JADES - ID 1 | 27143 | 53.13600 | -27.79848 | 5.94 ${}^{+0.04}_{-0.05}$ | 7.76 ${}^{+0.08}_{-0.08}$ | 3.01 ${}^{+3.53}_{-0.53}$ | -19.16 ${}^{+0.16}_{-0.15}$ | -2.20 ${}^{+0.51}_{-0.53}$ | 0.65
JADES - ID 1 | 36154 | 53.12488 | -27.78412 | 5.70 ${}^{+0.20}_{-0.05}$ | 8.23 ${}^{+0.15}_{-0.16}$ | 2.14 ${}^{+0.37}_{-0.28}$ | -19.05 ${}^{+0.18}_{-0.16}$ | -2.83 ${}^{+0.50}_{-0.52}$ | 0.60
JADES - ID 1 | 30094 | 53.12870 | -27.79826 | 5.48 ${}^{+0.03}_{-0.03}$ | 7.26 ${}^{+0.23}_{-0.23}$ | 0.78 ${}^{+0.12}_{-0.12}$ | -17.98 ${}^{+0.18}_{-0.16}$ | -3.22 ${}^{+0.52}_{-0.50}$ | 0.59
JADES - ID 1 | 17532 | 53.15530 | -27.80555 | 5.83 ${}^{+0.12}_{-0.07}$ | 7.11 ${}^{+0.11}_{-0.13}$ | 0.72 ${}^{+0.98}_{-0.17}$ | -17.49 ${}^{+0.16}_{-0.16}$ | -2.13 ${}^{+0.53}_{-0.51}$ | 0.59
JADES - ID 1 | 37094 | 53.13005 | -27.77839 | 5.60 ${}^{+0.26}_{-0.02}$ | 8.22 ${}^{+0.15}_{-0.20}$ | 3.94 ${}^{+0.67}_{-0.52}$ | -19.69 ${}^{+0.18}_{-0.17}$ | -2.60 ${}^{+0.53}_{-0.48}$ | 0.57
| … | … | … | … | … |  |  |  |
| … | … | … | … | … |  |  |  |


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

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

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.

###### Key words and phrases:

SARS-CoV-2, Epidemiological Modeling, Agent-Based Model, Compartmental Model,
Conceptual and Computational Complexity

###### 1991 Mathematics Subject Classification:

Primary: 92D30,9208; Secondary: 68W01.

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

∗ Corresponding author<EMAIL_ADDRESS>
Prateek Kunwar and Oleksandr Markovichenko

Department of Mathematics, University of Hawai‘i at Mānoa

Honolulu, HI 96822, USA

Monique Chyba∗ and Yuriy Mileyko

Department of Mathematics, University of Hawai‘i at Mānoa

Honolulu, HI 96822, USA

Alice Koniges

Data Science Institute, University of Hawai‘i at Mānoa

Honolulu, HI 96822, USA

Thomas Lee

Office of Public Health Studies, University of Hawai‘i at Mānoa

Honolulu, HI 96822, USA

##  1\. Introduction

SARS-CoV-2 (COVID-19) has impacted not only health, but the economy and how we
live daily life. However, it crept onto the world stage at the end of 2019 in
Wuhan, China, where health officials reported a cluster of pneumonia cases of
unknown cause. The first reported case of COVID-19 in the United States came
from an asymptomatic male who returned from China on January 15, 2020. By
January 19, Chinese officials closed off travel in and out of Wuhan and on
January 30, 2020 the World Health Organization (WHO) declared a global health
emergency.

COVID-19 was officially named on February 11, as it continued to spread across
Asia and Europe. While countries in those regions started to see massive
increases in cases, hospitalizations and fatalities during the initial months
of the outbreak, the United States did not report its first death until
February 29. By March 26, the United States led the world in confirmed cases.
While many European and Latin American countries were already in full shut
down due to COVID-19, the United States Government left the shut down to
individual states, resulting in a surge of cases and deaths. In July 2020, the
United States reached 68,000 daily cases for the first time.

During the Fall months, cases started to level off. However during this period
of relative calm, a more serious variant of COVID-19 was mutating and
spreading in the United Kingdom. In November 2020, the B.1.117 COVID-19
variant (UK variant) was detected in the United Kingdom and accounted for 43
percent of all COVID-19 cases by December 2020. In South Africa, another
COVID-19 variant, B.1.351, emerged independently of the UK variant around the
same time. The UK variant is associated with increased transmission and risk
of death, while the South African variant shows evidence that it may decrease
the neutralization by some monoclonal and polyclonal antibodies. With the
recent surge of cases in India, it was inevitable that India would have its
own variants of concern, primarily the B.1.617. The WHO has added it to its
list of variants of concern and England already identified a sub strain of the
Indian variant in early May.

The accelerated research, production, and distribution of the COVID-19
vaccines have had an impact on slowing the spread of COVID-19 in the areas
that had access to enough supply of the vaccine. As of June 1, 2021, only 5.5
percent of the world’s population have been vaccinated. While countries such
as the United States, Israel, and England have at least 38 percent of their
country fully vaccinated, helping to prevent future spread of COVID-19. Yet
many first world countries have yet to even break 10 percent of their
population fully vaccinated, including Japan, South Korea, and Canada. While
the mRNA vaccines allow for rapid adjustments due to variants, the possibility
of a mutation that escapes the current mRNA vaccine remains possible, though
unlikely. This possibility will remain until both industrialized and under-
developed countries increase their percentage of population fully vaccinated.

Mathematicians have found themselves at the front seat of this race against
COVID-19. Indeed, modeling is a powerful tool to address key questions such
as: when and which non-pharmaceutical mitigation measures should be
implemented? how to allocate efficiently vaccines? impact from the new
variants? when can we relax travel restrictions? However, there is still a lot
of unanswered questions and challenges regarding the outcome of several models
as well as their limitations. It is unclear at this time if there is a
“better” model, and while most of the challenges in epidemiological
forecasting come from incomplete data and impossibility to model people’s
behavior, there is still the question of what model to use when and for what
purpose.

