key: cord-0712012-w98847ai authors: Kohanovski, I.; Obolski, U.; Ram, Y. title: Inferring the effective start dates of non-pharmaceutical interventions during COVID-19 outbreaks date: 2020-05-26 journal: nan DOI: 10.1101/2020.05.24.20092817 sha: 76444627c36aae432b0bfa0d6a9d5bf5826cfb4f doc_id: 712012 cord_uid: w98847ai During February and March 2020, several countries implemented non-pharmaceutical interventions, such as school closures and lockdowns, with variable schedules, to control the COVID-19 pandemic caused by the SARS-CoV-2 virus. Overall, these interventions seem to have successfully reduced the spread of the pandemic. We hypothesise that the official and effective start date of such interventions can significantly differ, for example due to slow adoption by the population, or due to unpreparedness of the authorities and the public. We fit an SEIR model to case data from 12 countries to infer the effective start dates of interventions and contrast them with the official dates. We find both late and early effects of interventions. For example, Italy implemented a nationwide lockdown on Mar 11, but we infer the effective date on Mar 16 ({+/-}0.47 days 95% CI). In contrast, Spain announced a lockdown on Mar 14, but we infer an effective start date on Mar 8 ({+/-}1.08 days 95% CI). We discuss potential causes and consequences of our results. official dates, and find that they include both late and early effects of NPIs on infection dynamics. We conclude by demonstrating how differences between the official and effective start of NPIs can confound assessments of the effectiveness of the NPIs in a simple epidemic control framework. Several studies have described the effects of non-pharmaceutical interventions in different geographical regions 7, 9, 14 . Some of these studies have assumed that the parameters of the epidemiological model change at a specific date (Eq. 6), and set the change date τ to the official NPI date τ * (Table 1) . They then fit the model once for time t < τ * and once for time t ≥ τ * . For example, Li et al. 14 estimate the infection dynamics in China before and after τ * , which is set at Jan 23, 2020. Thereby, they effectively estimate the transmission and reporting rates before and after τ * separately. Here, we estimate the joint posterior distribution of the effective start date of the NPIs τ and the transmission and reporting rates before and after τ from the entire data, rather than splitting the data at τ. We then estimate the marginal posterior probability of τ by marginalising the joint posterior, and estimateτ as the posterior median. We compare the posterior predictive plots of a model with a free τ with those of a model with τ fixed at τ * and a model without τ (i.e. transmission and reporting rates are constant). The model with free τ clearly produces better and less variable predictions ( Figure S4a ). When we compare the models using WAIC (Eq. 9, Table S1), the model with a free τ is preferred in 8 The difference betweenτ the effective and τ * the official start of NPIs is shown for different regions. The effective dates in Italy and Wuhan are significantly delayed compared to the official dates, whereas in Denmark, France, Sweden, Spain, and Germany, the effective date is earlier than the official date. Here,τ is the posterior median, see Table 2 . τ * is the last NPI date (Table 1) . Thin and bold lines show 95% and 75% credible intervals, respectively (i.e. interval in which P(|τ −τ| | X) = 0.95 and 0.75.) (although only narrowly for 5 of the 8). The exceptions are Austria, Belgium, Norway, and United Kingdom. We compare the official τ * and effectiveτ start of NPIs and find that in most regions the effective start of NPIs significantly differs from the official date (Figure 1 ), that is, the credible interval onτ does not include τ * (Figure 1 ). The exceptions are, as with the comparison to the simpler models, Austria, Belgium, and United Kingdom, as well as Switzerland (see below). Norway also has a relatively wide credible interval, maybe because it has the longest duration between the first and last NPIs (Table 1) . In the following, we describe our findings in more detail. Late effective start of NPIs. In both Wuhan, China, and in Italy we estimate that the effective start of NPIsτ is significantly later than the official date τ * . In Italy, the first case was officially confirmed on Feb 21. School closures were implemented on Mar 5 7 , a lockdown was declared in Northern Italy on Mar 8, with social distancing implemented in the rest of the country, and the lockdown was extended to the entire nation on Mar 11 9 . That is, the first and last official NPI dates are Mar 8 and Mar 11. However, we estimate the effective dateτ at Mar 16 (±0.47 days 95% CI ; Figure 2 ). Similarly, in Wuhan, China, a lockdown was ordered on Jan 23 14 , but we estimate the effective start of NPIs to be more than a week later, at Feb 2 (±2.85 days 95% CI; Figure 2 ). Posterior density, (P( X) * Italy J a n 1 1 J a n 1 4 J a n 1 7 J a n 2 0 J a n 2 3 J a n 2 6 J a n 2 9 Early effective start of NPIs. In contrast, in other regions we estimate