June  2022, 17(3): 359-384. doi: 10.3934/nhm.2022011

A study of computational and conceptual complexities of compartment and agent based models

1. 

Department of Mathematics, University of Hawai'i at Mānoa, Honolulu, HI 96822, USA

2. 

Data Science Institute, University of Hawai'i at Mānoa, Honolulu, HI 96822, USA

3. 

Office of Public Health Studies, University of Hawai'i at Mānoa, Honolulu, HI 96822, USA

* Corresponding author: chyba@hawaii.edu

Received  June 2021 Revised  November 2021 Published  June 2022 Early access  March 2022

Fund Project: The first five authors are supported by NSF grant No. 2030789

The ongoing COVID-19 pandemic highlights the essential role of mathematical models in understanding the spread of the virus along with a quantifiable and science-based prediction of the impact of various mitigation measures. Numerous types of models have been employed with various levels of success. This leads to the question of what kind of a mathematical model is most appropriate for a given situation. We consider two widely used types of models: equation-based models (such as standard compartmental epidemiological models) and agent-based models. We assess their performance by modeling the spread of COVID-19 on the Hawaiian island of Oahu under different scenarios. We show that when it comes to information crucial to decision making, both models produce very similar results. At the same time, the two types of models exhibit very different characteristics when considering their computational and conceptual complexity. Consequently, we conclude that choosing the model should be mostly guided by available computational and human resources.

Citation: Prateek Kunwar, Oleksandr Markovichenko, Monique Chyba, Yuriy Mileyko, Alice Koniges, Thomas Lee. A study of computational and conceptual complexities of compartment and agent based models. Networks and Heterogeneous Media, 2022, 17 (3) : 359-384. doi: 10.3934/nhm.2022011
References:
[1]

S. AnsumaliS. KaushalA. KumarM. K. Prakash and M. Vidyasagar, Modelling a pandemic with asymptomatic patients, impact of lockdown and herd immunity, with applications to SARS-CoV-2, Annu Rev Control, 50 (2020), 432-447.  doi: 10.1016/j.arcontrol.2020.10.003.

[2]

J. Balisacan, M. Chyba and C. Shanbrom, Two new compartmental epidemiological models and their equilibria, COVID-19 SARS-CoV-2 Preprints from MedRxiv and BioRxiv, (2021). doi: 10.1101/2021.09.03.21263050.

[3]

C. Branas et al., Flattening the curve before it flattens us: hospital critical care capacity limits and mortality from novel coronavirus (SARS-CoV2) cases in US counties, medRxiv, (2020).

[4]

M. G. Burch, K. A. Jacobsen, J. H. Tien and G. A. Rempala, Network-based analysis of a small ebola outbreak, Math. Biosci. Eng., 14 (2017), 67–77, arXiv: 1511.02362. doi: 10.3934/mbe.2017005.

[5]

S. Cauchemez et al., Role of Social Networks in Shaping Disease Transmission During a Community Outbreak of 2009 H1N1 Pandemic Influenza, Proceedings of the National Academy of Sciences, 108 (2011), 2825–2830.

[6]

Center for Infectious Disease Research and Policy, Coroner: First US COVID-19 death occurred in early February, https://www.cidrap.umn.edu/news-perspective/2020/04/coroner-first-us-covid-19-death-occurred-early-february.

[7]

Centers for Disease Control and Prevention, Emergence of SARS-CoV-2 B.1.1.7 Lineage-United States, December 29, 2020-January 12, 2021, Morbidity and Mortality Weekly Report (MMWR), 70 (2021), 95–99, https://www.cdc.gov/mmwr/volumes/70/wr/mm7003e2.htm.

[8]

S. Chib and E. Greenberg, Understanding the metropolis-hastings algorithm, The american statistician, 49 (1995), 327-335. 

[9]

M. Chyba, Y. Mileyko, O. Markovichenko, R. Carney and A. Koniges, Epidemiological model of the spread of COVID-19 in Hawai'is chanllenging fight against the didease, The Ninth International Conference on Global Health Challenges, Proceedings, (2020), 32–38.

[10]

K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two Delays, J. Math. Biol., 35 (1990), 240-258.  doi: 10.1007/s002850050051.

