Bibliography

The threshold and the reproduction number

• Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London A 115(772):700-721. doi:10.1098/rspa.1927.0118.
  The compartmental model and its threshold; the homogeneous benchmark every result here is measured against.
• Diekmann, O., Heesterbeek, J. A. P. and Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28(4):365-382. doi:10.1007/BF00178324.
  R0 as the leading eigenvalue of the next-generation operator; the immunity factor is the rate at which that eigenvalue falls as susceptibles are depleted.
• Wallinga, J. and Lipsitch, M. (2007). How generation intervals shape the relationship between growth rates and reproductive numbers. Proceedings of the Royal Society B 274(1609):599-604. doi:10.1098/rspb.2006.3754.
  The conversion from growth rate to R used throughout, with a gamma generation interval.

Heterogeneity and the herd-immunity threshold

• Britton, T., Ball, F. and Trapman, P. (2020). A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2. Science 369(6505):846-849. doi:10.1126/science.abc6810.
  Age and activity groups lower the disease-induced herd immunity level well below 1 - 1/R0; persistent heterogeneity.
• Tkachenko, A. V., Maslov, S., Elbanna, A., Wong, G. N., Weiner, Z. J. and Goldenfeld, N. (2021). Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity. Proceedings of the National Academy of Sciences 118(17):e2015972118. doi:10.1073/pnas.2015972118.
  Introduces the immunity factor and its time dependence; fits about 4 to New York City and Chicago. The closed forms on the time-varying activity page are a solvable case of their integral.
• Tkachenko, A. V., Maslov, S., Wang, T., Elbanna, A., Wong, G. N. and Goldenfeld, N. (2021). Stochastic social behavior coupled to COVID-19 dynamics leads to waves, plateaus, and an endemic state. eLife 10:e68341. doi:10.7554/eLife.68341.
  Each person's activity follows a mean-reverting process with a correlation time; the closest prior model to activity that changes over time.
• Gomes, M. G. M., Ferreira, M. U., Corder, R. M., King, J. G. et al. (2022). Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold. Journal of Theoretical Biology 540:111063. doi:10.1016/j.jtbi.2022.111063.
  The gamma power law with exponent 1 + CV^2 for susceptibility and 1 + 2 CV^2 for exposure, fitted to European first waves.
• Lewontin, R. C. and Cohen, D. (1969). On population growth in a randomly varying environment. Proceedings of the National Academy of Sciences 62(4):1056-1060. doi:10.1073/pnas.62.4.1056.
  The mean of a randomly growing population can grow while almost every realization declines: the distinction between the mean and typical growth that the growth convexity rests on.

Contacts

• Mossong, J., Hens, N., Jit, M. et al. (2008). Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Medicine 5(3):e74. doi:10.1371/journal.pmed.0050074.
  POLYMOD: one-day contact diaries of 7,290 people in eight countries; the source of the within-age activity variance.
• Prem, K., van Zandvoort, K., Klepac, P. et al. (2021). Projecting contact matrices in 177 geographical regions: an update and comparison with empirical data for the COVID-19 era. PLOS Computational Biology 17(7):e1009098. doi:10.1371/journal.pcbi.1009098.
  Age contact matrices by country; the age part of the immunity factor.
• Gimma, A., Munday, J. D., Wong, K. L. M. et al. (2022). Changes in social contacts in England during the COVID-19 pandemic between March 2020 and March 2021 as measured by the CoMix survey: a repeated cross-sectional study. PLOS Medicine 19(3):e1003907. doi:10.1371/journal.pmed.1003907.
  The CoMix design: up to ten fortnightly rounds per participant, and the convention of capping daily contacts at 50.
• Wong, K. L. M., Gimma, A., Coletti, P. et al. (2023). Social contact patterns during the COVID-19 pandemic in 21 European countries: evidence from a two-year study. BMC Infectious Diseases 23:268. doi:10.1186/s12879-023-08214-y.
  CoMix in 21 countries; the data for out-of-sample predictions of the immunity factor.
• Jarvis, C. I., Van Zandvoort, K., Gimma, A. et al. (2020). Quantifying the impact of physical distance measures on the transmission of COVID-19 in the UK. BMC Medicine 18:124. doi:10.1186/s12916-020-01597-8. Data: Gimma, A., Wong, K. L. M., Van Zandvoort, K., Jarvis, C. I. and Edmunds, W. J. (2024), CoMix social contact data (UK), doi:10.5281/zenodo.13684044.
  The first CoMix round; the survey then asked each participant about their contacts up to eight or ten times, two weeks apart, from 2020 to 2022. The source of the persistent and switching parts of activity.
• Davies, N. G., Klepac, P., Liu, Y. et al. (2020). Age-dependent effects in the transmission and control of COVID-19 epidemics. Nature Medicine 26(8):1205-1211. doi:10.1038/s41591-020-0962-9.
  Lower susceptibility under 20; the sensitivity case on the age page.

Serology

• Pollán, M., Pérez-Gómez, B., Pastor-Barriuso, R. et al. (2020). Prevalence of SARS-CoV-2 in Spain (ENE-COVID): a nationwide, population-based seroepidemiological study. The Lancet 396(10250):535-544. doi:10.1016/S0140-6736(20)31483-5.
  Provincial seroprevalence; the provincial tables of rounds 3 and 4 come from the Ministry of Health reports.
• Stringhini, S., Wisniak, A., Piumatti, G. et al. (2020). Seroprevalence of anti-SARS-CoV-2 IgG antibodies in Geneva, Switzerland (SEROCoV-POP): a population-based study. The Lancet 396(10247):313-319. doi:10.1016/S0140-6736(20)31304-0.
  Weekly population serology in Geneva, April-May 2020.
• Ward, H., Atchison, C., Whitaker, M. et al. (2021). SARS-CoV-2 antibody prevalence in England following the first peak of the pandemic. Nature Communications 12:905. doi:10.1038/s41467-021-21237-w.
  REACT-2 round 1 by region, June-July 2020.
• Castro Dopico, X., Muschiol, S., Christian, M. et al. (2021). Seropositivity in blood donors and pregnant women during the first year of SARS-CoV-2 transmission in Stockholm, Sweden. Journal of Internal Medicine 290(3):666-676. doi:10.1111/joim.13304.
  Weekly Stockholm serology through the first year.
• Buss, L. F., Prete, C. A., Abrahim, C. M. M. et al. (2021). Three-quarters attack rate of SARS-CoV-2 in the Brazilian Amazon during a largely unmitigated epidemic. Science 371(6526):288-292. doi:10.1126/science.abe9728.
  Monthly blood-donor serology in Manaus, corrected for waning; the case where control was weak.

Data

• The New York Times, COVID-19 data: US county cases and deaths, 2020-21, counted by the date reported; archived 24 March 2023.
• Instituto de Salud Carlos III, CNE: Spanish daily admissions and deaths by province.
• Folkhälsomyndigheten, Socialstyrelsen and Svenska Intensivvårdsregistret: Swedish deaths, intensive-care admissions and serology reports.
• UK Health Security Agency coronavirus dashboard archive and ONS: English deaths and admissions by region.
• Swiss Federal Office of Public Health: Geneva deaths and admissions.
• Brazil's mortality information system (SIM): all-cause deaths in Manaus, 2015-2020.
• World Bank population by age, 2020; US Census county population estimates, 2019.