an effective start of NPIsτ that is earlier then the official date τ * (Figure 1 ). In Spain, social distancing was encouraged starting on Mar 8 7 , but mass gatherings still occurred on Mar 8, including a march of 120,000 people for the International Women's Day, and a football match between Real Betis and Real Madrid (final score: 2-1) with a crowd of 50,965 in Seville. A national lockdown was only announced on Mar 14 7 . Nevertheless, we estimate the effective start of NPIτ on Mar 8-9 (±1.08 days 95 %CI), rather than Mar 14 ( Figure 3 ). Similarly, in France we also estimate the effective start of NPIsτ on Mar 8 or 9 (±6.27 days 95% CI, Figure 3 ). Although the credible interval is wider compared to Spain, spanning from Mar 2 to Mar 15, the official lockdown start at Mar 17 is later still, and even the earliest NPI, banning of public events, only started on Mar 13 7 . 4 . CC-BY 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) The copyright holder for this preprint this version posted May 26, 2020. . Interestingly, the effect of NPIsτ in both France and Spain is estimated to have started on Mar 8-9, although the official NPI dates differ significantly: the first NPI in France is only one day before the last NPI in Spain. The number of daily cases was similar in both countries until Mar 8, but diverged by Mar 13, reaching significantly higher numbers in Spain ( Figure S2 ). This may suggest correlations between effective starts of NPIs due to global or international events. Like a Swiss watch. We find one case in which the official and effective dates match: Switzerland ordered a national lockdown on Mar 20, after banning public evens and closing schools on Mar 13 and 14 7 . Indeed, the posterior medianτ is Mar 20 (±8.46 days 95% CI), and the posterior distribution shows two density peaks: a smaller one between Mar 10 and Mar 14, and a bigger one between Mar 17 and Mar 22 ( Figure S3 ). It's also worth mentioning that Switzerland was the first to mandate self isolation of confirmed cases 7 . The success of nonpharmaceutical interventions is assessed by health officials using various metrics, such as the decline in the growth rate of daily cases. These assessments are made a specific number of days after the intervention began, to accommodate for the expected serial interval 3 (i.e. time between successive cases in a chain of transmission), which is estimated at about 4-7 days 9 . However, a significant difference between the beginning of the intervention and the effective change in transmission rates can invalidate assessments that assume a serial interval of 4-7 days and neglect the late or early population response to the NPI. This is illustrated in Figure 4 using data and parameters from Italy: a lockdown was officially ordered on Mar 10 (τ * ), but its late effect on the infection dynamics starts on Mar 16 (τ). If health officials assumed the dynamics to immediately change at τ * , they will have expected the number of cases be within the red lines (posterior predictions assuming τ = τ * ). This would have lead to a significant underestimation, which might have been interpreted by as ineffectiveness of the NPI, leading to further escalations. However, the number of cases would actually follow the blue lines (posterior predictions using τ =τ), which corresponds well to the real data (stars). . CC-BY 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. The copyright holder for this preprint this version posted May 26, 2020. (Table 2 ). We have inferred the effective start date of NPIs in several geographical regions using an SEIR epidemiological model and an MCMC parameter estimation framework. We find examples of both late and early effect of NPIs ( Figure 1 ). For example, in Italy and in Wuhan, China, the effective start of the lockdowns seems to have occurred five days or more after the official date ( Figure 2 ). This difference might be explained by low compliance: In Italy, for example, the government intention to lockdown Northern provinces leaked to the public, resulting in people leaving those provinces 9 . Late effect of NPIs may also be due to the time required by both the government and the citizens to organise for a lockdown, and for the new guidelines to be adopted by the population. In contrast, in most investigated countries (e.g., Spain and France), we inferred reduced transmission rates even before official lockdowns were implemented ( Figure 3 ). An early effective date might be due to early adoption of social distancing and similar behavioural adaptations in parts of the population. Adoption of these behaviours may occur via media and social networks, rather than official government recommendations and instructions, and may have been influenced by increased risk perception due to domestic or international COVID-19-related reports. Indeed, the evidence supports a change in infection dynamics (i.e. a model with fixed or free τ) even for Sweden (Table S1, Figure S4a) , where a lockdown was not implemented * . Attempts to asses the effect of NPIs 3,7 generally assume a seven-day delay between the implementation * Sweden banned public events on Mar 12, encouraged social distancing on Mar 16, and closed schools on Mar 18 7 . 