Forecasting Doubling Time of SARS-CoV-2 using Hawkes and SQUIDER Models
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Mar 29, 2024
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Abstract
The rate of spread of an emerging epidemic is frequently characterized via the doubling time, which is the time it takes for the number of cases to double. This paper explores different ways to estimate doubling time, and investigates the estimation of doubling time in relationship to parameters in the HawkesN model and the SQUIDER (Susceptible, Quarantine, Undetected Infected, Infected, Dead, Exposed, Recovered) model. We observe an approximately exponential relationship between the productivity parameter κ in the HawkesN model and doubling time. We also evaluate the performance of the models in forecasting doubling times and compare to empirical doubling times using daily reported statewide totals for SARS-CoV-2 infections in California, and find that the HawkesN model forecasts doubling times more accurately, with 3.6% smaller root mean squared errors in Spring 2020, 79.4% smaller root mean squared errors in Autumn 2020, and 5.4% smaller root mean squared errors in Summer 2021. The HawkesN and SQUIDER models appear to forecast daily rate doubling times accurately at most times, though the SQUIDER forecasts of daily rate doubling times are far more volatile and thus occasionally have much larger errors, particularly in Fall 2020.
Keywords:
Contagious diseases, epidemics, Hawkes model, Point process, SARS-Cov-2, Self-exciting
Article Details
How to Cite
KAPLAN, Andrew; KRESIN, Conor; SCHOENBERG, Frederic.
Forecasting Doubling Time of SARS-CoV-2 using Hawkes and SQUIDER Models.
Medical Research Archives, [S.l.], v. 12, n. 3, mar. 2024.
ISSN 2375-1924.
Available at: <https://esmed.org/MRA/mra/article/view/5137>. Date accessed: 22 dec. 2024.
doi: https://doi.org/10.18103/mra.v12i3.5137.
Section
Research Articles
The Medical Research Archives grants authors the right to publish and reproduce the unrevised contribution in whole or in part at any time and in any form for any scholarly non-commercial purpose with the condition that all publications of the contribution include a full citation to the journal as published by the Medical Research Archives.
References
1. Centers for Disease Control and Prevention. 2009 H1N1 Early Outbreak and Disease Characteristics. https://www.cdc.gov/h1n1flu/surveillanceqa.htm. Published 2009. Accessed Jan, 2022.
2. Centers for Disease Control and Prevention. Coronavirus Disease 2019 (COVID-19). https://www.cdc.gov/dotw/covid-19/index.html. Published 2021. Accessed Jan, 2022.
3. Merler S, Ajelli M, Fumanelli L, Vespignani A. Containing the accidental laboratory escape of potential pandemic influenza viruses. BMC Medicine 2013; 11:252.
4. Khan ZS, Van Bussel F, Hussain F. A predictive model for Covid-19 spread - with application to eight US states and how to end the pandemic. Epidemiology & Infection 2020;148:e249. doi: 10.1017/S0950268820002423.
5. Rizoiu MA, Mishra S, Kong Q, Carman M, Xie L. SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Processes. Proceedings of the 2018 World Wide Web Conference 2018; 419-428.
6. Hawkes A. Spectra of some self-exciting and mutually exciting point processes. Biometrika 1971; 58(1): 83-90.
7. Daley DJ, Vere-Jones D. An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods, 2nd ed. 2003; Springer, New York.
8. Mohler G, Schoenberg F, Short MB, Sledge D. Analyzing the impacts of public policy on COVID-19 transmission: a case study of the role of model and dataset selection using data from Indiana. Statistics and Public Policy 2021; 8(1): 1-8.
9. Schoenberg F. Estimating Covid-19 transmission time using Hawkes point processes. Annals of Applied Statistics, 2023; 17(4): 3349-3362.
10. Ogata Y. Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association 1988; 83(401): 9-27.
11. Ogata Y. Space-time point process models for earthquake occurrences. Ann. Inst. Statist. Math 1998; 50(2): 379-402.
12. Mohler GO, Short MB, Brantingham, PJ, Schoenberg FP, Tita GE. Self-exciting point process modeling of crime. J. Amer. Statist. Assoc. 2011; 106(493): 100-108.
