The phasing out of protective measures by governments and public health
agencies, despite continued seriousness of the coronavirus pandemic, leaves
individuals who are concerned for their health with two basic options over which they have control: 1) minimize risk of
infection by being vaccinated and by wearing a face mask when
appropriate, and 2) minimize risk of transmission upon infection by
self-isolating. For the latter to be effective, it is essential to have an
accurate sense of the probability of infectivity as a function of time following the onset of symptoms.
Epidemiological considerations suggest that the period of infectivity
follows a lognormal distribution. This proposition is tested empirically by
construction of the lognormal probability density function and cumulative
distribution function based on quantiles of infectivity reported by several
independent investigations. A comprehensive examination of a prototypical ideal clinical study, based on general
statistical principles (the Principle of Maximum Entropy and the Central
Limit Theorem) reveals that the probability of infectivity is a lognormal
random variable. Subsequent evolution of
new variants may change the parameters of the distribution, which can be
updated by the methods in this paper, but the form of the probability function
is expected to remain lognormal as this is the most probable distribution
consistent with mathematical requirements and available information.
References
[1]
Berstein, A. (2021) Coronavirus: How Long Is a Person Contagious? Medical News Today. https://www.medicalnewstoday.com/articles/how-long-is-a-person-contagious-with-coronavirus
[2]
Sun, L.H. and Achenbach, J. (2022) When You Have Covid, Here’s How You Know You Are No Longer Contagious. Washington Post.
https://www.washingtonpost.com/health/2022/08/01/covid-contagious-period-isolation
[3]
Gillespie, C. (2022) How Long after Having COVID-19 Are You Contagious? Health.
https://www.health.com/condition/infectious-diseases/coronavirus/how-long-after-coronavirus-are-you-contagious
[4]
Ries, J. (2022) How Long before Someone with COVID-19 Isn’t Contagious? Healthline.
https://www.healthline.com/health-news/how-long-before-someone-with-covid-19-isnt-contagious
[5]
Marshall, M. (2022) How Long Is Someone with COVID Contagious? CBS News Boston.
https://www.cbsnews.com/boston/news/how-long-is-someone-with-covid-contagious
CDC (2022) Ending Isolation and Precautions for People with COVID-19: Interim Guidance.
https://www.cdc.gov/coronavirus/2019-ncov/hcp/duration-isolation.html
[8]
Johnston, C., Hughes, H., Lingard, S., Hailey, S. and Healy, B. (2022) Immunity and Infectivity in Covid-19. The BMJ, 378, e061402.
https://doi.org/10.1136/bmj-2020-061402
[9]
Tsao, J., Kussman, A. and Segovia, N.A. (2022) Prevalence of Positive Rapid Antigen Tests after 7-Day Isolation Following SARS-Cov-2 Infection in College Athletes during Omicron Variant Predominance. JAMA Network Open, 5, e2237149.
https://doi.org/10.1001/jamanetworkopen.2022.37149
[10]
Wu, Y., Kang, L. and Guo, Z. (2022) Incubation Period of COVID-19 Caused by Unique SARS-Cov-2 Strains: A Systematic Review and Meta-Analysis. JAMA Network Open, 5, e2228008. https://doi.org/10.1001/jamanetworkopen.2022.28008
[11]
Calabuig, J.M., Garcia-Raffi, L.M., Garcia-Valiente, A. and Sanchez-Perez, E.A. (2021) Kaplan-Meier Type Survival Curves for COVID-19: A Health Data Based Decision-Making Tool. Frontiers in Public Health, 9, Article ID: 646863.
https://doi.org/10.3389/fpubh.2021.646863
[12]
Schlager, T. and Whillans, A.V. (2022) People Underestimate the Probability of Contracting the Coronavirus from Friends. Humanities and Social Sciences Communications, 9, 59. https://doi.org/10.1057/s41599-022-01052-4
[13]
Silverman, M.P. (2022) Exact Statistical Distribution and Correlation of Human Height and Weight: Analysis and Experimental Confirmation. Open Journal of Statistics, 12, 743-787. https://doi.org/10.4236/ojs.2022.125044
[14]
Silverman, M.P. and Lipscombe, T.C. (2022) Exact Statistical Distribution of the Body Mass Index (BMI): Analysis and Experimental Confirmation. Open Journal of Statistics, 12, 324-356. https://doi.org/10.4236/ojs.2022.123022
[15]
Limpert, E., Stahel, W.A. and Abbt, M. (2001) Log-Normal Distributions across the Sciences: Keys and Clues, BioScience, 51, 341-352.