[11]

E. Cuevas, An agent-based model to evaluate the COVID-19 transmission risks in facilities, Comput Biol Med, 121 (2020), 103827.  doi: 10.1016/j.compbiomed.2020.103827.

[12]

S. Y. Del ValleJ. M. Hyman and N. Chitnis, Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases, Mathematical Biosciences and Engineering, 10 (2013), 1475-1497.  doi: 10.3934/mbe.2013.10.1475.

[13]

N. M. Ferguson et al., Report 9 - Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, MRC Centre for Global Infectious Disease Analysis, Imperial College, (2020).

[14]

M. Gatto et al., Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, (2020).

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G. Giordano et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, (2020).

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C. GroendykeD. Welch and D. R. Hunter, A network-based analysis of the 1861 hagelloch measles data, Biometrics, 68 (2012), 755-765.  doi: 10.1111/j.1541-0420.2012.01748.x.

[17]

W. T. Harvey et al., SARS-CoV-2 variants, spike mutations and immune escape, Nature Reviews Microbiology, 19 (2021), 409–424.

[18]

Hawaii Department of Health COVID Dashboard, https://health.hawaii.gov/coronavirusdisease2019/what-you-should-know/current-situation-in-hawaii/.

[19]

Hawaii Population Model. Hawai'i Data Collaborative, https://www.hawaiidata.org/hawaii-population-model, (2021).

[20]

Hawaii Safe Travels Digital Platform, https://hawaiicovid19.com/travel/data/.

[21]

E. M. Hendrix, et al., Introduction to Nonlinear and Global Optimization, Springer Optimization and Its Applications, 37. Springer, New York, 2010. doi: 10.1007/978-0-387-88670-1.

[22]

H. W. Hethcote, Three basic epidemiological models, Applied Mathematical Ecology (Trieste, 1986), 18 (1989), 119-144.  doi: 10.1007/978-3-642-61317-3_5.

[23]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[24]

Z. Jin et al., Modelling and analysis of influenza A (H1N1) on networks, BMC public health, 11 (2011), 1–9.

[25]

D. G. Kendall, Deterministic and stochastic epidemics in closed populations, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, (1956), 149–165.

[26]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ.-The problem of endemicity, Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55-83. 

[27]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, R. Soc. Lond. A, 115 (1927), 700-721. 

[28]

C. C. Kerr et al., Covasim: An agent-based model of COVID-19 dynamics and interventions, medRxiv, (2020).

[29]

M. Y. Li and J. S. Muldoweney, Global stability of a SEIR epidemic model with vertical transmission, SIAM J. Appl.Math., 62 (2001), 58-69.  doi: 10.1137/S0036139999359860.

[30]

J. O. Lloyd-SmithA. P. Galvani and W. M. Getz, Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Royal Society, 270 (2003), 1979-1989.  doi: 10.1098/rspb.2003.2481.

[31]

E. Mathieu, H. Ritchie and E. Ortiz-Ospina, et al., Coronavirus (COVID-19) Vaccinations, A Global Database of COVID-19 Vaccinations. Nat Hum Behav, (2021), https://ourworldindata.org/covid-vaccinations.

[32]

L. H. Nguye et al., Risk of COVID-19 among front-line health-care workers and the general community: A prospective cohort study, The Lancet Public Health, 5 (2020), e475–e483.

[33]

Ö. Ozme et al., Analyzing the impact of modeling choices and assumptions in compartmental epidemiological models, Simulation (SAGE journals), 92 (2016), 459–472. doi: 10.1177/0037549716640877.

[34]

M. ParkA. R. CookJ. T. LimY. Sun and B. L. Dickens, A systematic review of covid-19 epidemiology based on current evidence, J. Clin. Med., 9 (2020), 967.  doi: 10.3390/jcm9040967.

[35]

K. Prem et al., The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study, The Lancet Public Health, 5 (2020), e261–e270.

[36]

A. RadulescuC. Williams and K. Cavanagh, Management strategies in a SEIR-type model of COVID 19 community spread, Sci Rep, 10 (2020), 21256.  doi: 10.1038/s41598-020-77628-4.

[37]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts that Shaped the Field, (1998), 273–282. doi: 10.1145/280811.281008.