13. Park J, Schoenberg FP, Brantingham PJ, Bertozzi AL. Investigating clustering and violence interruption in gang-related violent crime data using spatial-temporal point processes with covariates. J. Amer. Statist. Assoc. 2021; 116 (536): 1674-1687.
14. Meyer S, Elias J, Hohle M. A space-time conditional intensity model for invasive meningococcal disease occurrence. Biometrics 2012; 68(2): 607-616.
15. Meyer, S., Held L. (2014), Power-law models for infectious disease spread. AoAS, 2014; 8(3): 1612-1639.
16. Clements RA, Schoenberg FP, Schorlemmer D. Residual analysis for space-time point processes with applications to earthquake forecast models in California. Annals of Applied Statistics 2011; 5(4): 2549-2571.
17. Clements RA, Schoenberg FP, Veen A. Evaluation of space-time point process models using super-thinning. Environmetrics 2013; 23(7): 606-616.
18. Zechar JD, Schorlemmer D, Werner MJ, Gerstenberger MC, Rhoades DA, Jordan TH. Regional Earthquake Likelihood Models I: First-order results. Bull. Seismol. Soc. Am. 2013; 103: 787-798.
19. Bray A, Wong K, Barr CD, Schoenberg FP (2014). Voronoi cell based residual analysis of spatial point process models with applications to Southern California earthquake forecasts. Annals of Applied Statistics 8(4), 2247-2267.
20. Gordon JS, Clements RA, Schoenberg FP, Schorlemmer D. Voronoi residuals and other residual analyses applied to CSEP earthquake forecasts. Spatial Statistics 2015; 14:133-150.
21. Schorlemmer D, Werner MJ, Marzocchi W, Jordan TH, Ogata Y, Jackson DD, Mak S, Rhoades DA, Gerstenberger MC, Hirata N, Liukis M, Maechling PJ, Strader A, Taroni M, Wiemer S, Zechar JD, Zhuang J. The Collaboratory for the Study of Earthquake Predictability: achievements and priorities. Seismological Research Letters 2018; 89(4):1305-1313.
22. Yang AS. Modeling the Transmission Dynamics of Pertussis Using Recursive Point Process and SEIR model. PhD thesis, UCLA, 2019. Los Angeles, CA.
23. Park J, Chaffee A, Harrigan R, Schoenberg FP. A non-parametric Hawkes model of the spread of Ebola in West Africa. J Appl. Stat. 2022; 49(3): 621-637.
24. Kresin C, Schoenberg F, Mohler G. Comparison of Hawkes and SQUIDER models for the spread of Covid-19. Advances and Applications in Statistics 2021;74: 83-106.
25. Chiang WH, Liu X, Mohler G. Hawkes process modeling of COVID-19 with mobility leading indicators and spatial covariates. Int. J. Forecast. 2022; 38(2): 505-520.
26. Bertozzi AL, Franco E, Mohler G, Short MB, Sledge D. The challenges of modeling and forecasting the spread of COVID-19. Proceedings of the National Academy of Sciences 2020; 117(29): 16732-16738.
27. Schoenberg FP, Hoffmann M, Harrigan, R. A recursive point process model for infectious diseases. AISM 2019; 71(5): 1271-1287. https://doi.org/10.1007/s10463-018-0690-9.
28. Kelly JD, Harrigan RJ, Park J, Hoff NA, Lee SD, Wannier R, Selo B, Mossoko M, Njokolo B, Okitolonda-Wemakoy E, Mbala-Kingebeni P, Rutherford GW, Smith TB, Ahuka-Mundeke S, Muyembe-Tamfum JJ, Rimoin AW, Schoenberg FP. Real-time predictions of the 2018-2019 Ebola virus disease outbreak in the Democratic Republic of Congo using Hawkes point process models. Epidemics 2019; 28, 100354. doi: 10.1016/j.epidem.2019.100354.