6 . CC-BY 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. The copyright holder for this preprint this version posted May 26, 2020. . of the intervention and the observable change in dynamics, due to the characteristic serial interval of COVID-19 9 . However, late and early effects such as we have inferred may bias these assessments and lead to wrong conclusions about the effects of NPIs (Figure 4) . We have found that the evidence supports a model in which the parameters change at a specific time point τ over a model without such a change-point in 9 out of 12 regions (i.e. free or fixed model in Table S1 ). It could be interesting to check if the evidence favors a model with two change-points, rather than one. Two such change-points could reflect escalating NPIs (e.g. school closures followed by lockdowns), or an intervention followed by a relaxation. However, interpretation of such models will be harder, as two change-points can also reflect a mix of NPIs and other events, such as changing weather, news of new treatments, and international outbreaks. As several countries begin to relieve lockdowns and ease restrictions, we expect similar delays and advances to occur: in some countries the population will behave as if restrictions were eased even before the official date, and in some countries the population will continue to self-restrict even after restrictions are officially removed. We have inferred the effective start date of NPIs and found that they often differ from the official dates. Our results highlight the complex interaction between personal, regional, and global determinants of behavioral response to an epidemic. Therefore, we emphasize the need to further study variability in compliance and behavior over both time and space. This can be accomplished both by surveying differences in compliance within and between populations 2 , and by incorporating specific behavioral models into epidemiological models 1,5,18 . Data. We use daily confirmed case data X = (X 1 , . . . , X T ) from 12 regions during Jan-Mar 2020. These incidence data summarise the number of individuals X t tested positive for SARS-CoV-2 RNA (using RT-qPCR) at each day t. Data for Wuhan, China retrieved from Pei and Shaman 15 , data for 11 European countries retrieved from Flaxman et al. 7 . Where there were multiple sequences of days with zero confirmed cases (e.g. France), we cropped the data to begin with the last sequence so that our analysis focuses on the first sustained outbreak rather than isolated imported cases. For official NPI dates see Table 1 . SEIR model. We model SARS-CoV-2 infection dynamics by following the number of susceptible S, exposed E, reported infected I r , unreported infected I u , and recovered R individuals in a population of size N. This model distinguishes between reported and unreported infected individuals: the reported infected are those that have enough symptoms to eventually be tested and thus appear in daily case reports, to which we fit the model. This model is inspired by Li et al. 14 Susceptible (S) individuals become exposed due to contact with reported or unreported infected individuals (I r or I u ) at a rate β t or µβ t , respectively. The parameter 0 < µ < 1 represents the decreased transmission rate from unreported infected individuals, who are often subclinical or even asymptomatic 6, 17 . The transmission rate β t ≥ 0 may change over time t due to behavioural changes of both susceptible and infected individuals. Exposed individuals, after an average incubation period of Z days, become reported infected with probability α t or unreported infected with probability (1 − α t ). The reporting rate 0 < α t < 1 may also change over time due to changes in human behaviour. Infected individuals remain infectious for an average period of D days, after which they either recover, or become ill enough to be quarantined. In either case, they no longer infect other individuals, and therefore effectively become recorved (R). The model is described by the following set of equations: 7 . CC-BY 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. The copyright holder for this preprint this version posted May 26, 2020. (1) The initial numbers of exposed E(0) and unreported infected I u (0) are free model parameters (i.e. inferred from the data), whereas the initial number of reported infected and recovered is assumed to be zero, I r (0) = R(0) = 0, and the number of susceptible is Likelihood function. For a given vector θ of model parameters the expected cumulative number of reported infected individuals (I r ) until day t, following Eq. 1, is We assume that reported infected individuals are confirmed and therefore observed in the daily case report of day t with probability p t (note that an individual can only be observed once, and that p t may change over time, but t is a specific date rather than the time elapsed since the individual was infected). We denote by X t the observed number of confirmed cases in day t, and byX t the cumulative number of confirmed cases