[38]

A. RizzoB. Pedalino and M. Porfiri, A network model for ebola spreading, Elsevier, 394 (2016), 212-222.  doi: 10.1016/j.jtbi.2016.01.015.

[39]

R. Ross, An Application of the Theory of Probabilities to the Study of a priori Pathometry. Part Ⅰ, Proceedings of the Royal Society of London Series A, 92 (1916), 204-230. 

[40]

R. Ross and H. P. Hudson, An application of the theory of probabilities to the study of a priori pathometry. Part Ⅱ, Proceedings of the Royal Society of London Series A, 93 (1917), 212-225. 

[41]

G. Sabetian et al., COVID-19 infection among healthcare workers: A cross-sectional study in southwest Iran, Virology Journal, 18 (2021), 58.

[42]

T. C. Schelling, Dynamic models of segregation, The Journal of Mathematical Sociology, 1 (1971), 143-186. 

[43]

T. C. Schelling, Micromotives and Macrobehavior, WW Norton & Company, 1978.

[44]

The New York Times, New Coronavirus Cases in U.S. Soar Past 68,000, Shattering Record, https://www.nytimes.com/2020/07/10/world/coronavirus-updates.html.

[45]

The New York Times, The U.S. Now Leads the World in Confirmed Coronavirus Cases, https://www.nytimes.com/2020/03/26/health/usa-coronavirus-cases.html.

[46]

A. Truszkowska et al., High-resolution agent-based modeling of COVID-19 spreading in a small town, Advanced Theory and Simulations, 4 (2021), 2000277. doi: 10.1002/adts.202000277.

[47]

World Health Organization, Listings of WHO's response to COVID-19, https://www.who.int/news/item/29-06-2020-covidtimeline.

[48]

F. Zhou, et al., Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study, The Lancet, (2020).

show all references

References:
[1]

S. AnsumaliS. KaushalA. KumarM. K. Prakash and M. Vidyasagar, Modelling a pandemic with asymptomatic patients, impact of lockdown and herd immunity, with applications to SARS-CoV-2, Annu Rev Control, 50 (2020), 432-447.  doi: 10.1016/j.arcontrol.2020.10.003.

[2]

J. Balisacan, M. Chyba and C. Shanbrom, Two new compartmental epidemiological models and their equilibria, COVID-19 SARS-CoV-2 Preprints from MedRxiv and BioRxiv, (2021). doi: 10.1101/2021.09.03.21263050.

[3]

C. Branas et al., Flattening the curve before it flattens us: hospital critical care capacity limits and mortality from novel coronavirus (SARS-CoV2) cases in US counties, medRxiv, (2020).

[4]

M. G. Burch, K. A. Jacobsen, J. H. Tien and G. A. Rempala, Network-based analysis of a small ebola outbreak, Math. Biosci. Eng., 14 (2017), 67–77, arXiv: 1511.02362. doi: 10.3934/mbe.2017005.

[5]

S. Cauchemez et al., Role of Social Networks in Shaping Disease Transmission During a Community Outbreak of 2009 H1N1 Pandemic Influenza, Proceedings of the National Academy of Sciences, 108 (2011), 2825–2830.

[6]

Center for Infectious Disease Research and Policy, Coroner: First US COVID-19 death occurred in early February, https://www.cidrap.umn.edu/news-perspective/2020/04/coroner-first-us-covid-19-death-occurred-early-february.

[7]

Centers for Disease Control and Prevention, Emergence of SARS-CoV-2 B.1.1.7 Lineage-United States, December 29, 2020-January 12, 2021, Morbidity and Mortality Weekly Report (MMWR), 70 (2021), 95–99, https://www.cdc.gov/mmwr/volumes/70/wr/mm7003e2.htm.

[8]

S. Chib and E. Greenberg, Understanding the metropolis-hastings algorithm, The american statistician, 49 (1995), 327-335. 

[9]

M. Chyba, Y. Mileyko, O. Markovichenko, R. Carney and A. Koniges, Epidemiological model of the spread of COVID-19 in Hawai'is chanllenging fight against the didease, The Ninth International Conference on Global Health Challenges, Proceedings, (2020), 32–38.

[10]

K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two Delays, J. Math. Biol., 35 (1990), 240-258.  doi: 10.1007/s002850050051.