29. Schoenberg F. Nonparametric estimation of variable productivity Hawkes processes. Environmetrics 2022; 33(6): e2747.
30. Wallinga J, Teunis P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Amer. J. Epidemiology 2004; 160(6):509-516.
31. Cauchemez S, Boelle PY, Donnelly CA, Ferguson NM, Thomas G, Leung GM, Hedley AJ, Anderson RM, Valleron AJ. Real-time estimates in early detection of SARS. Emerging infectious diseases 2006; 12(1):110.
32. Farrington C, Kanaan M, Gay N. Branching process models for surveillance of infectious diseases controlled by mass vaccination. Biostatistics 2003; 4(2):279-295.
33. Meyer S, Held L, Hohle, M. Spatio-Temporal Analysis of Epidemic Phenomena Using the R Package surveillance. Journal of Statistical Software 2017; 77 (11): 1-55.
34. Centers for Disease Control and Prevention. Quarantine and Isolation. https://www.cdc.gov/coronavirus/2019-ncov/your-health/quarantine-isolation.html. Published 2021. Accessed Dec, 2021.
35. Bo ̌hning D, Rocchetti I, Maruotti A, Hollinge H. Estimating the undetected infections in the Covid-19 outbreak by harnessing capture-recapture methods. Int. J. Infect. Dis. 2020; 97: 197-201.
36. Ogata Y. The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics 1978; 30(2): 243-261.
37. Kirchner M. Hawkes and INAR(∞) processes. Stochastic Processes and their Applications 2016; 26(8): 2494-2525.
38. Kirchner M. An estimation procedure for the Hawkes process. Quant. Financ. 2017; 17(4):571-595.
39. Newsom G, Padilla A. Executive Order N-33-20. Executive Department State of California. https://covid19.ca.gov/img/Executive-Order-N-33-20.pdf. Published 2020. Accessed Jan, 2022.
40. California Department of Public Health. COVID-19 Time-Series Metrics by County and State, California Open Data Portal. https://data.chhs.ca.gov/dataset/covid-19-time-series-metrics-by-county-and-state . Published 2022. Accessed Aug, 2021.
41. Bajema KL, Wiegand RE, Cuffe K, Patel SV, Iachan R, Lim T, Lee A, Moyse D, Havers F, Harding L, Fry AM, Hall AJ, Martin K, Biel M, Deng Y, Meyer WA, Mathur M, Kyle T. Estimated SARS-CoV-2 Seroprevalence in the US as of September 2020. JAMA Intern Med. 2021; 181(4):450-460.
42. Wermer E, Stein J. Trump administration pushing to block new money for testing, tracing and CDC in upcoming coronavirus relief bill. Washington Post, 2020;July 18: 1-2.
43. Centers for Disease Control and Prevention. COVIDView – Key Updates for Week 3, ending January 30, 2021. https://www.cdc.gov/coronavirus/2019-ncov/covid-data/covidview/index.html. Published 2021. Accessed Jan, 2022.
44. Hogan C, Garamani N, Sahoo M, Huang CH, Zehnder J, Pinsky B. Retrospective Screening for SARS-COV-2 RNA in California, USA, Late 2019. Emerg Infect Dis 2020; 26(10): 2487-2488.
45. Patel, N. (2020). Why the CDC botched its coronavirus testing. MIT Technology Review. www.technologyreview.com/2020/03/05/905484/ why-the-cdc-botched-its-coronavirus-testing . Published Mar 5, 2020. Accessed Jan, 2022.
46. Fontanet A, Cauchemez S. COVID-19 herd immunity: where are we? Nat Rev Immunol. 2020; 20(10): 583-584.
47. Centers for Disease Control and Prevention. Delta Variant: What We Know About the Science. https://www.cdc.gov/coronavirus/2019-ncov/variants/about-variants.html. Published 2021. Accessed Dec, 2021.
48. BianL,GaoQ,GaoF,WangQ,HeQ,WuX,MaoQ,XuM,LiangZ. Impact of the Delta variant on vaccine efficacy and response strategies. Expert Rev. Vaccines 2021; 20(10): 1201-1209.