There are primarily two different types of epidemiological models:
differential equation-based (EBM) and agent based (ABM). We here focus on two
such models: a discrete compartmentalized SEIR model that we developed and the
COVID-19 Agent-based Simulator (Covasim) [13] that we altered to include some
specific attributes. The first one is deterministic while the second one is
stochastic which makes them different in design, however it is expected that
their forecasting converge provides similar assumptions since they do model
the same pandemic. It is often stated that EBMs are simpler and faster to
compute, while ABMs are more detailed and computationally expensive.[13] Our
goal is to test this assumption concretely on a specific data set that we have
extensive knowledge of so we can determine for future studies the benefits and
pitfalls of each of these different simulation methods. In particular, we use
Honolulu county in the Hawaiian archipelago as a test-bed population for our
simulations. We show that rather than treating the two models as two distinct
ways to obtain the same results we should exploit advantages of both
throughout a pandemic simulation, particularly when the simulation is used in
a predictive real-time fashion. Throughout the current COVID-19 pandemic, most
results and forecasting come from one model but not a combination of both. We
consider what can be learned from running both models side-by-side; taking and
applying the best of each model using the measured data. We also note the
conceptual and computational requirements of each of the models for this
particular test-bed population.

The outline of the paper is as follows. In section 2 we introduce the concepts
of compartmental and agent based models and explain how adding features such
as travelers and vaccines in the models increases conceptual complexity. We
also illustrate our model on data from Honolulu County. Section 3 discusses
conceptual and computational complexity for both models, as well as
optimization for data fitting. A benchmark example is provided to analyse
performance of both models. We end the paper with a conclusion, section 4.

##  2\. Epidemiological Models

Infectious disease modeling has been used by epidemiologists and
mathematicians for years, however the COVID-19 pandemic has really highlighted
the importance of understanding the purpose and functionality of these models.
At its most core function, the intent of these epidemiological models is to
estimate the spread of the infectious disease across a population based on
some core epidemiology determinants: incubation period, duration of infectious
period, population size, and R0 (the reproductive value). Our specific
implementations of these two models use more variables and complex
interactions than other more standard implementations [5, 19, 20, 27]. The
purpose of this added functionality is to better account for the nuances and
assumptions in a real-world situation.

The fundamental difference between the two types of models is that EBM
captures aggregate behavior over the population while ABM captures agent
interactions and progression of the disease over each individual. EBM are
typically less computational expensive, and easier to use since they require
less information. However, they provide only large-scale information on the
spread of the disease compared to ABM models that can take into account
spatial information in much more detail.

###  2.1. Compartmental Models

The origin of compartmental models, also called Equation Based Models (EBMs),
in the study of infectious diseases is from the works of Ross (1916) [23],
Ross and Hudson (1917)[22], Kermack and McKendrick in 1927[17] and Kendall
(1956) [12]. In those models, the major assumption is that the population is
divided into compartments based on the nature of the evolution of the disease.
The population size is typically assumed to be fixed and by design is assumed
homogeneous within each compartment. The basic model consists of three
compartments - Susceptible(S), Infected(I) and Removed(R). Variations of this
model may include compartments such as Exposed(E) or a loop back into
susceptible in case of diseases with no immunity against re-infection. See
Fig. 1, where the hazard rate $\lambda(t)=\beta\frac{I(t)}{N}$, with $\beta$
denoting the baseline transmission rate and $N$ denoting the (fixed) total
population.

Figure 1. Basic SEIR Model diagram and parameters.

Variants of the SIR model can be differentiated based their treatment of vital
dynamics. Vital dynamics refers to life outcomes such as births and deaths
[11]. In very simple models (or with diseases that are not prolonged) the SIR
model without vital dynamics is ideal. In models that are more detailed, or if
the disease is present in the population for a prolonged period of time, the
SIR model with vital dynamics is more suitable. However, the duration of the
“prolonged period of time” is at least 10 years [1]. It is too early to tell
where COVID-19 falls regarding the possibility of it being endemic in
particular pockets in the world. Historically, the SIR model has been used to
estimate the impact of highly infectious diseases, such as smallpox [6].

Of the various compartmental models, we use one inspired by [15], which is
based on a standard discrete and deterministic SEIR model. Some classical SEIR
models include Cooke and Driessche [4], who introduced the SEIR model with two
delays and Li et al. [14] studied global dynamics with both non-linear and
standard incidence rates. We assume a given population is divided into four
compartments: Susceptible (not currently infected), Exposed (infected with no
symptoms), Infected (infected with symptoms), and Removed (recovered or
deceased). We subdivide the entire population into two additional groups: the
general community (C), healthcare workers (H). This is motivated by the fact
that healthcare workers are potentially more exposed to a virus but also use
better protection, and therefore should interact differently with the
community during a pandemic. In addition, during the severe acute respiratory
syndrome (SARS) epidemic in 2003, healthcare workers formed a large fraction
of the infected population [15]. This has also been which suggested to be the
case for COVID-19 [16, 24].

In our model, Exposed and Infected (in each population group) are split into
multiple stages per day to better reflect the progression of the disease.
Individuals in isolation (including hospitalization) are similarly
distinguished. The dynamics of the two population groups are essentially the
same and are represented using the diagram in Fig. 2 with variables described
in Tab. 1.

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 1. SEIR basic model variables. Isolated accounts for quarantine and
hospitalization.