until end of day t,X Therefore, at day t the number of reported infected yet-to-be confirmed individuals is (Y t (θ) −X t−1 ). We therefore assume that X t conditioned onX t−1 is Poisson distributed, such that Hence, the likelihood function L(θ | X) for a parameter vector θ given the confirmed case data X = (X 1 , . . . , X T ) is defined by the probability to observe X given θ, NPI model. To model non-pharmaceutical interventions (NPIs), we set the start of the NPIs to day τ and where 0 < λ < 1. The values for p t follow Li et al. 14 , who estimated the average time between infection and reporting in Wuhan, China, at 9 days before the start of NPIs and 6 days after start of NPIs. Parameter estimation. To estimate the model parameters from the daily case data X, we apply a Bayesian inference approach. We start our model ∆t days 9 before the outbreak (defined as consecutive days with increasing confirmed cases) in each country. The model in Eq. 1 is parameterised by the vector θ, where 8 . CC-BY 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. The copyright holder for this preprint this version posted May 26, 2020. . The likelihood function is defined in Eq. 5. The posterior distribution of the model parameters P(θ | X) is estimated using the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) 11 implemented in the emcee Python package 8 . We defined the following prior distributions on the model parameters P(θ): where the prior for τ is a truncated normal distribution shaped so that the date of the first and last NPI, τ 0 and τ * (Table 1) , are at minus and plus one standard deviation, and taking values only between 1 and T − 2, where T is the number of days in the data X. We also tested an uninformative uniform prior, Uniform(1, T − 2). WAIC (Eq. 9) of a model with this uniform prior was either higher, or lower by less than 2, compared to WAIC of a model with the truncated normal prior. The uninformative prior resulted in non-negligible posterior probability for unreasonable τ values, such as Mar 1 in the United Kingdom. This was probably due to MCMC chains being stuck in low posterior regions of the parameter space. We therefore decided to use the more informative truncated normal prior for τ. Other priors follow Li et al. 14 , with the following exceptions. λ is used to ensure transmission rates are lower after the start of the NPIs (λ < 1). We checked values of ∆t larger than five days and found they generally produce lower likelihood and unreasonable parameter estimates, and therefore chose Uniform (1, 5) as the prior for ∆t. Model comparison. We perform model selection using two methods. First, we compute WAIC (widely applicable information criterion) 10 , where E[·] and V[·] are the expectation and variance operators taken over the posterior distribution P(θ | X). We compare models by reporting their relative WAIC; lower is better (Table S1) . A minority (<5%) of MCMC chains that fail to fully converge can lead to overestimation of the variance (the second term in Eq. 9). Therefore, we exclude from the computation of WAIC chains with mean log-likelihood that is three standard deviations or more from the overall mean. We also plot posterior predictions: we sample 1,000 parameter vectors from the posterior distribution P(θ | X), use these parameter vectors to simulate the SEIR model (Eq. 1), and plot the simulated dynamics ( Figure S4a ). Both the accuracy (i.e. overlap of data and prediction) and the precision (i.e. the tightness of the predictions) are good ways to visually compare models. Source code. We use Python 3 with the NumPy, Matplotlib, SciPy, Pandas, Seaborn, and emcee packages. All source code will be publicly available under a permissive open-source license at github.com/yoavramlab/EffectiveNPI. Samples from the posterior distributions will be deposited on FigShare. 9 . CC-BY 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. The copyright holder for this preprint this version posted May 26, 2020. 10 . CC-BY 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. The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.24.20092817 doi: medRxiv preprint . CC-BY 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. Table 1 . CC-BY 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) The copyright holder for this preprint this version posted May 26, 2020. 14 . CC-BY 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. 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This work was supported in part by the Israel Science Foundation 552/19 and 1399/17. Wuhan, China J a n 1 1 J a n 1 6 J a n 2 1 J a n 2 6 J a n 3 1 Daily cases J a n 1 1 J a n 1 6 J a n 2 1 J a n 2 6 J a n 3 1 Daily cases Figure S4 . Posterior prediction plots. Markers represent data (X). Black line represent a smoothing of the data points using a Savitzky-Golay filter. Colored lines represent posterior predictions from a model with fixed τ in red, and free τ in blue. These predictions are made by drawing 1,000 samples from the parameter posterior distribution and then generating a daily case count using the SEIR model in Eq. 1. Note the differences in the y-axis scale.