[11]

E. Cuevas, An agent-based model to evaluate the COVID-19 transmission risks in facilities, Comput Biol Med, 121 (2020), 103827.  doi: 10.1016/j.compbiomed.2020.103827.

[12]

S. Y. Del ValleJ. M. Hyman and N. Chitnis, Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases, Mathematical Biosciences and Engineering, 10 (2013), 1475-1497.  doi: 10.3934/mbe.2013.10.1475.

[13]

N. M. Ferguson et al., Report 9 - Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, MRC Centre for Global Infectious Disease Analysis, Imperial College, (2020).

[14]

M. Gatto et al., Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, (2020).

[15]

G. Giordano et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, (2020).

[16]

C. GroendykeD. Welch and D. R. Hunter, A network-based analysis of the 1861 hagelloch measles data, Biometrics, 68 (2012), 755-765.  doi: 10.1111/j.1541-0420.2012.01748.x.

[17]

W. T. Harvey et al., SARS-CoV-2 variants, spike mutations and immune escape, Nature Reviews Microbiology, 19 (2021), 409–424.

[18]

Hawaii Department of Health COVID Dashboard, https://health.hawaii.gov/coronavirusdisease2019/what-you-should-know/current-situation-in-hawaii/.

[19]

Hawaii Population Model. Hawai'i Data Collaborative, https://www.hawaiidata.org/hawaii-population-model, (2021).

[20]

Hawaii Safe Travels Digital Platform, https://hawaiicovid19.com/travel/data/.

[21]

E. M. Hendrix, et al., Introduction to Nonlinear and Global Optimization, Springer Optimization and Its Applications, 37. Springer, New York, 2010. doi: 10.1007/978-0-387-88670-1.

[22]

H. W. Hethcote, Three basic epidemiological models, Applied Mathematical Ecology (Trieste, 1986), 18 (1989), 119-144.  doi: 10.1007/978-3-642-61317-3_5.

[23]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[24]

Z. Jin et al., Modelling and analysis of influenza A (H1N1) on networks, BMC public health, 11 (2011), 1–9.

[25]

D. G. Kendall, Deterministic and stochastic epidemics in closed populations, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, (1956), 149–165.

[26]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ.-The problem of endemicity, Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55-83. 

[27]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, R. Soc. Lond. A, 115 (1927), 700-721. 

[28]

C. C. Kerr et al., Covasim: An agent-based model of COVID-19 dynamics and interventions, medRxiv, (2020).

[29]

M. Y. Li and J. S. Muldoweney, Global stability of a SEIR epidemic model with vertical transmission, SIAM J. Appl.Math., 62 (2001), 58-69.  doi: 10.1137/S0036139999359860.

[30]

J. O. Lloyd-SmithA. P. Galvani and W. M. Getz, Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Royal Society, 270 (2003), 1979-1989.  doi: 10.1098/rspb.2003.2481.

[31]

E. Mathieu, H. Ritchie and E. Ortiz-Ospina, et al., Coronavirus (COVID-19) Vaccinations, A Global Database of COVID-19 Vaccinations. Nat Hum Behav, (2021), https://ourworldindata.org/covid-vaccinations.

[32]

L. H. Nguye et al., Risk of COVID-19 among front-line health-care workers and the general community: A prospective cohort study, The Lancet Public Health, 5 (2020), e475–e483.

[33]

Ö. Ozme et al., Analyzing the impact of modeling choices and assumptions in compartmental epidemiological models, Simulation (SAGE journals), 92 (2016), 459–472. doi: 10.1177/0037549716640877.

[34]

M. ParkA. R. CookJ. T. LimY. Sun and B. L. Dickens, A systematic review of covid-19 epidemiology based on current evidence, J. Clin. Med., 9 (2020), 967.  doi: 10.3390/jcm9040967.

[35]

K. Prem et al., The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study, The Lancet Public Health, 5 (2020), e261–e270.

[36]

A. RadulescuC. Williams and K. Cavanagh, Management strategies in a SEIR-type model of COVID 19 community spread, Sci Rep, 10 (2020), 21256.  doi: 10.1038/s41598-020-77628-4.

[37]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Seminal Graphics: Pioneering Efforts that Shaped the Field, (1998), 273–282. doi: 10.1145/280811.281008.