By choosing the probabilities of moving from one stage to the other, we are
able to model the observation (according to the Centers for Disease Control
and Prevention (CDC) as well as other sources) that about 40% of people who
contract SARS-CoV-2 remain asymptomatic and that the incubation period for
those who do develop symptoms is between 2 to 14 days after exposure, with the
mean period being between 4 and 6 days[18]. Those who do not develop symptoms
after 14 days are moved directly from the Exposed compartment to the Removed
compartment. The quarantine sub-compartment $E_{q,i}$ is also broken down into
14 stages and is used to model the effect of contact tracing and the reduced
transmission rate for quarantined individuals. The infected individuals go
through 5 stages, of which the first two represent the first two days of being
symptomatic whereas the last three represent the phase where the immune system
is fighting the disease. Since the last three stages can go on for more than
one day each, there is a variability in the number of days any given person
can spend at each stage. Our model assumes that the symptomatic phase of the
illness lasts at least 5 days. This can be seen in Figure 2 as the feedback
loops in $I_{2},I_{3}$ and $I_{4}$ where only $20\%$ of people in each of
these compartments move on to the next one. The remaining $80\%$ stay in the
same compartment for the next iteration.

Figure 2. Diagram of our basic compartmental model.

The parameters in Fig. 2 are described in Tab. 2. The parameter $\beta$, the
basal transmission rate, is optimized to fit the data. Addition of
compartments, mask mandate, contact tracing, travelers and vaccinations modify
$\beta$ according to equations (14), (16), (25), (27), (37) and (39) to obtain
the hazard rate $\lambda_{i}$, $(i=c,h,v)$ which governs the dynamics of the
evolution of disease for each category. We introduce parameters $p_{i}$ for
the probability to develop symptoms on day $i$, and refine them such that if
symptoms do develop, it takes between 2 to 14 days, with a mean between 4 and
6 days[18], while assuming that about 40% of all infections remain
asymptomatic. The values of $q_{s,i}$ are chosen to reflect the prediction
that symptomatic individuals are likely to quarantine, with a probability of
0.1, 0.4, 0.8, 0.9 and 0.99 for the first five days of symptoms. For
healthcare population these probabilities are assumed to be slightly higher at
0.2, 0.5, 0.9, 0.98 and 0.99 for the first five days of symptoms. In addition,
the parameter $r$ is the probability of transitioning from one stage of the
illness to the next (with the final stage being recovery or death). Based on
prior work [3], we chose $r$ to yield an expected length of illness of 17
days.

The equations for the dynamics are given in (1)-(13).

$\displaystyle S(t+1)$ $\displaystyle=e^{-\lambda(t)}S(t)$ (1) $\displaystyle
E_{0}(t+1)$ $\displaystyle=(1-e^{-\lambda(t)})S(t)$ (2) $\displaystyle
E_{i}(t+1)$ $\displaystyle=(1-p_{i-1})(1-q_{a,i-1})E_{i-1}(t),\quad
i=1,\ldots,13$ (3) $\displaystyle E_{q,i}(t+1)$
$\displaystyle=(1-p_{i-1})(q_{a,i-1}E_{i-1}(t)+E_{q,i-1}(t)),\quad
i=1,\ldots,13$ (4) $\displaystyle I_{0}(t+1)$
$\displaystyle=\sum_{i=0}^{13}p_{i}(1-q_{a,i})E_{i}(t)$ (5) $\displaystyle
I_{1}(t+1)$ $\displaystyle=(1-q_{s,0})I_{0}(t)$ (6) $\displaystyle I_{2}(t+1)$
$\displaystyle=(1-q_{s,1})I_{1}(t)+(1-r)(1-q_{s,2})I_{2}(t)$ (7)
$\displaystyle I_{j}(t+1)$
$\displaystyle=r(1-q_{s,j-1})I_{j-1}(t)+(1-r)(1-q_{s,j})I_{j}(t),\quad j=3,4$
(8) $\displaystyle I_{q,0}(t+1)$
$\displaystyle=\sum_{i=0}^{13}p_{i}(q_{a,i}E_{i}(t)+E_{q,i}(t))$ (9)
$\displaystyle I_{q,1}(t+1)$ $\displaystyle=I_{q,0}(t)+q_{s,0}I_{0}(t)$ (10)
$\displaystyle I_{q,2}(t+1)$
$\displaystyle=I_{q,1}(t)+q_{s,1}I_{1}(t)+(1-r)(q_{s,2}I_{2}(t)+I_{q,2}(t))$
(11) $\displaystyle I_{q,j}(t+1)$
$\displaystyle=r(q_{s,j-1}I_{j-1}(t)+I_{q,j-1}(t))+(1-r)(q_{s,j}I_{j}(t)+I_{q,j}(t)),\quad
j=3,4$ (12) $\displaystyle R(t+1)$
$\displaystyle=R(t)+rI_{4}(t)+rI_{q,4}(t)+(1-p_{13})E_{13}(t)+(1-p_{13})E_{q,13}(t)$
(13)

As we mentioned, a crucial part of the dynamics relates to the hazard rate.
For the general community, group C, we have

$\lambda_{c}(t)=\beta(1-p_{mp}(1-p_{me}))\Big{[}(I_{c}+\varepsilon
E_{c})+\gamma((1-\nu)I_{c,q}+\varepsilon E_{c,q})+\\\ \rho[(I_{h}+\varepsilon
E_{h})+\gamma((1-\nu)I_{h,q}+\varepsilon E_{h,q})]]\Big{]}/N_{c},$ (14)

where we suppressed the dependency on $t$ on the right for convenience. We use
sub-indices $c$ (community), and $h$ (healthcare workers) to indicate the
appropriate group. The subscript $q$ indicates quarantined and isolated
individuals. Here $p_{me}$ and $p_{mp}$ represent mask efficiency and mask
compliance, respectively. Mask efficiency is chosen to reflect a reduction in
transmission of $75\%$. $N_{c}$ denotes the mixing pool for the general
community, computed as

$N_{c}(t)=S_{c}+E_{c}+I_{c}+R_{c}+\rho(S_{h}+E_{h}+I_{h}+R_{h}).$ (15)

where variables $E$ and $I$ here represent the sum over all the stages within
these compartments. For the healthcare worker group, we have

$\lambda_{h}(t)=\rho\lambda_{c}+\beta\eta\Big{[}(I_{h}+\varepsilon
E_{h})+\kappa\nu(I_{h,q}+I_{c,q})\Big{]}/N_{h},$ (16)

where $N_{h}(t)=S_{h}+E_{h}+I_{h}+R_{h}$.