49. U.S. Food & Drug Administration (2021). Coronavirus (COVID-19) Update: FDA Authorizes Pfizer-BioNTech COVID-19 Vaccine for Emergency Use in Adolescents in Another Important Action in Fight Against Pandemic. https://www.fda.gov/news-events/press-announcements/ coronavirus-covid-19-update-fda-authorizes-pfizer-biontech-covid-19vaccine-emergency-use. Published May 10, 2021. Accessed Jan, 2022.
50. Johns Hopkins University and Medicine. Cumulative Cases by Days since 50th Confirmed Case, Coronavirus Resource Center. https://coronavirus.jhu.edu/data/cumulative-cases . Published 2022. Accessed Jan, 2022.
51. Hamady A, Lee J, Loboda Z. Waning antibody responses in COVID-19: what can we learn from the analysis of other coronaviruses? Infection 2022;50(1): 11-25.
52. Lewis PAW, Shedler GS. Simulation of nonhomogeneous poisson processes by thinning. Naval Research Logistics Quarterly 1979; 26(3), 403-413.
2. Centers for Disease Control and Prevention. Coronavirus Disease 2019 (COVID-19). https://www.cdc.gov/dotw/covid-19/index.html. Published 2021. Accessed Jan, 2022.
3. Merler S, Ajelli M, Fumanelli L, Vespignani A. Containing the accidental laboratory escape of potential pandemic influenza viruses. BMC Medicine 2013; 11:252.
4. Khan ZS, Van Bussel F, Hussain F. A predictive model for Covid-19 spread - with application to eight US states and how to end the pandemic. Epidemiology & Infection 2020;148:e249. doi: 10.1017/S0950268820002423.
5. Rizoiu MA, Mishra S, Kong Q, Carman M, Xie L. SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Processes. Proceedings of the 2018 World Wide Web Conference 2018; 419-428.
6. Hawkes A. Spectra of some self-exciting and mutually exciting point processes. Biometrika 1971; 58(1): 83-90.
7. Daley DJ, Vere-Jones D. An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods, 2nd ed. 2003; Springer, New York.
8. Mohler G, Schoenberg F, Short MB, Sledge D. Analyzing the impacts of public policy on COVID-19 transmission: a case study of the role of model and dataset selection using data from Indiana. Statistics and Public Policy 2021; 8(1): 1-8.
9. Schoenberg F. Estimating Covid-19 transmission time using Hawkes point processes. Annals of Applied Statistics, 2023; 17(4): 3349-3362.
10. Ogata Y. Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association 1988; 83(401): 9-27.
11. Ogata Y. Space-time point process models for earthquake occurrences. Ann. Inst. Statist. Math 1998; 50(2): 379-402.
12. Mohler GO, Short MB, Brantingham, PJ, Schoenberg FP, Tita GE. Self-exciting point process modeling of crime. J. Amer. Statist. Assoc. 2011; 106(493): 100-108.
13. Park J, Schoenberg FP, Brantingham PJ, Bertozzi AL. Investigating clustering and violence interruption in gang-related violent crime data using spatial-temporal point processes with covariates. J. Amer. Statist. Assoc. 2021; 116 (536): 1674-1687.
14. Meyer S, Elias J, Hohle M. A space-time conditional intensity model for invasive meningococcal disease occurrence. Biometrics 2012; 68(2): 607-616.
15. Meyer, S., Held L. (2014), Power-law models for infectious disease spread. AoAS, 2014; 8(3): 1612-1639.
16. Clements RA, Schoenberg FP, Schorlemmer D. Residual analysis for space-time point processes with applications to earthquake forecast models in California. Annals of Applied Statistics 2011; 5(4): 2549-2571.
17. Clements RA, Schoenberg FP, Veen A. Evaluation of space-time point process models using super-thinning. Environmetrics 2013; 23(7): 606-616.