[38]

A. RizzoB. Pedalino and M. Porfiri, A network model for ebola spreading, Elsevier, 394 (2016), 212-222.  doi: 10.1016/j.jtbi.2016.01.015.

[39]

R. Ross, An Application of the Theory of Probabilities to the Study of a priori Pathometry. Part Ⅰ, Proceedings of the Royal Society of London Series A, 92 (1916), 204-230. 

[40]

R. Ross and H. P. Hudson, An application of the theory of probabilities to the study of a priori pathometry. Part Ⅱ, Proceedings of the Royal Society of London Series A, 93 (1917), 212-225. 

[41]

G. Sabetian et al., COVID-19 infection among healthcare workers: A cross-sectional study in southwest Iran, Virology Journal, 18 (2021), 58.

[42]

T. C. Schelling, Dynamic models of segregation, The Journal of Mathematical Sociology, 1 (1971), 143-186. 

[43]

T. C. Schelling, Micromotives and Macrobehavior, WW Norton & Company, 1978.

[44]

The New York Times, New Coronavirus Cases in U.S. Soar Past 68,000, Shattering Record, https://www.nytimes.com/2020/07/10/world/coronavirus-updates.html.

[45]

The New York Times, The U.S. Now Leads the World in Confirmed Coronavirus Cases, https://www.nytimes.com/2020/03/26/health/usa-coronavirus-cases.html.

[46]

A. Truszkowska et al., High-resolution agent-based modeling of COVID-19 spreading in a small town, Advanced Theory and Simulations, 4 (2021), 2000277. doi: 10.1002/adts.202000277.

[47]

World Health Organization, Listings of WHO's response to COVID-19, https://www.who.int/news/item/29-06-2020-covidtimeline.

[48]

