key: cord-0757026-kyflzrn0 authors: Naude, J.; Mellado, B.; Choma, J.; Correa, F.; Dahbi, S.; Dwolatzky, B.; Dwolatzky, L.; Hayasi, K.; Lieberman, B.; Maslo, C.; Monnakgotla, K.; Ruan, X.; Stevenson, F. title: Worldwide Effectiveness of Various Non-Pharmaceutical Intervention Control Strategies on the Global COVID-19 Pandemic: A Linearised Control Model date: 2020-05-05 journal: nan DOI: 10.1101/2020.04.30.20085316 sha: 9af2fccfec7288912a02348c42c49a6f8fb3abde doc_id: 757026 cord_uid: kyflzrn0 Background COVID-19 is a virus which has lead to a global pandemic. Worldwide, more than 100 countries have imposed severe restrictions regarding freedom of movement amongst their citizens in a bid to slow the spread of the virus. These restrictions, which are part of a set of non-pharmaceutical interventions, have recently been classified by the Oxford COVID-19 Government Response Tracker (OxCGRT) team and a nominal index measure has been defined for use by the wider international community. We address the use of this index measure to establish the degree and characteristics of control of the transmission rate of the virus within a representative sample of countries in the World and states in the United States of America. Methods Country specific, Susceptible-Infected-Recovered-Deaths (SIRD) models with latent dynamics were constructed using publicly available data for 23 countries and 25 states of the United States of America. Each of the models were linearised and classical frequentist error propagation was applied to them individually. The time varying, observable model parameters were extracted for each day that data was made available. The OxCGRT stringency index, p, was used to regress against these model parameters. The regression of the transmission rate as a function of p in each locale was through a linear parameter _s. In addition, macroscopic indices from the World Bank were used to explore inter-country variation in the measured parameters. Results The world average was _s=0.01 (95% CI 0.0102 - 0.0112) with an ensemble standard deviation of 0.0017 (95% C.I. 0.0014 - 0.0021), strongly indicating a universal behavior. While lockdown measures have been successful in curbing the spread, our study indicates that removing them too swiftly will result in the resurgence of the spread within one to two months. Reducing the stringency index by 10 will delay reaching the apex by about 6 months, where reducing it by 20 will delay by only four months. During the post-lockdown period it is essential to increase _s. For the system to remain sub-critical, the rate with which _s increases should outpace that of the decrease of the stringency index. The spread of the virus is found to be insensitive to the Gini index and other socio-economic indexes. The typical adjustment time to see the effects of control varied between 1.49 days for Peru and 38.09 days for Sweden. In the United States, the typical adjustment time to see the effects of control varied between 1.41 days for Colorado to 15.91 days for Ohio. Interpretation Given the measured characterisations of each locale, the effects of any change in non-pharmaceutical intervention may be anticipated and predictions can be made regarding the possible case load which is specific to that environment. This is accomplished by specifying an acceptable level of transmission, {beta}_f, given the prevailing economic and social constraints which uniquely determines an overall stringency of intervention level p. As a policy maker, there are possible intervention combinations to choose from and a combination must be selected that achieves p or greater. models with latent dynamics were constructed using publicly available data for 23 countries and 25 states of the United States of America. Each of the models were linearised and classical frequentist error propagation was applied to them individually. The time varying, observable model parameters were extracted for each day that data was made available. The OxCGRT stringency index, p, was used to regress against these model parameters. The regression of the transmission rate as a function of p in each locale was through a linear parameter α s . In addition, macroscopic indices from the World Bank were used to explore inter-country variation in the measured parameters. Results The world average was α s = 0.01 (95% CI 0.0102 -0.0112) with an ensemble standard deviation of 0.0017 (95% C.I. 0.0014 -0.0021), strongly indicating a universal behavior. While lockdown measures have been successful in curbing the spread, our study indicates that removing them too swiftly will result in the resurgence of the spread within one to two months. Reducing the stringency index by 10 will delay reaching the apex by about 6 months, where reducing it by 20 will delay by only four months. During the post-lockdown