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,…,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,…,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

Figure 3 represents the optimized fit obtained from the SEIR model for
Honolulu County.

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 [7] Included are the optimized transmission rates (aligned with
non pharmaceutical mitigation measures that were taken by the State of
Hawai‘i).

On October 15, 2020, the State of Hawai‘i implemented the Safe travel program
which brought back tourists to the islands as well as allowed more residents
to travel to the mainland. Travelers are an important component of spread of a
disease, especially for more isolated locations such as islands or
archipelagos (among which for instance New Zealand, Iceland, Japan and
Polynesia). To implement travelers, we introduce daily travelers (T) and
consider two broad categories of travelers - tourists and returning residents.
The returning residents are assumed to behave similar to the existing
community members whereas the tourists are assumed to have different behaviour
and form a new group (V). For tourists, we assume a 25% higher basal
transmission rate to account for their risk-taking behaviour. We also assume a
50% reduced interaction of tourists with the community. This is reflected in
the parameter $\rho_{v}$ which can be seen in the expression for hazard rate,
$\lambda_{c}$. To account for some form of safe travel protocol for every
region, we make further assumptions about the testing rate $\phi_{1}$, false
negative rate $\phi_{2}$, prevalence $\phi_{3}$ and fraction of untested
travelers quarantining $\phi_{4}$. See Diagram 4 for a visual of our
assumptions.

Figure 4. Sub-Diagram for travelers assumptions.

From these assumptions we compute the coefficients for the number of arriving
travelers distributed in each compartment as $v_{1},v_{2},v_{3}$:

$\displaystyle v_{1}=$
$\displaystyle\phi_{1}(1-\phi_{2})+(1-\phi_{1})(1-\phi_{3})$ (17)
$\displaystyle v_{2}=$
$\displaystyle(1-\phi_{1})\phi_{3}(1-\phi_{4})+\phi_{1}\phi_{2}$ (18)
$\displaystyle v_{3}=$ $\displaystyle(1-\phi_{1})\phi_{3}\phi_{4}$ (19)

For simplicity we assume the untested unexposed travelers go directly into the
susceptible group even though they might quarantine for 10 days. This is to
avoid introducing a new variable of susceptible quarantine individuals.

Figure 5. Diagram of our compartmental model with travelers.

The dynamics equations become

$\displaystyle S(t+1)$
$\displaystyle=e^{-\lambda(t)}S(t){\color[rgb]{1,0,0}+v_{1}T}$ (21)
$\displaystyle E_{0}(t+1)$
$\displaystyle=(1-e^{-\lambda(t)})S(t){\color[rgb]{1,0,0}+\frac{v_{2}}{5}T}$
(22) $\displaystyle E_{i}(t+1)$
$\displaystyle=(1-p_{i-1})(1-q_{a,i-1})E_{i-1}(t){\color[rgb]{1,0,0}+\frac{v_{2}}{5}T,\quad
i=1,\ldots,4}$ (23) $\displaystyle E_{q,i}(t+1)$
$\displaystyle=(1-p_{i-1})(q_{a,i-1}E_{i-1}(t)+E_{q,i-1}(t)){\color[rgb]{1,0,0}+\frac{v_{3}}{5}T,\quad
i=1,\ldots,5}$ (24)

where the terms in red account for travelers and $T$ is the average number of
travelers entering the region per day. With these assumptions and changes in
the model, the equation for the hazard rates get modified to:

$\lambda_{c}(t)=\beta(1-p_{mp}(1-p_{me}))\Big{[}(I_{c}+\varepsilon
E_{c})+\gamma((1-\nu)I_{c,q}+\varepsilon E_{c,q})+\\\ \rho[(I_{h}+\varepsilon
E_{h})+\gamma((1-\nu)I_{h,q}+\varepsilon E_{h,q})]+\\\
{\color[rgb]{1,0,0}\rho_{v}[(I_{v}+\varepsilon
E_{v})+\gamma((1-\nu)I_{v,q}+\varepsilon E_{v,q})]]\Big{]}/N_{c}},$ (25)

where:

$N_{c}(t)=S_{c}+E_{c}+I_{c}+R_{c}+\rho(S_{h}+E_{h}+I_{h}+R_{h}){\color[rgb]{1,0,0}+\rho_{v}(S_{v}+E_{v}+I_{v}+R_{v})}.$
(26)

The expression for $\lambda_{h}$ remains unchanged with the introduction of
travelers, however, we now have a new hazard rate, $\lambda_{v}$ that governs
the dynamic of the tourist group (V):

$\lambda_{v}(t)=\frac{\rho_{v}N_{c}\lambda_{c}+\beta_{v}(1-p_{m_{p}}(1-p_{m_{e}}))[(I_{v}+\varepsilon
E_{v})+\gamma((1-\nu)I_{v,q}+\varepsilon E_{v,q})]}{\rho_{v}N_{c}+N_{v}},$
(27)

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

Since the travelers add to the susceptible population, we have to make certain
assumptions on how and when they are removed from the model once they leave
the region. The number of susceptible travelers is given by
$v_{1}e^{-\lambda(t)}T$, and we remove a fraction of them to reflect them
leaving the region (represented by the $\alpha$ in Fig. 5). Tourists are
removed from the R compartment proportionally to incoming tourists that got
exposed.

Figure 6 represents the optimized fit obtained from the SEIR model for
Honolulu County including the period from October 15 to December 27, 2020 when
travelers were re-introduced through the safe travel program. The parameters
in Table 3 were used for Honolulu county to find $v_{1},v_{2},v_{3}$ as
described earlier. The per day average influx of travelers and returning
residents are modeled as piece-wise linear functions over two week intervals
from October 15, 2020 to April 25, 2021. The data is obtained from [9] and the
values are shown in Table 4.

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

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.