18. Zechar JD, Schorlemmer D, Werner MJ, Gerstenberger MC, Rhoades DA, Jordan TH. Regional Earthquake Likelihood Models I: First-order results. Bull. Seismol. Soc. Am. 2013; 103: 787-798.
19. Bray A, Wong K, Barr CD, Schoenberg FP (2014). Voronoi cell based residual analysis of spatial point process models with applications to Southern California earthquake forecasts. Annals of Applied Statistics 8(4), 2247-2267.
20. Gordon JS, Clements RA, Schoenberg FP, Schorlemmer D. Voronoi residuals and other residual analyses applied to CSEP earthquake forecasts. Spatial Statistics 2015; 14:133-150.
21. Schorlemmer D, Werner MJ, Marzocchi W, Jordan TH, Ogata Y, Jackson DD, Mak S, Rhoades DA, Gerstenberger MC, Hirata N, Liukis M, Maechling PJ, Strader A, Taroni M, Wiemer S, Zechar JD, Zhuang J. The Collaboratory for the Study of Earthquake Predictability: achievements and priorities. Seismological Research Letters 2018; 89(4):1305-1313.
22. Yang AS. Modeling the Transmission Dynamics of Pertussis Using Recursive Point Process and SEIR model. PhD thesis, UCLA, 2019. Los Angeles, CA.
23. Park J, Chaffee A, Harrigan R, Schoenberg FP. A non-parametric Hawkes model of the spread of Ebola in West Africa. J Appl. Stat. 2022; 49(3): 621-637.
24. Kresin C, Schoenberg F, Mohler G. Comparison of Hawkes and SQUIDER models for the spread of Covid-19. Advances and Applications in Statistics 2021;74: 83-106.
25. Chiang WH, Liu X, Mohler G. Hawkes process modeling of COVID-19 with mobility leading indicators and spatial covariates. Int. J. Forecast. 2022; 38(2): 505-520.
26. Bertozzi AL, Franco E, Mohler G, Short MB, Sledge D. The challenges of modeling and forecasting the spread of COVID-19. Proceedings of the National Academy of Sciences 2020; 117(29): 16732-16738.
27. Schoenberg FP, Hoffmann M, Harrigan, R. A recursive point process model for infectious diseases. AISM 2019; 71(5): 1271-1287. https://doi.org/10.1007/s10463-018-0690-9.
28. Kelly JD, Harrigan RJ, Park J, Hoff NA, Lee SD, Wannier R, Selo B, Mossoko M, Njokolo B, Okitolonda-Wemakoy E, Mbala-Kingebeni P, Rutherford GW, Smith TB, Ahuka-Mundeke S, Muyembe-Tamfum JJ, Rimoin AW, Schoenberg FP. Real-time predictions of the 2018-2019 Ebola virus disease outbreak in the Democratic Republic of Congo using Hawkes point process models. Epidemics 2019; 28, 100354. doi: 10.1016/j.epidem.2019.100354.
29. Schoenberg F. Nonparametric estimation of variable productivity Hawkes processes. Environmetrics 2022; 33(6): e2747.
30. Wallinga J, Teunis P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Amer. J. Epidemiology 2004; 160(6):509-516.
31. Cauchemez S, Boelle PY, Donnelly CA, Ferguson NM, Thomas G, Leung GM, Hedley AJ, Anderson RM, Valleron AJ. Real-time estimates in early detection of SARS. Emerging infectious diseases 2006; 12(1):110.
32. Farrington C, Kanaan M, Gay N. Branching process models for surveillance of infectious diseases controlled by mass vaccination. Biostatistics 2003; 4(2):279-295.
33. Meyer S, Held L, Hohle, M. Spatio-Temporal Analysis of Epidemic Phenomena Using the R Package surveillance. Journal of Statistical Software 2017; 77 (11): 1-55.
34. Centers for Disease Control and Prevention. Quarantine and Isolation. https://www.cdc.gov/coronavirus/2019-ncov/your-health/quarantine-isolation.html. Published 2021. Accessed Dec, 2021.