https://doi.org/10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2
[16]
O’Connor, P. and Kleyner, A. (2011) Practical Reliability Engineering. John Wiley & Sons, New York, 33-41. https://doi.org/10.1002/9781119961260
[17]
Sartwell, P.E. (1950) The Distribution of Incubation Periods of Infectious Disease. The American Journal of Hygiene, 51, 310-318.
https://doi.org/10.1093/oxfordjournals.aje.a119397
[18]
Lee, H., Kim, K., Choi, K., Hong, S., Son, H. and Ryu, S. (2020) Incubation Period of the Coronavirus Disease 2019 (COVID-19) in Busan, South Korea. Journal of Infection and Chemotherapy, 26, 1011-1013. https://doi.org/10.1016/j.jiac.2020.06.018
[19]
Karlin, S. (1966) A First Course in Stochastic Processes. Academic, New York, 189-201.
[20]
Silverman, M.P. (2014) A Certain Uncertainty: Nature’s Random Ways. Cambridge University Press, Cambridge, 495-506. https://doi.org/10.1017/CBO9781139507370
[21]
Altman, D.G. (1999) Practical Statistics for Medical Research. Chapman & Hall/CRC, London, 368-371.
[22]
van Kampen, J.J.A., van de Vijver, D.A.M.C., Fraaij, P.L.A., et al. (2021) Duration and Key Determinants of Infectious Virus Shedding in Hospitalized Patients with Coronavirus Disease-2019. Nature Communications, 12, Article No. 267.
https://doi.org/10.1038/s41467-020-20568-4
[23]
Davies, M, Bramwell, L.R., Jeffery, N., Bunce, B., et al. (2022) Persistence of Clinically Relevant Levels of SARS-CoV2 Envelope Gene Subgenomic RNAs in Non-Immuno-Compromised Individuals. International Journal of Infectious Diseases, 116, 418-425. https://doi.org/10.1016/j.ijid.2021.12.312
[24]
Hafter, N. (2021) What’s the Difference between a PCR and Antigen COVID-19 Test? UMass Chan Medical School, Worcester.
https://www.umassmed.edu/news/news-archives/2021/11/whats-the-difference-between-a-pcr-and-antigen-covid-19-test
[25]
Hakki, S., Zhou, J., Jonnerby, J., Singanayagam, A., et al. (2022) Onset and Window of SARS-Cov-2 Infectiousness and Temporal Correlation with Symptom Onset: A Prospective, Longitudinal, Community Cohort Study. The Lancet Respiratory Medicine, 10, 1061-1073. https://doi.org/10.1016/S2213-2600(22)00226-0
[26]
Imperial College London (2022, August 18) Real-World Study Details Average Duration of Infectiousness for COVID-19. ScienceDaily.
https://www.sciencedaily.com/releases/2022/08/220818190625.htm
[27]
Ke, R., Martinez, P.P., Smith, R.L., Gibson, L.L., et al. (2022) Daily Longitudinal Sampling of SARS-Cov-2 Infection Reveals Substantial Heterogeneity in Infectiousness. Nature Microbiology, 7, 640-652. https://doi.org/10.1038/s41564-022-01105-z
[28]
Rosenkrantz, R.D. (1989) Where Do We Stand on Maximum Entropy (1978). In: Rosenkrantz, R.D., Ed., E. T. Jaynes: Papers on Probability, Statistics, and Statistical Physics, Springer, Berlin, 210-314. https://doi.org/10.1007/978-94-009-6581-2_10
[29]
Jaynes, E.T. (1957) Information Theory and Statistical Mechanics. Physical Review, 106, 620-630. https://doi.org/10.1103/PhysRev.106.620
[30]
Jaynes, E.T. (1957) Information Theory and Statistical Mechanics, II. Physical Review, 108, 171-190. https://doi.org/10.1103/PhysRev.108.171
[31]
Shannon, C.E. (1948) A Mathematical Theory of Communication. Bell System Technical Journal, 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
[32]
Silverman, M.P. (2015) Cheating or Coincidence? Statistical Method Employing the Principle of Maximum Entropy for Judging Whether a Student Has Committed Plagiarism. Open Journal of Statistics, 5, 143-157.
https://doi.org/10.4236/ojs.2015.52018
[33]
Sivia, D.S. and Skilling, J. (2006) Data Analysis: A Bayesian Tutorial. Oxford University Press, Oxford, 120-124.