Then in December 2020, the world started vaccinating. For simplicity our model
assumes one type of vaccine requiring two doses. Expending to multiple
vaccines, possibly some requiring only one dose, follows the same conceptual
framework by adding more compartments. To implement vaccination in the EBM
model, we introduce new compartments $NV1$, $NV2$, $\bar{E}$, $\bar{E_{q}}$,
$\bar{I}$ and $\bar{I_{q}}$. While $NV1$ and $NV2$ represent the number of
people who have received the first and second dose of vaccination
respectively, the $\bar{E},\bar{I},\bar{E_{q}},\bar{I_{q}}$ keeps track of
exposures and infections between vaccine doses and post vaccination. They have
the exact same sub-structure as the Exposed and Infected compartments used for
non-vaccinated groups but parameters might vary. We introduce the new
parameter $\psi=1/\delta$ where $\delta$ is the time gap between doses, as
well as $\mu_{1}$ and $\mu_{2}$ to account for possible reduced susceptibility
of vaccinated individuals. We also assume an 80% reduction in transmissibility
for vaccinated population, which is represented by the parameter $\omega$
which multiplies the contribution of vaccinated compartment to the hazard rate
as seen in Equation 37. We assume a 95% protection as a result of vaccination.
This is modeled as $\prod_{i=0}^{13}(1-\bar{p}_{i})=0.95$. This reduces the
probability of vaccinated individuals developing symptoms and also reduces the
probability of severe infections. The flow diagram with the vaccinated
compartments is shown in Fig. 7.

In the EBM, we assume that 100% of healthcare population are completely
vaccinated by December 27, which is when the community starts receiving
vaccination. For this, we begin vaccinating Healthcare population on December
22 with an average of 2500 people being vaccinated everyday. The proportion of
community population that is vaccinated is based on daily averages from data
from [7]
The introduction of new compartments results in our equations being modified
to:

$\displaystyle S(t+1)$
$\displaystyle=e^{-\lambda(t)}S(t)-{\color[rgb]{1,0,0}NV}$ (28)
$\displaystyle{\color[rgb]{1,0,0}NV1(t+1)}$
$\displaystyle{\color[rgb]{1,0,0}=(1-\psi)(1-(1-e^{-\lambda(t)})\mu_{1}))NV1(t)+NV}$
(29) $\displaystyle{\color[rgb]{1,0,0}NV2(t+1)}$
$\displaystyle{\color[rgb]{1,0,0}=\psi(1-(1-e^{-\lambda(t)})\mu_{1}))NV1(t)+(1-(1-e^{-\lambda(t)})\mu_{2}))NV2(t)}$
(30) $\displaystyle E_{0}(t+1)$ $\displaystyle=(1-e^{-\lambda(t)})S(t)$ (31)
$\displaystyle{\color[rgb]{1,0,0}\bar{E}_{0}(t+1)}$
$\displaystyle{\color[rgb]{1,0,0}=(1-e^{-\lambda(t)})(\mu_{1}NV1(t)+\mu_{2}NV2(t))}$
(32) $\displaystyle E_{q,i}(t+1)$
$\displaystyle=(1-p_{i-1})(q_{a,i-1}E_{i-1}(t)+E_{q,i-1}(t))$ (33)
$\displaystyle{\color[rgb]{1,0,0}\bar{E}_{q,i}(t+1)}$
$\displaystyle{\color[rgb]{1,0,0}=(1-\bar{p}_{i-1})(q_{a,i-1}\bar{E}_{i-1}(t)+\bar{E}_{q,i-1}(t))}$
(34) $\displaystyle{\color[rgb]{1,0,0}\bar{I}_{0}(t+1)}$
$\displaystyle{\color[rgb]{1,0,0}=\sum_{i=0}^{13}\bar{p}_{i}(1-q_{a,i})\bar{E}_{i}(t)}$
(35) $\displaystyle{\color[rgb]{1,0,0}\bar{I}_{q,0}(t+1)}$
$\displaystyle{\color[rgb]{1,0,0}=\sum_{i=0}^{13}\bar{p}_{i}(q_{a,i}\bar{E}_{i}(t)+\bar{E}_{q,i}(t))}$
(36)

where the terms in red account for vaccination where $NV$ represents the daily
average of number of people receiving the first dose of vaccination. With
these assumptions and changes in the model, the equation for the hazard rates
get modified to:

$\lambda_{c}(t)=\beta(1-p_{mp}(1-p_{me}))\Big{[}(I_{c}+\varepsilon
E_{c})+\gamma((1-\nu)I_{c,q}+\varepsilon E_{c,q})+\\\ \rho[(I_{h}+\varepsilon
E_{h})+\gamma((1-\nu)I_{h,q}+\varepsilon E_{h,q})]+\\\
{\color[rgb]{1,0,0}\omega((\bar{I}_{c}+\varepsilon\bar{E}_{c})+\gamma((1-\nu)\bar{I}_{c,q}+\varepsilon\bar{E}_{c,q}))}+\\\
{\color[rgb]{1,0,0}\omega((\bar{I}_{h}+\varepsilon\bar{E}_{h})+\gamma((1-\nu)\bar{I}_{h,q}+\varepsilon\bar{E}_{h,q}))}]\Big{]}/N_{c},$
(37)

where:

$N_{c}(t)=S_{c}+E_{c}+I_{c}+R_{c}+\rho(S_{h}+E_{h}+I_{h}+R_{h})\\\
{\color[rgb]{1,0,0}+NV1_{c}+NV2_{c}}$ (38)

and $\omega$ is the reduction in transmissibility due to vaccination. If for
simplicity we assume that by the start of the vaccination period for community
population, the entire healthcare population is fully vaccinated, their hazard
rate is modified to:

$\lambda_{h}(t)=\rho\lambda_{c}+\beta\eta{\color[rgb]{1,0,0}\omega}\Big{[}(I_{h}+\varepsilon
E_{h})+\kappa\nu(I_{h,q}+I_{c,q})\Big{]}/N_{h},$ (39)

When combining travelers and vaccines, we assume that a proportion $\theta$ of
all arriving travelers are fully vaccinated. These are moved directly into the
NV2 compartment (for respective categories returning resident and tourists).
The remaining $1-\theta$ are distributed between Susceptible, Exposed and
Exposed quarantine compartment based on our prior assumptions from the model
including travelers.