35. Bo ̌hning D, Rocchetti I, Maruotti A, Hollinge H. Estimating the undetected infections in the Covid-19 outbreak by harnessing capture-recapture methods. Int. J. Infect. Dis. 2020; 97: 197-201.
36. Ogata Y. The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics 1978; 30(2): 243-261.
37. Kirchner M. Hawkes and INAR(∞) processes. Stochastic Processes and their Applications 2016; 26(8): 2494-2525.
38. Kirchner M. An estimation procedure for the Hawkes process. Quant. Financ. 2017; 17(4):571-595.
39. Newsom G, Padilla A. Executive Order N-33-20. Executive Department State of California. https://covid19.ca.gov/img/Executive-Order-N-33-20.pdf. Published 2020. Accessed Jan, 2022.
40. California Department of Public Health. COVID-19 Time-Series Metrics by County and State, California Open Data Portal. https://data.chhs.ca.gov/dataset/covid-19-time-series-metrics-by-county-and-state . Published 2022. Accessed Aug, 2021.
41. Bajema KL, Wiegand RE, Cuffe K, Patel SV, Iachan R, Lim T, Lee A, Moyse D, Havers F, Harding L, Fry AM, Hall AJ, Martin K, Biel M, Deng Y, Meyer WA, Mathur M, Kyle T. Estimated SARS-CoV-2 Seroprevalence in the US as of September 2020. JAMA Intern Med. 2021; 181(4):450-460.
42. Wermer E, Stein J. Trump administration pushing to block new money for testing, tracing and CDC in upcoming coronavirus relief bill. Washington Post, 2020;July 18: 1-2.
43. Centers for Disease Control and Prevention. COVIDView – Key Updates for Week 3, ending January 30, 2021. https://www.cdc.gov/coronavirus/2019-ncov/covid-data/covidview/index.html. Published 2021. Accessed Jan, 2022.
44. Hogan C, Garamani N, Sahoo M, Huang CH, Zehnder J, Pinsky B. Retrospective Screening for SARS-COV-2 RNA in California, USA, Late 2019. Emerg Infect Dis 2020; 26(10): 2487-2488.
45. Patel, N. (2020). Why the CDC botched its coronavirus testing. MIT Technology Review. www.technologyreview.com/2020/03/05/905484/ why-the-cdc-botched-its-coronavirus-testing . Published Mar 5, 2020. Accessed Jan, 2022.
46. Fontanet A, Cauchemez S. COVID-19 herd immunity: where are we? Nat Rev Immunol. 2020; 20(10): 583-584.
47. Centers for Disease Control and Prevention. Delta Variant: What We Know About the Science. https://www.cdc.gov/coronavirus/2019-ncov/variants/about-variants.html. Published 2021. Accessed Dec, 2021.
48. BianL,GaoQ,GaoF,WangQ,HeQ,WuX,MaoQ,XuM,LiangZ. Impact of the Delta variant on vaccine efficacy and response strategies. Expert Rev. Vaccines 2021; 20(10): 1201-1209.
49. U.S. Food & Drug Administration (2021). Coronavirus (COVID-19) Update: FDA Authorizes Pfizer-BioNTech COVID-19 Vaccine for Emergency Use in Adolescents in Another Important Action in Fight Against Pandemic. https://www.fda.gov/news-events/press-announcements/ coronavirus-covid-19-update-fda-authorizes-pfizer-biontech-covid-19vaccine-emergency-use. Published May 10, 2021. Accessed Jan, 2022.
50. Johns Hopkins University and Medicine. Cumulative Cases by Days since 50th Confirmed Case, Coronavirus Resource Center. https://coronavirus.jhu.edu/data/cumulative-cases . Published 2022. Accessed Jan, 2022.
51. Hamady A, Lee J, Loboda Z. Waning antibody responses in COVID-19: what can we learn from the analysis of other coronaviruses? Infection 2022;50(1): 11-25.
52. Lewis PAW, Shedler GS. Simulation of nonhomogeneous poisson processes by thinning. Naval Research Logistics Quarterly 1979; 26(3), 403-413.