# American Institute of Mathematical Sciences

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
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##### References:
Basic SEIR Model diagram and parameters
Diagram of our basic compartmental model
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)
Sub-Diagram for travelers assumptions
Diagram of our compartmental model with travelers
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
Diagram of our compartmental model including vaccines
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
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)
Diagram of basic Covasim simulation algorithm
In green is the Covasim model fit for Honolulu county from March 6 to October 15, 2020. Included are the optimized transmission rates
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
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
Increasing computational cost of the compartmental model. For each number of simulation days, an average of 15 simulations is taken
Compartmental(blue) and Covasim (green) fit from March 6, 2020 to April 25, 2021 including travelers and vaccines
SEIR (blue) and Covasim(green) benchmark scenarios forecasting spread and vaccination
The mean of 20 covasim simulations (black) with simulations with highest (red), lowest (magenta), earliest (blue) and latest (green) peaks
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
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
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
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
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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