Figure 7. Diagram of our compartmental model including vaccines.

Figure 8 including travelers and vaccines. The values of parameters used for
Honolulu county for the model including vaccines are shown in Table 5. The
parameter values and data for travelers is the same as shown in Table 3 and
Table 4 respectively.

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

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: Zoom in on period where both
vaccination and travelers are included with the corresponding basal
transmission rates.

###  2.2. Agent-Based Model

Compartmental models focus on directly capturing the collective behavior of
groups of people, and are typically derived using estimates of aggregate (or
limiting) behavior of a large number of individuals under (many) simplifying
assumptions. In contrast, agent based models (ABMs) focus on capturing the
behavior of a single individual, referred to as an agent. Such individual
behavior can often be described using fairly simple rules, however the
collective behavior may still exhibit complicated phenomena. As a related
example from physics, one could imagine modeling diffusion using the standard
PDE versus modeling the Brownian motion of each particle. It should be noted,
however, that ABMs have been used in social, economic, and biological sciences
as early as the 80s [21] , and some models could be tracked to the 70s [25,
26]. Popularity of ABMs exploded in the 90s, when the computational power
significantly increased and became widely available.

d $S$ can be a product $\mathbb{R}^{m}\times\mathbb{Z}_{2}^{k}$, where
$\mathbb{R}^{m}$ captures continuous variables related to the disease, such as
age and susceptibility to infection, and $\mathbb{Z}_{2}^{k}$ captures binary
variables, such as presence of infection and comorbidities. Of course, one can
add other categorical variables if needed.

The evolution of the system state through time can be regarded as a function
$g:T\to A^{S}$ or, equivalently, $f:T\times A\to S$, with $f(t,a)=g(t)(a)$,
where $T$ is our time set, which we shall assume to be discrete, i.e.
$T=\mathbb{N}$. The function $f$ can be defined deterministically or it can be
a realization of a stochastic process. Importantly, one does not focus on the
whole $f$ and instead defines how an individual value, $f(t+1,a)$ is obtained
when $f(t,a)$ for all $a\in A$ is known. Of course, a change in the state of
an agent is unlikely to depend on all of the other agents. Typically, each
agent has an associated subset of agents that may affect its state due to
“interaction”. We shall refer to such a subset as a _contact set_ of an agent.
In the case of pandemic modeling, a contact set consists of people who
actually interact with a given individual. This example also suggests that a
contact set of an agent $a\in A$, which we shall denote $N(a)$, may have an
additional structure to better reflect interactions within different contexts.
For example, interactions at work may be different from those at home.
Mathematically, this may be represented as a disjoint union,
$N(a)=\sqcup_{i=1}^{n}{N_{i}(a)}$. Also, a contact set may be time dependent,
thus yielding $N(t,a)=\sqcup_{i=1}^{n}{N_{i}(t,a)}$, $t\in T$.

Considering that interactions between agents are typically symmetric, it is
convenient to represent all $N(a)$, $a\in A$, using an undirected graph, so
that each separate $N(a)$ is just a collection of adjacent vertices. More
specifically, we let $A$ be the vertex set of our graph, and let
$E\subset\\{\\{a,b\\}|a,b\in A\\}$ be the set of edges. We shall refer to such
a graph as a _contact network_. We can then define a contact set of an agent
$a\in A$ as $N(a)=\\{b\in A|\exists\\{a,b\\}\in E\\}$. Additional structure
and time dependency of contact sets are obtained by defining multiple,
possibly time dependent interaction networks with the same vertex set $A$ and
different edge sets $E_{i}(t)$, $i=1,\ldots,n$.

In the deterministic case, the state of an agent $a\in A$ at time step $t+1$
is defined as a function of time $t$ and the states $f(t,b)$ where $b$ ranges
over $N_{i}(t)$, $i=1,\ldots,n$. In the stochastic case, one computes the
conditional probability distribution for $f(t+1,a)$, conditioned on the above
$f(t,a)$ (and possibly $t$), and then samples from it. When modeling a
pandemic, this often simplifies to computing the probability of infection
given an uninfected individual, or computing the probability of developing
symptoms, given an infected but asymptomatic individual, etc. Then a
(pseudo)random number is generated to determine the actual state transition
(e.g. an individual gets infected, develops symptoms, etc). While the
transition between states can be mathematically quite complicated and non-
Markovian, the actual implementation is often fairly straightforward.

An example of a contact network for an agent based model of a pandemic is
shown in Fig. 9. We should note that contact networks constitute one of the
most important aspects of agent based models and their construction can be a
challenging problem. But once the network construction is done, handling
changes to agents’ states due to newly available information can be readily
implemented as tweaks to the interaction network and/or states of agents. For
example, Fig. 9 shows that adding vaccinated individuals is conceptually quite
simple. Of course, the process governing state transitions also needs to be
updated.

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).

####  2.2.1. Covasim

For our simulations, we use the open source COVID-19 Agent-based Simulator
(Covasim) [13]. Covasim provides several ways to construct contact networks,
with the default choice resulting in a network consisting of four “social
layers” (i.e. edge sets) analogous to the ones shown in Fig. 9. The state of a
Covasim agent consists of $39$ variables which include demographic
information, individual susceptibility, and variables representing intrahost
viral dynamics (along with viral-load-based transmissibility). By design,
Covasim is a stochastic agent-based model. Thus, as mentioned above, each step
is focused on computing multiple transition probabilities.

