hotspots such as Manaus [10]. However country-level herd immunity thresholds as 88 estimated here are not likely to be homogeneous over every locality. In hotspots where 89 the basic reproduction number R 0 is higher than the population average, the herd 90 immunity threshold and overshoot of this threshold will be correspondingly higher, with 91 or without heterogeneity. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted September 28, 2020. . . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted September 28, 2020. . https://doi.org/10.1101/2020.09.26.20202267 doi: medRxiv preprint

Containing Papers of a 101 Mathematical and Physical Character

Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe

On the spread of epidemics in a closed heterogeneous population

Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd 110 immunity threshold. medRxiv

A SEIR model with variable susceptibility or 113 exposure to infection

Estimating the infection fatality ratio in England

Did lockdowns really save 3 million COVID-19 deaths, as Flaxman et al. 120 Claim? Nicholas Lewis

Predicting the Trajectory of Any COVID19 124

Epidemic From the Best Straight Line. medRxiv. Cold Spring Harbor Laboratory 125

The end of exponential growth: The decline in the spread of 127 coronavirus. The Times of Israel