Spatial Sampling Design to Improve the Efficiency of the Estimation of the Critical Parameters of the SARS-CoV-2 Epidemic

Abstract

Given the urgent informational needs connected with the diffusion of infection with regard to the COVID-19 pandemic, in this article, we propose a sampling design for building a continuous-time surveillance system. Compared with other observational strategies, the proposed method has three important elements of strength and originality: (1) it aims to provide a snapshot of the phenomenon at a single moment in time, and it is designed to be a continuous survey that is repeated in several waves over time, taking different target variables during different stages of the development of the epidemic into account; (2) the statistical optimality properties of the proposed estimators are formally derived and tested with a Monte Carlo experiment; and (3) it is rapidly operational as this property is required by the emergency connected with the diffusion of the virus. The sampling design is thought to be designed with the diffusion of SAR-CoV-2 in Italy during the spring of 2020 in mind. However, it is very general, and we are confident that it can be easily extended to other geographical areas and to possible future epidemic outbreaks. Formal proofs and a Monte Carlo exercise highlight that the estimators are unbiased and have higher efficiency than the simple random sampling scheme.

Keywords

Sampling design SARS-CoV-2 diffusion Health surveillance system Unbiasedness Efficiency

References

ABC. 2020. “Random coronavirus testing to begin in Canberra next week at drive-through centre and clinic”. ABC News. Available at: https://www.abc.net.au/news/2020-04-03/random-coronavirus-testing-begins-in-canberra/12119364 (accessed April 2020).

Aguilar, J.B., J.S. Faust, L.M. Westafer, and J.B. Gutierrez. 2020. “Investigating the Impact of Asymptomatic Carriers on C0VID-19”, medXiv. DOI: https://doi.org/10.1101/2020.03.18.20037994.

Alleva, G. 2017. “The new role of sample surveys in official statistics”. ITACOSM 2017, The 5th Italian Conference on Survey Methodology, June 14, 2017. Bologna IT. Available at: https://www.istatit/it/files//2015/10/Alleva_ITAC0SM_14062017.pdf (accessed April 2020).

Alleva G. 2020. Contributo per la 12° Commissione permanente Igiene e sanità del Senato della Repubblica. May 27, 2020. Roma, IT. https://www.senato.it/application/xmanager/projects/leg18/attachments/documento_evento_procedura_commissione/files/000/135/501/GI0RGI0_ALLEVA.pdf (accessed May 2020).

Alleva, G., G. Arbia, P.D. Falorsi, G. Pellegrini, and A. Zuliani. 2020. A sampling design for reliable estimates of the SARS-CoV-2 epidemic’s parameters. Calling for a protocol using panel data. https://web.uniroma1.it/memotef/sites/default/files/Proposal.pdf (accessed April 2020).

Alleva G., and A. Zuliani. 2020. “Coronavirus: chiarezza sui dati”, Bancaria. Available at: https://www.bancaria.it/livello-2/archivio-sommari/gli-ultimi-sommari-di-bancaria/-bancaria-giugno-2020/covid-19-chiarezza-sui-dati/.

Arbia, G. 1994. “Selection techniques in sampling spatial units”, Quaderni di statistica e matematica applicata alle scienze economico-sociali, XVI(1–2): 81–91.

Arbia, G. 2020. A note on early epidemiological analysis of coronavirus disease 2019 outbreak using crowdsourced data. arXiv.

Arbia, G. and G. Lafratta. 1997. “Evaluating and updating the sample design: the case of the concentration of SO₂ in Padua”, Journal of Agricultural, Biological and Environmental Statistics, 2, 4: 451–466. DOI: https://doi.org/10.2307/1400514.

10.

Arbia, G., and G. Lafratta. 2002. “Spatial sampling designs optimized under anisotropic superpopulation models”, Journal of the Royal Statistical Society series c – Applied Statistics, 51, 2: 223–23.

11.

Ascani, P. 2020. Technical Note on the methods of the data collection phase for a proposal of sampling design for reliable estimates of the epidemic’s parameters of SARS-CoV-2. Available at: https://web.uniroma1.it/memotef/sites/default/files/TechNote.pdf (accessed May 2020).