While Covasim comes with its own demographic data set, it can also incorporate
user-supplied demographic information, and we chose to use the data from the
Hawai‘i Population Model developed by the Hawai‘i Data Collaborative [8]. In
fact, Covasim allows the user to easily customize multiple aspects of the
initialization step. We customized the population size along with the number
of initially infected people as well as multiple parameters affecting the
simulation, such as probability of asymptomatic infection, probability of
isolation upon the onset of symptoms, etc. Since Covasim is a stochastic
model, output quantities of interest (e.g. new daily infections, the number of
hospitalizations, etc.) are averaged over multiple simulations, typically 15
to 20. A user can supply a seed for the (pseudo)random number generator,
typically keeping it fixed during the testing phase and then properly
resetting the seed before each simulation. In most cases we employed the
default construction of the contact network with its four social layers:
household, work, school, and community. However, it is fairly straightforward
to alter the default construction, for example by increasing the average
number of contacts among young adults in the community layer. We actually did
implement the latter to better capture the fact that young people are
typically more socially active.

Customization of Covasim simulation steps is done through so called
“interventions,” which are procedures that can be supplied to and then called
by the main simulation routine. Covasim comes pre-packages with several
interventions, allowing the user to take into account changes in the baseline
transmission rate due to imposed mitigation measures, such as a lockdown, as
well as incorporate testing and contact tracing with customized isolation
probabilities. One can also write custom interventions, which we did to
implement the Hawai‘i vaccination protocol. The latter was necessary because
the vaccine intervention bundled with Covasim could not properly capture the
necessary dynamics of administering available COVID-19 vaccines.

Once Covasim runs its initialization step, it iterates over the given number
of days and for each iteration uses the constructed contact network along with
the provided parameters and interventions to calculate the probability that an
agent gets infected. Additionally, the state of each already infected agent
gets updated (e.g. switching from an asymptomatic infection to a symptomatic
one) according to the prognoses pre-calculated during the initialization step.

Figure 10. Diagram of basic Covasim simulation algorithm.

Incorporating the effect of travelers on the spread of the disease is
important for Hawai‘i, but it had to be done in an indirect way, since Covasim
keeps the total number of agents fixed throughout the simulation. Hence, we
added a fifth social layer of contacts representing workers in the hospitality
industry and a new custom intervention which increased the baseline
transmission rate within the new layer based on the number of tourists
traveling to Hawai‘i as well as the nationwide infection rates.

Figure 11 represents the optimized fit obtained from the Covasim model for
Honolulu County.

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: zoom in on the period October 15,
2020 - April 25, 2021 including the corresponding transmission rates.

It is important to note that while both SEIR and Covasim models denote by
$\beta$ the parameter representing the baseline transmission rate of the
disease, these two parameters are, in fact, different quantities. In Covasim,
it can be regarded as the probability of a susceptible person getting infected
when a contact with an infected individual occurs and no information about
other modifiers is given (i.e. layer-specific transmission rate factor,
individual transmissibility, and individual susceptibility are all equal to
$1$). In the standard discrete SEIR model, this parameter also affects the
fraction of newly infected individuals, with the latter given by
$1-\exp{(-\beta\frac{I}{N})}$, but its relation to the probability of
infection per contact is indirect and depends on multiple simplifying
assumptions. This relation becomes even more complicated in our expanded SEIR
models.

##  3\. Computational Versus Conceptual Complexity, and Data Fitting

In this section we discuss the advantages and limitations of both models.

###  3.1. Conceptual Complexity

While computationally efficient, compartmental models present another
difficulty: if the model itself needs to be modified to take into account
newly discovered features of the pandemic, such modifications can be highly
non-trivial. The reason for the difficulty is the aggregate nature of the
interactions. Some of these interactions are quite intricate and are based on
a series of complicated assumptions. Consequently, the conceptual complexity
of the model may become a hurdle. This can be observed from the diagrams in
figures 2, 5 and 8 where we illustrate the incorporation of additional
compartments into the model. Adding new variants to the compartmental model
would drastically increase the already existing complexity of the interactions
between different compartments. A compartmental model is also quite limiting
when it comes to the inclusion of demographic, ethnic and other essential
information, as it would again require one to introduce numerous compartments.

For agent based models, incorporating new attributes for individuals, such as
age, ethnicity or vaccination status, is conceptually fairly simple, since
each individual is represented by an agent. Thus, a simple augmentation of the
variables representing the state of each agent should be done, and the network
remains unchanged. For instance, taking into account a new variant of SARS-
CoV-2 would amount to simply adding variables that characterize the variant to
the agent state space.

###  3.2. Computational Complexity

Simulation of agent based models may be a computationally expensive process if
the number of agents (i.e., the population) is large and the computation is
not appropriately parallelized. Each time step requires an iteration over each
interaction edge, and the state of each agent needs to be evaluated and
possibly updated. Thus, the total time complexity of each time step is
$O(|E|+|A|)$. Here $|E|$ denotes the number of interaction edges and $|A|$
denotes the number of agents. Multiplying this already substantial number by
the number of time steps, $N$, results in a fairly large computational cost
$O(N(|E|+|A|))$. We should also mention that interventions may also increase
the time complexity. Fortunately, in many cases such computations can be
parallelized, and employing compute clusters and/or multicore nodes can
provide a significant speed-up. However, this potential parallelism is not
currently exploited in our simulations, although the 15-20 simulations needed
to average the results (as mentioned earlier) are run parallel.

If $|E|$ is proportional to $|A|$, then one can expect a linear increase in
computational time with respect to the number of days and the number of
agents. Figure 13 shows this for our simulations. Here, no extra parallelism
is employed. These are run on a single Haswell node of the NERSC Cori system.
Each node has two sockets, and each socket is populated with a 2.3 GHz 16-core
Haswell processor (Intel Xeon Processor E5-2698 v3). Each core supports 2
hyper-threads, and has two 256-bit-wide vector units. To produce the results
in the picture, we used Covasim without additional parallelism and performed
the computations on a single core. One can see that the increase in
computational time with the number of days is roughly linear, and a reference
straight line is included (top figure). The scaling of the computational time
with respect to the number of agents is also close to being linear (bottom
figure).

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.

In the future, we plan to investigate the role of parallelism in reducing the
computational time of agent based epidemiological models. Possible speedup is
crucial for large population sizes since, as shown in Fig. 13, already for 50
million agents one has to wait more than 2 hours to perform required
computations on a single node. Moreover, larger populations may require
employing distributed computations, as the total state of the model may not
fit into the RAM of a single node.