F. Zhou, et al., Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study, The Lancet, (2020).

Figure 1.  Basic SEIR Model diagram and parameters
Figure 2.  Diagram of our basic compartmental model
Figure 3.  Blue denotes the SEIR model fit for Honolulu county from March 6 to October 15, 2020. The dots represent the actual daily cases from the Hawai'i DOH dashboard [18]. In red are the optimized transmission rates $ \beta $ (aligned with non pharmaceutical mitigation measures that were taken by the State of Hawai'i)
Figure 4.  Sub-Diagram for travelers assumptions
Figure 5.  Diagram of our compartmental model with travelers
Figure 6.  Top: Honolulu County fit from March 6, 2020 including travelers from October 15 to December 27, 2020. Bottom: Honolulu County fit zoomed in for period of October 15 - December 27 with the value for the basal transmission rate
Figure 7.  Diagram of our compartmental model including vaccines
Figure 8.  Top: Honolulu County fit from March 6, 2020 including travelers starting October 15, 2020 and vaccination starting December 27, 2020. Simulation runs through April 25, 2021. Bottom: Zoomed in on the period where both vaccination and travelers are included with the corresponding basal transmission rates
Figure 9.  Sample contact network representing individuals in the population as nodes and the interactions for possible viral transmission among them as edges. The different colors refer to four different types of contacts or individuals in the population (LHS). The vaccinated individuals will have a reduced transmission which is reflected in their state (RHS)
Figure 10.  Diagram of basic Covasim simulation algorithm
Figure 11.  In green is the Covasim model fit for Honolulu county from March 6 to October 15, 2020. Included are the optimized transmission rates
Figure 12.  Top: Covasim fit for Honolulu county from March 6 to April 25, 2021 with travelers and vaccine included. Bottom: zoomed in on the period October 15, 2020 - April 25, 2021 including the corresponding transmission rates
Figure 13.  Increase in simulation time as a function of problem size for serial Covasim runs. For the top graph, we set the population size equal to $ 1 $ million and run 15 simulations for each scenario with a different number of days starting from 50 to 500. For the bottom graph one we set the number of days equal to 150 and run 15 simulations for each scenario with different population sizes, ranging from a thousand to a million, with one additional point for a population of size $ 50 $ million
Figure 14.  Increasing computational cost of the compartmental model. For each number of simulation days, an average of 15 simulations is taken
Figure 15.  Compartmental(blue) and Covasim (green) fit from March 6, 2020 to April 25, 2021 including travelers and vaccines
Figure 16.  SEIR (blue) and Covasim(green) benchmark scenarios forecasting spread and vaccination
Figure 17.  The mean of 20 covasim simulations (black) with simulations with highest (red), lowest (magenta), earliest (blue) and latest (green) peaks
Table 1.  SEIR basic model variables. Isolated accounts for quarantine and hospitalization
Description of the variables in the EBM model.
Variable Description
$ S(t) $ Number of total susceptible individuals
$ E_{i}(t) $ (resp. $ E_{q, i}(t) $) Number of asymptomatic infected individuals $ i $ days after exposure who are not quarantined (resp. qurantined)
$ I_{j}(t) $ (resp. $ I_{q, j}(t) $), $ i=0, 1 $ Number of symptomatic infected individuals $ i $ days after the onset of symptoms who are not isolated (resp. isolated)
$ I_{j}(t) $ (resp. $ I_{q, j}(t) $), $ j=3, 4, 5 $ Number of symptomatic infected individuals at the nominal stage $ i $ of the illness that has not been isolated (resp. isolated). Note that a person can stay at a given stage for several days
$ R(t) $ Number of removed (recovered or deceased) individuals
Description of the variables in the EBM model.
Variable Description
$ S(t) $ Number of total susceptible individuals
$ E_{i}(t) $ (resp. $ E_{q, i}(t) $) Number of asymptomatic infected individuals $ i $ days after exposure who are not quarantined (resp. qurantined)
$ I_{j}(t) $ (resp. $ I_{q, j}(t) $), $ i=0, 1 $ Number of symptomatic infected individuals $ i $ days after the onset of symptoms who are not isolated (resp. isolated)
$ I_{j}(t) $ (resp. $ I_{q, j}(t) $), $ j=3, 4, 5 $ Number of symptomatic infected individuals at the nominal stage $ i $ of the illness that has not been isolated (resp. isolated). Note that a person can stay at a given stage for several days
$ R(t) $ Number of removed (recovered or deceased) individuals
Table 2.  Parameters intrinsic to COVID-19
Parameter, meaning Value
$ \beta $, basal transmission rates optimized to fit data