12.

Bassi F., G. Arbia, and P.D. Falorsi. 2020. “Observed and estimated prevalence of Covid-19 in Italy: How to estimate the total cases from medical swabs data, from medical sbabs data”. Science of the Total Environment, 764: 142799. DOI: https://doi.org/10.1016/j.scitotenv.2020.142799.

13.

Borchers, D. 2009. “A non-technical overview of spatially explicit capture-recapture models”. Journal of Ornithology, 152: 435–444. DOI: https://doi.org/10.1007/s10336-010-0583-z.

14.

Chughtai, A.A., and A.A. Malik. 2020. “Is Coronavirus disease (C0VID-19) case fatality ratio underestimated?”. Global Biosecurity, 1(3). DOI: http://doi.org/10.31646/gbio.56.

15.

Cliff, A.D., P. Haggett, J.K. Ord, and F.R. Verfey. 1981. Spatial Diffusion: an Historical Geography of Epidemics in an Island Community 14. Cambridge University Press.

16.

Cochran, W.G. 1977. Sampling Techniques. Wiley. New York.

17.

Deville, J.-C., and Y. Tillé. 2004. “Efficient Balanced Sampling: the Cube Method”, Biometrika 91: 893–912. DOI: https://doi.org/10.1093/biomet/91.4.893.

18.

Deville, J.-C., and Y. Tillé. 2005. “Variance approximation under balanced sampling”, Journal of Statistical Planning and Inference 128: 569–591. DOI: https://doi.org/10.1016Zj.jspi.2003.11.011.

19.

Dewatripont, M., M. Goldman, E. Muraille, and J.-P. Platteau. 2020. “Rapidly identifying workers who are immune to COVID-19 and virus-free is a priority for restarting the economy”, VoxEU. Available at: https://voxeu.org/article/rapidly-identifying-workers-who-are-immune-covid-19-and-virus-free-priority-restarting-economy (accessed March 2020).

20.

Di Gennaro Splendore, L. 2020. “Random testing, quality of data and lack of information: COVID-19”. Data and Policy. Available at: https://medium.com/data-policy/random-testing-quality-of-data-and-lack-of-information-covid-19-a6e09a398d1d (accessed April 2020).

21.

Dorfman, R. 1943. The Detection of Defective Members of Large Populations, The Annals of Mathematical Statistics 14 (4): 436–440. DOI: http://dx.doi.org/10.1214/aoms/1177731363.

22.

Eurostat. 2017. European statistics Code of Practice – revised edition. Available at: https://ec.europa.eu/eurostat/web/products-catalogues/-/KS-02-18-142 (accessed May 2020)

23.

Falorsi P.D., and P. Righi. 2015. “Generalized framework for defining the optimal inclusion probabilities of one-stage sampling designs for multivariate and multi-domain surveys”. Survey methodology 41: 215–236. Available at: https://www150.statcan.gc.-ca/n1/en/catalogue/12-001-X201500114149.

24.

Falorsi P.D., P. Righi, and P. Lavallée. 2019. “Cost optimal sampling for the integrated observation of different populations”. Survey methodology 45(3): 485–511. Available at: https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X201900300004.

25.

Ferretti, L, C. Wymant, M. Kendall, L. Zhao, A. Nurtay, L. Abeler-Dorner, M. Parker, D. Bonsall, and C. Fraser. 2020. “Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing”, Science. DOI: http://dx.doi.org/10.1126/science.abb6936.

26.

Fuggetta M. 2020. “Testing for the Base Rate”. Bayes. Available at: http://massimofuggetta.com/2020/04/28/testing-for-the-base-rate/ (accessed April 2020).

27.

Gros, D. 2020. “Creating an EU ’Corona Panel’: Standardised European sample tests to uncover the true spread of the coronavirus”. VoxEU. Available at: https://voxeu.org/article/standardised-european-sample-tests-uncover-true-spread-coronavirus (accessed May 2020).

28.