period it is essential to increase α s . For the system to remain sub-critical, the rate with which α s increases should outpace that of the decrease of the stringency index. The spread of the virus is found to be insensitive to the Gini index and other socio-economic indexes. The typical adjustment time to see the effects of control, b −1 r varied between 1.49 days for Peru and 38.09 days for Sweden. In the United States, the typical adjustment time to see the effects of control, b −1 r varied between 1.41 days for Colorado to 15.91 days for Ohio. Interpretation Given the measured characterisations of each locale, the effects of any change in non-pharmaceutical intervention may be anticipated and predictions can be made regarding the possible case load which is specific to that environment. This is accomplished by specifying an acceptable level of transmission, β f , given the prevailing economic and social constraints which uniquely determines an overall stringency of intervention level p. As a policy maker, there are possible intervention combinations to choose from and a combination must be selected that achieves p or greater. COVID-19. It is sensitive enough to differentiate between situations where 48 an intervention measure is suggested by government but not enforced 12 . At the time of writing; America did not have a stringency index published. A 50 best fit, using the methodology in Hale et al. 12 and publicly available coarse 51 grained intervention information, was employed to align the results. This is 52 described fully in Appendix A. 53 Daily parameter estimates of the models were extracted from the data using Reliable estimates of asymptomatic cases due to either extensive randomised 148 testing or exhaustive testing in closed populations are presented in Table 1 . The roll-out of any country-wide control measure is subject to random Study Asympt. positive tests Diamond Princess 15 [n=4, 062] 55% (95% CI 52%-59%) Vo Italy 24 [n=3,300] 50% -75% Japanese nationals evacuated 25 66% (95% CI 54% -77%) This was modeled with the kernel function, conditioned on the control mea-162 sure p: where β f (p) is the asymptotic value of the observed daily transmission rate, b r 164 is the typical adjustment rate and b 0 is used to model the initial transmission 165 rate before the control p is applied. A characteristic plot is depicted in Figure 166 2. As a starting point, it is hypothesised that the typical adjustment rate b r is 168 characteristic of a nation and that β f (p) is dependent on the stringency of 169 control p through: where α s is the effect of stringency on transmission rate. The kernel function for γ(t) is: regardless of control. The clinical and physical justification for this model 173 is that the inherent properties of the recovery process do not change with 174 non-pharmaceutical control, assuming the hospital system has not been over-175 whelmed yet. The kernel function for δ(t) is: The OxCGRT has developed a valuable database for comparing countries 219 response strategies. 31 The database contains the following levels of control 220 (coded using ordinal numbers) and timing for 139 countries: 221 10 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10. 1101 3. S3 -Cancel public events was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10. 1101 dynamics are dictated by γ(t) = γ 0 = const., whatever the value of p. This is 243 true provided that the health care system has not been overwhelmed; which 244 is the purpose of the control system. In the absence of data on the population dynamics during withdrawal of 246 some NPI's, our model is conservative and uses the same typical adjustment 247 rate for the withdrawal of NPI's as for the addition of NPIs. The research output, using the foregoing methodology are shown in Ta-250 ble 2 for various countries and in Table 3 for various States of the US. Results are expressed in terms of γ 0 , d 0 , β 0 , b −1 r , p max and α s in Table 2 . Table 3 252 does not display γ 0 , as the data for the number of recoveries has not been 253 reported since late March. The first salient feature of Tables 2 and 3 is that the evolution of the The results of α s display some deviations from the average value, espe-267 cially in some States of the US. Section 4.4 will touch upon an analysis of 268 these outliers and how these appear to be related to population sparsity. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. dynamics are significantly less understood. The framework proposed here 277 allows for the modeller to factorise latent and observable dynamics, with the 278 intent to address the evolution of the latter without major hindrance from 279 limited knowledge of the former. In this setup the temporal evolution of 280 observable dynamics are weakly coupled to that that of the latent dynamics 281 through the term φI L (see Figure 1 ). In practice, the presence of this term 282 implies that the total population of observed infections will be bound by 283 the parameter φ, which can be viewed as the fraction of true infections that 284 13 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10.1101/2020.04.30.20085316 doi: medRxiv preprint Figure 5 : Implications of the model on observed transmission rate and stringency index for representative countries. become symptomatic enough to warrant testing and, therefore, become clas-285 sified as an observed infection upon a positive result. Provided that γ ≤ γ L 286 the maximum number of observed infections will be limited to φN . The temporal evolution will depend on the initial parameters relevant 288 to the spread and the parameters that are chosen to depend on time. For 289 illustration purposes, we chose to freeze d to the last values measured, or d 0 , 290 as reported in Tables 2 and 3 . From Eqs. 1 and 2 one gets for the post-lockdown period: where t = 0 corresponds to the end of the lockdown period and the enacting 293 of new interventions, such that p < p max . The implementation of new less 294 restrictive measures will yield ∆p = p − p max < 0, where β(t, p) > β f . It is 295 reasonable to infer that the typical adjustment time leading to the asymptotic 296 value of β f may also apply here. For illustration purposes, we choose the configuration of parameters ob-298 tained with Italian data, where φ = 0.1 is used (see Section 2.2. The scenario 299 14 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10.1101/2020.04.30.20085316 doi: medRxiv preprint approach is adopted. The country-to-country variation, which is of order of 315 10%, as discussed in Section 4.5, is considered to be a more realistic estimate 316 of the potential deviation from the linear behavior assumed in Eq. (2). A 317 15 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. country's healthcare system is not overwhelmed: One can assume that γ c and d c display a weak time dependence in that 336 they primarily depend on medical advances, rather than on policy interven-337 tions. In this setup the condition for the system to remain sub-critical can 338 17 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. be expressed as follows: where the temporal partial derivatives are evaluated at the point of critical-340 ity defined by v c (t). The inequality 7, while seemingly straightforward from 341 the mathematical standpoint, it has serious consequences for policy makers. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. where I tot denotes the total expected number of positive cases for t → ∞, 365 ξ I is the slope of the exponential growth that characterises the first phase was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10. 1101 where containment measures have proven effective in curbing the spread. These include countries in all continents and with a wide span in terms of 371 socio-economic development, inequality and population density. and 50, respectively. It is found that the parameter ξ I is almost insensitive to 380 the Gini index. This is illustrated in Figure 11 where the red line correspond 381 to a first order polynomial that is consistent with zero slope. In order to was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10.1101/2020.04.30.20085316 doi: medRxiv preprint cient living area, and durability of housing. Out of the 67 countries under 393 scrutiny, 18 report a significant fraction of urban population living in slums. In this sample of countries the fraction ranges from 8% to 53%. No signifi-395 cant correlation is found between this index and ξ I . In addition, the average 396 value of ξ I for these 18 countries is compatible with that of the rest of the 397 ensemble studied. As per the physical picture underlying the model used to describe the 399 spread, it is expected that population density should play a significant role. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020 . . https://doi.org/10.1101 Some countries, such as Switzerland and most prominently Sweden, have 429 achieved containment without the application of stringent lockdown mea-430 sures. In these countries citizens are allowed to go out for walks while re-431 specting social distancing. The value of α s for Sweden is particularly high. To this end, we lack the evidence that would support the above mentioned 453 non-linear behavior. As pointed out in Section 4.2, it is paramount to closely 454 monitor the evolution of α s as NPIs are released. It is remarkable that α s has such a stable value across locales and over 457 these different scales; from states in the US to entire countries α s ≈ 0.01. As proof of this assertion, Figure 12 is given by: µ αs is the empirical mean and Z is a normalisation constant. Equation (9) 466 is an inverse gamma distribution and it may be used to calculate the range 467 of probable ensemble variances of α s across the world, given the sample of 468 locales used in this study. The results are that the standard deviation of the ensemble σ αs is 0.0017 470 (95% CI 0.0014 -0.0021). 