Computational complexity of compartmental models is typically $O(NC)$, where
$NC$ is the number of compartments, which is significantly less
computationally intensive for the same problem. As expected, Fig. 14 shows a
linear increase in the computational time with respect to the number of days.
For this benchmark, a basic model with community and healthcare compartments
is used without any interventions, travelers, or vaccinations. It was run on
single Haswell node of the NERSC Cori system. Note that scaling with respect
to the population is not relevant for a compartmental model since the
population is aggregated. Generally agent based models are likely easier to
parallelize, however we have not explored this because of the Python-based
code in Covasim and its reliance on Python specific libraries which make
hands-on parallelization techniques on a node, such as OpenMP, more difficult.

Figure 14. Increasing computational cost of the compartmental model. For each
number of simulation days, an average of 15 simulations is taken.

###  3.3. Data Fitting

There are differences in data fitting methodology, implementation and results
for the two models we study. Our SEIR model was fitted to available data from
the State of Hawai‘i based on daily infections using a classical gradient-
based optimization method. The latter is possible because we can explicitly
compute the gradient with respect to the parameters. Thus obtained values of
the baseline transmission rate for the SEIR model allowed us to calculate
appropriate values of the corresponding parameter in Covasim. Fitting an agent
based model to data directly is a far more complicated task. In most cases,
one needs to resort to a very general global optimization technique such as
the Metropolis-Hastings algorithm or the genetic algorithm [2, 10]. Such a
procedure is very computationally costly and does not always produce a good
fit [10].

Figure 15 shows the optimized fit obtained from the SEIR and the Covasim model
for Honolulu County. We can see that qualitatively both fits agree very well
in certain places, including the spike in August.

Figure 15. Compartmental(blue) and Covasim (green) fit from March 6, 2020 to
April 25, 2021 including travelers and vaccines.

While the models agree well when fitting specific data as for Honolulu County,
it is unclear if the agreement will holds as we run the model without fitting
data. In Fig. 16 we show simulation of a benchmark scenario to visualize the
impact of the vaccines for both models. We start with a basic model with no
interventions, vaccinations or travelers. For both models, we assume a
population of 1 million, with 100 infections on day 1. In EBM this is
accomplished by setting $E_{0}(0)$=100 for the community population and
choosing a single beta value for the simulations. We forecast two scenarios,
the first one has no vaccines and shows comparatively how the two models
forecast over a year. For the second scenario, we assume 2500 individuals are
vaccinated per day, a 95% vaccine protection for developing symptoms (this
means that $\bar{p}_{i}$ are chosen such that
$\Pi_{i=0}^{13}(1-\bar{p}_{i})=0.95$), and a 80% reduction in transmissibility
(which corresponds to $\omega=0.2$). We also assume the vaccine requires only
one shot. We can observe that the model forecasting agrees even with no
fitting and overall the models behave similarly.

Figure 16. SEIR (blue) and Covasim(green) benchmark scenarios forecasting
spread and vaccination.

It is important to note that the Covasim curve above is actually the mean of
20 simulation curves. In Fig. 17 you can see the mean with 4 individual
simulations curves: one with the highest peak (red), one with the lowest peak
(magenta), with the earliest peak (blue), and with the latest peak (green).
Taking the values (and times of occurrence) of all the peaks of our $20$
simulations and computing their means for the scenario without vaccines, we
get the mean peak value of $520$ and the mean time of occurrence of $341.05$,
while the value of the peak of the mean of all simulations is $463.6$
occurring at time $321$. For the second scenario, the mean value of the peaks
$114.95$ with the mean time of occurrence of $178.15$, while the value of the
peak of the mean of the simulations is $102.05$ occurring at time $172$. These
observations suggest that the differences in the curves in Fig. 16 are at
least partially caused by somewhat a simplistic computation of the averages in
Covasim.

Figure 17. The mean of 20 covasim simulations (black) with simulations with
highest (red), lowest (magenta), earliest (blue) and latest (green) peaks.

##  4\. Conclusion

In this paper, we look at two types of epidemiological models and analyze
their complexity both in terms of conceptual design and computational time.
Our conclusion is that the decision to use one model versus the other ones
depends on the objectives, available data as well as access to resources. If
used properly, these two types of models offer similar outcomes for the spread
of the disease at the population level. This is not surprising, it was indeed
observed in [28] over data from the 1918 Influenza pandemic. Our example, see
Fig. 16 is designed to understand whether or not the two models agree when no
data fitting is performed. It is clearly the case, and demonstrates that
overall the models behave similarly.

Reflecting upon a year of modeling in the current COVID-19 pandemic, we
conclude that for the State of Hawai‘i both models played an important role.
Early in the pandemic the compartment model allowed us to fit the data, and
run some forecasting with limited data and mitigation measures that applied to
the entire population (such as stay-at-home orders). Once the pandemic
advanced and actions got more sophisticated with testing, contact tracing,
safe travel program, tier system for reopening strategy and eventually
vaccines it was clear that shifting to an agent based model was a better
strategy. It allowed us to mimic the age demographic for vaccines plan, and
include other attribute in an easier way. Computationally, because of the
small size population in the State of Hawai‘i it was not a major issue to use
the agent based model. However for a state like California with almost 40
millions people advanced computational resources are required to run an agent
based model.

Developing an hybrid model with some aspects that are agent based but some
also with aggregated population might provide in the end the ideal tool.

##  5\. Acknowledgments

This material is based upon work supported by the National Science Foundation
under Grant No. 2030789. This research used resources of the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility located at Lawrence Berkeley National
Laboratory, operated under Contract No. DE-AC02-05CH11231. The authors are
grateful to the the developers of Covasim (COVID-19 Agent-based Simulator), an
open-source model, for useful discussions on the use of their software and the
incorporation of new facets such as tourists and vaccines. Covasim is
supported by Bill and Melinda Gates through the Global Good Fund.

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