Hackenbroch, V. 2020. “Grobe Antikörperstudie soll Immunität der Deutschen gegen Covid-19 feststellen” Spiegel. Available at: https://www.spiegel.de/wissenschaft/medizin/coronavirus-grosse-antikoerper-studie-soll-immunitaet-der-deutschen-feststel-len-a-c8c64a33-5c0f-4630-bd73-48c17c1bad23?d=1585300132&sara_ecid=soci_upd_wbMbjhOSvViISjc8RPU89NcCvtlFcJ. (accessed May 2020).

29.

Hamer W.H. 1906. “Epidemic diseases in England”, Lancet, 1. DOI: https://doi.org/10.1016/S0140-6736(01)80187-2.

30.

Hansen N.H., N.W. Hurwitz, and W.G. Meadow. 1953. Sample Survey Method and Theory. Wiley, New York.

31.

Hartley, H.O. 1962. “Multiple Frame Surveys”, Proceedings of the Social Statistics Section, American Statistical Association, Alexandria, Va. 1962.

32.

Hartley, H.O. 1974. “Multiple Frame Methodology and Selected Applications”, Sankhya, 36: 99–118.

33.

Horvitz, D.G., and D.L. Thompson. 1952. “A generalisation of sampling without replacement from finite-universe”. J Amer Statist. Assoc. 47: 663–685. DOI: http://doi.org/0.1080/01621459.1952.10483446.

34.

ILO (International Labour Organization). 2020. “COVID-19 impact on the collection of labour market statistics”. https://ilostat.ilo.org/topics/covid-19/covid-19-impact-onlabour-market-statistics/ (accessed May 2020).

35.

Ioannidis, J. 2020. “A fiasco in the making? As the coronavirus pandemic takes hold, we are making decisions without reliable data”. Statnews. https://www.statnews.com/2020/03/17/afiasco-in-the-making-as-the-coronavirus-pandemic-takes-hold-we-are-making-decisions-withoutreliable-data/ (accessed Mar 2020).

36.

Istat (Istituto nazionale di statistica). 2020. “Primi risultati dell’indagine di sieroprevalenza sul SARS-CoV-2. https://www.istat.it/it/files//2020/08/ReportPrimiR-isultatiIndagineSiero.pdf (accessed Aug 2020).

37.

Kermack, W.O., and A.G. McKendrick. 1927. “A contributions to the mathematical theory of epidemics” Proceedings of the Royal society London 115: 700–721.

38.

Kiesl, H. 2016. “Indirect Sampling: A Review of Theory and Recent Applications”. AStA Wirtschafts und Sozialstatistisches Archiv. DOI: http://doi.org/10.10.1007/s11943-016-0183-3.

39.

Kish, L. 1965. Survey Sampling, Wiley. New York.

40.

Lavallée, P. 2007. Indirect Sampling, Springer series in statistics.

41.

Lavallée, P., and L.P. Rivest. 2012. “Capture-Recapture Sampling and Indirect Sampling”. Journal of Official Statistics 28(1): 1–27. Available at: https://www.scb.se/contentassets/ca21efb41fee47d293bbee5bf7be7fb3/capture150recapture-sampling-and-indirect-sampling.pdf. (accessed March 2022).

42.

Leung, G., and K. Leung. 2020. “Crowdsourcing data to mitigate epidemics, the lancet digital health”, The Lancet Digital Health. DOI: https://doi.org/10.1016/S2589-7500(20)30055-8.

43.

Li, R., S. Pei, B. Chen, Y. Song, T. Zhang, W. Yang, and J. Shaman. 2020. “Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARSCoV2)”, Science 368 (6490): 489–493. DOI: http://doi.org/10.1126/science.abb3221.

44.

Mizumoto, K., K. Kagaya, A., Zarebski, and G. Chowell. 2020. “Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020”. Euro surveillance: bulletin Europeen sur les maladies transmissibles = European communicable disease bulletin, 25(10), 2000180. DOI: https://doi.org/10.2807/1560-7917.ES.2020.25.10.2000180.

45.

Murthy M.N., and V.K. Sethi. 1965. “Self-Weighting Design at Tabulation Stage” Sankhya: The Indian Journal of Statistics, 27(1–2): 201–210.