24 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020 . . https://doi.org/10.1101 and, therefore, almost all locales will (in the limit as p → 100) eventually 476 extinguish the epidemic under tight enough control. The outliers which were 477 removed from the analysis show an even greater sensitivity and our models 478 predict that these locales are more easily controlled with softer NPIs. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . For each policy response measure S1-S7, OxCGRT use the ordinal value 512 (and add one if the policy is general rather than targeted). This creates a 513 score between 0 and 2 and for S5, and 0 and 3 for the other six responses 31 . The OxCGRT stringency index is given by: where p J is defined by: To this end, we coded the known levels of intervention in America to 527 match as nearly as possible, the OxCGRT system. We used the Institute 528 for Health Metrics and Evaluation (IHME) dashboard to obtain six dates at 529 which specific states imposed different NPIs 35 . In order to compare the US intervention data it was necessary to make 532 a stringency index for the US states that mimics that of the index that was 533 made for the World data by OxCGRT. The following decisions were made during the process of mapping the re-536 ported US NPIs to the OxCGRT index: 537 26 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020 . . https://doi.org/10.1101 where v i is a number out of 100 indicating the extent each of the inter- It is unclear that these may be directly attributable to the virus in that con-637 firmation would be required via post-mortem COVID-19 testing ? . Given 638 the existing testing burden posed on most countries, this sort of testing is 639 rare and, therefore, the uncertainty in this parameter will remain high 19 . Known recoveries, R : These are patients who are known to test pos-642 itive for COVID-19 and are known to have recovered fully from the virus. The recovery rate, γ, models how quickly known infections are resolved and 644 discharged out of the healthcare system. This rate is physically dependent 645 on treatment regime and the patient's own physical condition. Under the above conditions, the model explicitly caters explicitly for the 655 situation that the latent infections are asymptomatic or mild. It is trivial to show that S + I + I L + R L + R + D = N at every instant 657 30 All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10. 1101 in time. Furthermore, the model in Figure 1 implies that: is small compared with the size of the total population N . The derivatives with respect to time were approximated using first order 665 backward difference approximations at a daily level. Classical frequentist er-666 ror propagation was applied to this linear approximation using the Gaussian 667 process assumption. Theoretically, these approximations are valid provided that: was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. where σ x (t) is the noise estimate at day t, ∆x := All rights reserved. No reuse allowed without permission. was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. methods to shape β(t) to a form that is acceptable for the desired goals of 721 the pandemic control system eg. minimise total deaths, minimise the peak 722 load on the health care system, maximise the economic activity etc. These 723 aspects will be dealt with in detail in a follow up paper. icelandic population, New England Journal of Medicine (2020). was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10. 1101 county, washington, march 2020, MMWR. Morbidity and Mortality Weekly Report 69 (2020) 377-381. [27] Coronavirus, castiglione d'addaè un caso di studio: 'il 70% dei donatori 858 di sangueè positivo', online https://www.lastampa.it/topnews/primo-859 piano/2020/04/02/news/coronavirus-castiglione-d-adda-e-un-caso-di- (2016) was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint (which this version posted May 5, 2020. . https://doi.org/10. 1101 Who director-general's opening remarks at the 726 media briefing on covid-19 -11 Coronavirus disease 2019 (covid-19) situation report Ham-739 let Estimating the num-744 ber of infections and the impact of non-pharmaceutical interventions on 745 covid-19 First-wave covid-19 transmis-747 sibility and severity in china outside hubei after control measures, and 748 second-wave scenario planning: a modelling impact assessment How will country-based mitigation measures influence the course of the 752 covid-19 epidemic? 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