Factors modifying transmission rate
$ \varepsilon $, asymptomatic transmission (25% reduction in transmission) 0.75
$ \rho $, reduced healthcare worker interactions 0.8
$ \gamma $, quarantine (80% reduction in transmission) 0.2
$ \kappa $, hospital precautions 0.5
$ \eta $, healthcare worker precautions $ 0.2375 $
Population fractions
$ p_i $, $ i= 0, \ldots, 13$, onset of symptoms after day $ i $ 0.000792, 0.00198, 0.1056, 0.198, 0.2376, 0.0858, 0.0528, 0.0462, 0.0396, 0.0264, 0.0198, 0.0198, 0.0198, 0
$ q_{s, i} $, $ i= 0, \ldots, 4$, symptomatic quarantine after day/stage $ i $ C: 0.1, 0.4, 0.8, 0.9, 0.99;
H: 0.2, 0.5, 0.9, 0.98, 0.99
r, transition to next symptomatic day/stage 0.2
$ \nu $, symptomatic hospitalization 0.11
Parameter, meaning Value
$ \beta $, basal transmission rates optimized to fit data
Factors modifying transmission rate
$ \varepsilon $, asymptomatic transmission (25% reduction in transmission) 0.75
$ \rho $, reduced healthcare worker interactions 0.8
$ \gamma $, quarantine (80% reduction in transmission) 0.2
$ \kappa $, hospital precautions 0.5
$ \eta $, healthcare worker precautions $ 0.2375 $
Population fractions
$ p_i $, $ i= 0, \ldots, 13$, onset of symptoms after day $ i $ 0.000792, 0.00198, 0.1056, 0.198, 0.2376, 0.0858, 0.0528, 0.0462, 0.0396, 0.0264, 0.0198, 0.0198, 0.0198, 0
$ q_{s, i} $, $ i= 0, \ldots, 4$, symptomatic quarantine after day/stage $ i $ C: 0.1, 0.4, 0.8, 0.9, 0.99;
H: 0.2, 0.5, 0.9, 0.98, 0.99
r, transition to next symptomatic day/stage 0.2
$ \nu $, symptomatic hospitalization 0.11
Table 3.  Parameters intrinsic to travelers
Parameter, meaning Value
Factors modifying transmission rate
$ \rho_v $, reduced interaction of travelers with community 0.5
$ \phi_1 $, percentage of tested travelers vary by destination (0.86 for Honolulu)
$ \phi_2 $, false negative test 0.005 (assuming Nucleic Acid Amplification Test)
$ \phi_3 $, prevalence 0.05
$ \phi_4 $, untested, exposed into quarantine 0.99
Parameter, meaning Value
Factors modifying transmission rate
$ \rho_v $, reduced interaction of travelers with community 0.5
$ \phi_1 $, percentage of tested travelers vary by destination (0.86 for Honolulu)
$ \phi_2 $, false negative test 0.005 (assuming Nucleic Acid Amplification Test)
$ \phi_3 $, prevalence 0.05
$ \phi_4 $, untested, exposed into quarantine 0.99
Table 4.  Traveler Data for Honolulu County
Dates Average Tourists per day Average Returning Residents per day
Oct 15 - Oct 28 1353 692
Oct 29 - Nov 11 2124 716
Nov 12 - Nov 25 3051 967
Nov 26 - Dec 9 2028 951
Dec 10 - Dec 23 4724 1014
Dec 24 - Jan 6 2195 1018
Jan 7 - Jan 20 1522 1053
Jan 21 - Feb 3 1531 710
Feb 4 - Feb 18 2828 843
Feb 19 - Mar 4 2832 942
Mar 5 - Mar 19 4483 1017
Mar 20 - Apr 5 6263 1543
Apr 6 - Apr 20 6231 1087
Apr 21 - Apr 25 5683 1331
Dates Average Tourists per day Average Returning Residents per day
Oct 15 - Oct 28 1353 692
Oct 29 - Nov 11 2124 716
Nov 12 - Nov 25 3051 967
Nov 26 - Dec 9 2028 951
Dec 10 - Dec 23 4724 1014
Dec 24 - Jan 6 2195 1018
Jan 7 - Jan 20 1522 1053
Jan 21 - Feb 3 1531 710
Feb 4 - Feb 18 2828 843
Feb 19 - Mar 4 2832 942
Mar 5 - Mar 19 4483 1017
Mar 20 - Apr 5 6263 1543
Apr 6 - Apr 20 6231 1087
Apr 21 - Apr 25 5683 1331
Table 5.  Parameters intrinsic to vaccination used in our simulations
Parameter, meaning Value
Factors modifying transmission rate
$ \mu_1 $, reduced susceptibility after Dose 1 1 (we assume no reduction in susceptibility after dose 1)
$ \mu_2 $, reduced susceptibility after Dose 2 1 (we assume no reduction in susceptibility after dose 2)
$ \psi $, fraction of newly fully vaccinated to dose 1 vaccinated 1/21
$ \omega $, reduced transmissibility due to vaccination 0.20
$ \bar{p}_i $, $ i=0, 1, ...13 $, probability of onset of symptoms after day i (after vaccination) 0.000492, 0.001080, 0.002056, 0.0415, 0.002376, 0.000858, 0.000528, 0.000302, 0.00019, 0.00019, 0.00019, 0.00019, 0.00019, 0
$ \theta $, proportion of travelers assumed to be unvaccinated 1
$ NV $, vaccinations received per day 2500
Parameter, meaning Value
Factors modifying transmission rate
$ \mu_1 $, reduced susceptibility after Dose 1 1 (we assume no reduction in susceptibility after dose 1)
$ \mu_2 $, reduced susceptibility after Dose 2 1 (we assume no reduction in susceptibility after dose 2)
$ \psi $, fraction of newly fully vaccinated to dose 1 vaccinated 1/21
$ \omega $, reduced transmissibility due to vaccination 0.20
$ \bar{p}_i $, $ i=0, 1, ...13 $, probability of onset of symptoms after day i (after vaccination) 0.000492, 0.001080, 0.002056, 0.0415, 0.002376, 0.000858, 0.000528, 0.000302, 0.00019, 0.00019, 0.00019, 0.00019, 0.00019, 0
$ \theta $, proportion of travelers assumed to be unvaccinated 1
$ NV $, vaccinations received per day 2500
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