46.

Narain, R.D. 1951. “On sampling without replacement with varying probabilities”. Journal of the Indian Society of Agricultural Statistics 3: 169–174.

47.

ONS (Office for national Statistics). 2020. Coronavirus (COVID-19) Infection Survey pilot: England and Wales. Available at: https://www.ons.gov.uk/peoplepopulationand-community/healthandsocialcare/conditionsanddiseases/bulletins/coronaviruscovid19infectionsurveypilot/englandandwales14august2020. (accessed August 2020).

48.

Romania-insider.com. 2020. “Coronavirus in Romania: Over 10,000 Bucharest residents will be tested for Covid-19 as part of a study”. Romania-insider.com. Available at: https://www.romania-insider.com/coronavirus-romania-bucharest-testing-streinu-cercel. (accessed April 2020).

49.

Rossman, H., A. Keshet, S. Shilo, A. Gavrieli, T. Bauman, O. Cohen, R. Balicer, B. Geiger, Y. Dor, and E. Segal. 2020. “A framework for identifying regional outbreak and spread of COVID-19 from one- minute population-wide surveys”. Nature Medicine 26(5): 634–638. DOI: https://doi.org/10.1101/2020.03.19.20038844.

50.

Saunders-Hastings, P., B.Q. Quinn Hayes, R. Smith, and D. Krewski. 2017. “Control strategies to protect hospital resources during an influenza pandemic”. PloS one 12(6): e0179315. DOI: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0179315.

51.

Scott J. 2000. Social Network Analysis. A Handbook, London, Sage Publications.

52.

Singh, A.C., and F. Mecatti. 2011. “Generalized Multiplicity-Adjusted Horvitz-Thompson Estimation as a Unified Approach to Multiple Frame Surveys”. Journal of Official Statistics 27(4): 633–650. Available at: https://www.scb.se/contentassets/ca21efb41-fee47d293bbee5bf7be7fb3/generalized-multiplicity-adjusted-horvitz-thompson-estimation-as-a-unified-approach-to-multiple-frame-surveys.pdf (accessed Marsch 2022).

53.

Soper H.E. 1929. “Interpretation of periodicity in disease prevalence”, Journal of the Royal Statistical Society A 92: 34–73. DOI: https://doi.org/10.2307/2341437.

54.

Sudman, S., G. Monroe, M.G. Sirken, and C.D. Cowan. 1988. “Sampling Rare and Elusive Populations” Science 240(4855): 991–996. Available at: https://www.science.org/doi/10.1126/science.240.4855.991.

55.

Sun., K., J. Chen, and C. Viboud. 2020. “Early epidemiological analysis of coronavirus disease 2019 outbreak using crowdsourced data: a population level observational study”, The Lancet Digital Health. DOI: https://doi.org/10.1016/S2589-7500(20)30026-1.

56.

Tillé Y., and M. Wilhelm. 2017. “Probability Sampling Designs: Principles for Choice of Design and Balancing”. Statistical Science 32(2): 176–189. DOI: https://doi.org/10.1214/16-STS606.

57.

Thompson S.K., and G.A.F. Seber. 1996. Adaptive Sampling. Wiley Series in Probability and Statistics, New York.

58.

Vynnycky, E. 2010. An Introduction to Infectious Disease Modelling, edited by R.G. White. Oxford: Oxford University Press.

59.

Yang S., F.B. Keller, and L. Zheng. 2016. Social Network Analysis: Methods and Examples, Sage Publications, London.

60.

Yelin, I., N. Aharony, E. Shaer-Tamar, A. Argoetti, E. Messer, D. Berenbaum, E. Shafran, A. Kuzli., N. Gandali, T. Hashimshony, Y. Mandel-Gutfreund, M. Halberthal, Y. Geffen, M. Szwarcwort-Cohen, and R. Kishony. 2020. “Evaluation of C0VID-19 RT-qPCR test in multi-sample pools”, medRxiv. DOI: https://doi.org/10.1101/2020.03.26.20039438.