A simple SIR model with a large set of asymptomatic infectives

Giuseppe Gaeta; Giuseppe Gaeta

doi:10.3934/mine.2021013

Mathematics in Engineering

2021, Volume 3, Issue 2: 1-39. doi: 10.3934/mine.2021013

Previous Article Next Article

Research article

A simple SIR model with a large set of asymptomatic infectives

Giuseppe Gaeta ^{1,2
,
,}

1.
Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, I-20133 Milano, Italy
2.
SMRI, 00058 Santa Marinella, Italy

Received: 19 May 2020 Accepted: 23 May 2020 Published: 01 June 2020

There is increasing evidence that one of the most difficult problems in trying to control the ongoing COVID-19 epidemic is the presence of a large cohort of asymptomatic infectives. We develop a SIR-type model taking into account the presence of asymptomatic, or however undetected, infective, and the substantially long time these spend being infective and not isolated. We discuss how a SIR-based prediction of the epidemic course based on early data but not taking into account the presence of a large set of asymptomatic infectives would give wrong estimate of very relevant quantities such as the need of hospital beds, the time to the epidemic peak, and the number of people which are left untouched by the first wave and thus in danger in case of a second epidemic wave. In the second part of the note, we apply our model to the COVID-19 epidemics in Northern Italy. We obtain a good agreement with epidemiological data; according to the best fit of epidemiological data in terms of this model, only 10% of infectives in Italy is symptomatic.
- COVID,
- epidemiological models,
- SIR model,
- asymptomatic transmission,
- nonlinear dynamics
Citation: Giuseppe Gaeta. A simple SIR model with a large set of asymptomatic infectives[J]. Mathematics in Engineering, 2021, 3(2): 1-39. doi: 10.3934/mine.2021013

Related Papers:

Abstract

There is increasing evidence that one of the most difficult problems in trying to control the ongoing COVID-19 epidemic is the presence of a large cohort of asymptomatic infectives. We develop a SIR-type model taking into account the presence of asymptomatic, or however undetected, infective, and the substantially long time these spend being infective and not isolated. We discuss how a SIR-based prediction of the epidemic course based on early data but not taking into account the presence of a large set of asymptomatic infectives would give wrong estimate of very relevant quantities such as the need of hospital beds, the time to the epidemic peak, and the number of people which are left untouched by the first wave and thus in danger in case of a second epidemic wave. In the second part of the note, we apply our model to the COVID-19 epidemics in Northern Italy. We obtain a good agreement with epidemiological data; according to the best fit of epidemiological data in terms of this model, only 10% of infectives in Italy is symptomatic.