[34]
Silverman, M.P. (2012) Unpublished Notes on Thermal and Statistical Physics.
[35]
Wikipedia Contributors (2023) Truncated Normal Distribution.
https://en.wikipedia.org/w/index.php?title=Truncated_normal_distribution&oldid=1142956876
[36]
Taavitsainen, A. and Vanhanen, R. (2017) On the Maximum Entropy Distributions of Inherently Positive Nuclear Data. Nuclear Instruments and Methods in Physics Research A, 854, 156-162. https://doi.org/10.1016/j.nima.2016.11.061
[37]
Chou, Y. (1969) Statistical Analysis: With Business and Economic Applications. Holt, Rinehart, and Winston, New York, 242-244.
[38]
Hogg, R.V., McKean, J.W. and Craig, A.T. (2005) Introduction to Mathematical Statistics. Pearson/Prentice Hall, Upper Saddle River, 220-225.
[39]
Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011) Statistical Distributions. 4th Edition, Wiley, New York, 152-156. https://doi.org/10.1002/9780470627242
[40]
Arfken, G.B. and Weber, H.J. (2005) Mathematical Methods for Physicists. 6th Edition, Elsevier, New York, 83-85, 669-670, 975.
[41]
Silverman, M.P., Strange, W. and Lipscombe T.C. (2004) The Distribution of Composite Measurements: How to Be Certain of the Uncertainties in What We Measure. American Journal of Physics, 72, 1068-1081. https://doi.org/10.1119/1.1738426
[42]
Mood, A.M., Graybill, F.A. and Boes, D.C. (1963) Introduction to the Theory of Statistics. McGraw-Hill, New York, 181.
[43]
Taylor, J.R. (1997) An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, Sausalito, 60-62, 73-77, 146-148.
[44]
Wikipedia Contributors (2023, March 19) Maximum Entropy Probability Distribution. In Wikipedia, The Free Encyclopedia.
https://en.wikipedia.org/w/index.php?title=Maximum_entropy_probability_distribution&oldid=1145413438
Jain, U. (2020) Effect of COVID-19 on the Organs. Cureus, 12, e9540.
https://doi.org/10.7759/cureus.9540
[47]
Mayo Clinic, COVID-19: Long-Term Effects (2023).
https://www.mayoclinic.org/diseases-conditions/coronavirus/in-depth/coronavirus-long-term-effects/art-20490351
[48]
Sutherland, S. (2023) The Brain and Long Covid. Scientific American, 328, 26-33.
[49]
LaFraniere, S. and Welland, N. (2023) U.S. Plans to End Public Health Emergency for Covid in May. New York Times.
https://www.nytimes.com/2023/01/30/us/politics/biden-covid-public-health-emergency.html
[50]
Topol, E. (2023) Ground Truths (March 7): A Break from Covid Waves and a Breakthrough for Preventing Long Covid.
https://erictopol.substack.com/p/a-break-from-covid-waves-and-a-breakthrough
[51]
Allen, J., Almukhtar, S., Aufrichtig, A., Barnard, A., et al. (2023) Coronavirus in the U.S.: Latest Map and Case Count. New York Times.
https://www.nytimes.com/interactive/2021/us/covid-cases.html
[52]
Weber, L. and Achenbach, J. (2023) Covid Backlash Hobbles Public Health and Future Pandemic Response. The Washington Post.
https://www.washingtonpost.com/health/2023/03/08/covid-public-health-backlash
[53]
Harris, E., (2023) Most COVID-19 Deaths Worldwide Were among Older People. JAMA, 329, 704. https://jamanetwork.com/journals/jama/fullarticle/2801723
https://doi.org/10.1001/jama.2023.1554
[54]
CDC (2022), Isolation and Precautions for People with COVID-19. https://www.cdc.gov/coronavirus/2019-ncov/your-health/isolation.html