References

[1]	Gandhi M, Yokoe DS, Havlir DV (2020) Asymptomatic transmission, the Achilles' Heel of current strategies to control Covid-19. N Engl J Med 382: 1-3.
[2]	Arons MM, Hatfield KM, Reddy SC, et al. (2020) Presymptomatic SARS-CoV-2 infections and transmission in a skilled nursing facility. N Engl J Med 382: 1-10.
[3]	Bai Y, Yao L, Wei T, et al. (2020) Presumed asymptomatic carrier transmission of COVID-19. JAMA 323: 1406-1407.
[4]	Mason H (2020) Pre- and asymptomatic individuals contribute up to 80% of COVID-19 transmission. Univadis Medical News.
[5]	Furukawa NW, Brooks JT, Sobel J (2020) Evidence supporting transmission of severe acute respiratory syndrome coronavirus 2 while presymptomatic or asymptomatic. DOI: 10.3201/eid2607.201595.
[6]	Li Q, Guan XH, Wu P, et al. (2020) Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia. N Engl J Med 382: 1-9.
[7]	Yu X, Yang R (2020) COVID-19 transmission through asymptomatic carriers is a challenge to containment. Influenza Other Respir Viruses 2020: 1-2.
[8]	Park SW, Cornforth DM, Dushoff J, et al. (2020) The time scale of asymptomatic transmission affects estimates of epidemic potential in the COVID-19 outbreak. DOI: https://doi.org/10.1101/2020.03.09.20033514.
[9]	Yang Y, Liu YR, Luo FM, et al. (2020) COVID-19 Asymptomatic infection estimation. DOI: https://doi.org/10.1101/2020.04.19.20068072.
[10]	Day M (2020) Covid-19: Four fifths of cases are asymptomatic, China figures indicate. BMJ 369: m1375.
[11]	Treibel TA, Manisty C, Burton M, et al. (2020) COVID-19: PCR screening of asymptomatic healthcare workers at London hospital. DOI: https://doi.org/10.1016/S0140-6736(20)31100-4.
[12]	Zhang J, Tian S, Lou J, et al. (2020) Familial cluster of COVID-19 infection from an asymptomatic. Critical Care 24: 119.
[13]	Gronvall G, Asymptomatic spread makes COVID-19 tough to contain, John Hopkins University COVID resource, Available from: https://hub.jhu.edu/2020/05/12/gigi-gronvall-asymptomaticspread-covid-19-immunity-passports/.
[14]	Zhou R, Li FR, Chen FJ, et al. (2020) Viral dynamics in asymptomatic patients with COVID-19. DOI: https://doi.org/10.1016/j.ijid.2020.05.030.
[15]	He X (2020) Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat Med 26: 672-675.
[16]	How asymptomatic cases fuelled spread of coronavirus, Available from: https://timesofindia.indiatimes.com/india/how-asymptomatic-cases-fuelled-spread-ofcoronavirus/articleshow/75292506.cms.
[17]	Nishiura H, Kobayashi T, Yang Y, et al. (2020) The rate of underascertainment of novel coronavirus (2019-nCoV) infection: Estimation using Japanese passengers data on evacuation flights. J Clin Med 9: 419.
[18]	Nishiura H, Kobayashi T, Miyama T, et al. (2020) Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19). DOI: 10.1016/j.ijid.2020.03.020.
[19]	Mizumoto K, Kagaya K, Zarebski A, et al. (2020) Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020. Euro Surveill 25: 2000180.
[20]	Lavezzo E, Franchin E, Ciavarella C, et al. (2020) Suppression of COVID-19 outbreak in the municipality of Vò, Italy. medRxiv 2020.04.17.20053157.
[21]	MRC Centre for Global Infectious Disease Analysis, Available from: http://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/.
[22]	Press reports on UK Scientific Advisers audition by Parliament, Available from: https://www.theguardian.com/world/2020/mar/12/uk-governments-coronavirus-adviceand-why-it-gave-it; https://www.ft.com/content/38a81588-6508-11ea-b3f3-fe4680ea68b5; https://www.theguardian.com/politics/live/2020/mar/05/chief-medical-officer-chris-whittyquestioned-by-mps-live-news.
[23]	Li R, Pei S, Chen B, et al. (2020) Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2), Science 368: 489-493.
[24]	Lourenço J, Paton R, Ghafari M, et al. (2020) Fundamental principles of epidemic spread highlight the immediate need for large-scale serological survey to assess the stage of the SARS-CoV-2 epidemic, Available from: https://www.medrxiv.org/content/10.1101/2020.03.24.20042291v1.
[25]	Flaxman S, Mishra S, Gandy A, et al. (2020) Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries, Available from: http://www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/report-13-europe-npiimpact/.
[26]	Kermack WO, McKendrick AG (1927) A contributions to the mathematical theory of epidemics. Proc R Soc Lond A 115: 700-721.
[27]	Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics. II. - The problem of endemicity. Proc R Soc Lond A 138: 55-83.
[28]	Kermack WO, McKendrick AG (1933) Contributions to the mathematical theory of epidemics. III. - Further studies of the problem of endemicity. Proc. R. Soc. Lond. A 141: 94-122.
[29]	Hethcote HW (2000) The mathematics of infectious diseases. SIAM Review 42: 599-653.
[30]	Murray JD (2002) Mathematical Biology. I: An Introduction, Springer.
[31]	Edelstein-Keshet L (2005) Mathematical Models in Biology, SIAM.
[32]	Britton NF (2003) Essential Mathematical Biology, Springer.
[33]	Diekmann O, Heesterbeek H (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley.
[34]	Anderson RM, May RM (1992) Infectious Diseases of Humans: Dynamics and Control, Oxford UP.
[35]	Newman MEJ (2002) Spread of epidemic disease on networks. Phys Rev E 66: 016128.
[36]	Majumdar A, Mehta A (2020) Heterogeneous contact networks in COVID-19 spreading: The role of social deprivation. arXiv: 2005.00491.
[37]	Gatto M, Bertuzzo E, Mari L, et al. (2020) Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. PNAS 117: 10484-10491.
[38]	Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol 28: 356-382.
[39]	Dietz K (1993) The estimation of the basic reproduction number for infectious diseases. Stat Meth Med Res 2: 23-41.
[40]	Heesterbeek JAP (2002) A brief history of R0 and a recipe for its calculation. Acta Biotheor 50: 189-204.
[41]	The Novel Coronavirus Pneumonia Emergency Response Epidemiology Team (2020) Vital Surveillances: The Epidemiological Characteristics of an Outbreak of 2019 Novel Coronavirus Diseases (COVID-19) - China, 2020. China CDC Weekly 2: 113-122.
[42]	CDC-WHO joint COVID-19 mission, Available from: https://www.who.int/docs/defaultsource/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf.
[43]	WHO situation reports, Available from: https://www.who.int/emergencies/diseases/novelcoronavirus-2019/situation-reports.
[44]	Italian Government press releases, Available from: http://www.protezionecivile.gov.it/mediacomunicazione/comunicati-stampa.
[45]	Italian Health Ministry reports, Available from: http://www.salute.gov.it/nuovocoronavirus.
[46]	Giordano G, Blanchini F, Bruno R, et al. (2020) Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. DOI: https://doi.org/10.1038/s41591-020-0883-7.
[47]	Gaeta G (2020) Social distancing versus early detection and contacts tracing in epidemic management. arXiv: 2003.14102.
[48]	Gaeta G (2020) Data analysis for the COVID-19 early dynamics in Northern Italy. arXiv: 2003.02062.
[49]	Gaeta G (2020) Data Analysis for the COVID-19 early dynamics in Northern Italy. The effect of first restrictive measures. arXiv: 2003.03775.
[50]	Cadoni M (2020) How to reduce epidemic peaks keeping under control the time-span of the epidemic. arXiv: 2004.02189.
[51]	Cadoni M, Gaeta G (2020) How long does a lockdown need to be?. arXiv: 2004.11633.
[52]	Cadoni M, Gaeta G (2020) Size and timescale of epidemics in the SIR framework. arXiv: 2004.13024.
[53]	Google mobility reports, Available from: https: //www.google.com/covid19/mobility/.
[54]	Banerjee M, Tokarev A, Volpert V (2020) Immuno-epidemiological model of two-stage epidemic growth. Math Model Nat Phenom 15: 27.
[55]	Gaeta G (2020) Asymptomatic infectives and R0 for COVID. arXiv: 2003.14098.
[56]	Britton T, Ball F, Trapman P (2020) The disease-induced herd immunity level for Covid-19 is substantially lower than the classical herd immunity level. arXiv: 2005.03085.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)