# CoV-ABM: A stochastic discrete-event agent-based framework to simulate
spatiotemporal dynamics of COVID-19

Masoud Jalayer 1
<EMAIL_ADDRESS>
&Carlotta Orsenigo 1
<EMAIL_ADDRESS>
&Carlo Vercellis 1
<EMAIL_ADDRESS>

1 Department of Management, Economics and Industrial Engineering
Politecnico di Milano, via Lambruschini 4B, 20156, Milan, MI, Italy
Corresponding author

###### Abstract

The paper develops a stochastic Agent-Based Model (ABM) mimicking the spread
of infectious diseases in geographical domains. The model is designed to
simulate the spatiotemporal spread of SARS-CoV2 disease, known as COVID-19.
Our SARS-CoV2-based ABM framework (CoV-ABM) simulates the spread at any
geographical scale, ranging from a village to a country and considers unique
characteristics of SARS-CoV2 viruses such as its persistence in the
environment. Therefore, unlike other simulators, CoV-ABM computes the density
of active viruses inside each location space to get the virus transmission
probability for each agent. It also uses the local census and health data to
create health and risk factor profiles for each individual. The proposed model
relies on a flexible timestamp scale to optimize the computational speed and
the level of detail. In our framework each agent represents a person
interacting with the surrounding space and other adjacent agents inside the
same space. Moreover, families’ stochastic daily tasks are formulated to get
tracked by the corresponding family members. The model also formulates the
possibility of meetings for each subset of friendships and relatives. The main
aim of the proposed framework is threefold: to illustrate the dynamics of
SARS-CoV diseases, to identify places which have a higher probability to
become infection hubs and to provide a decision-support system to design
efficient interventions in order to fight against pandemics. The framework
employs SEIHRD dynamics of viral diseases with different intervention
scenarios. The paper simulates the spread of COVID-19 in the State of
Delaware, United States, with near one million stochastic agents. The results
achieved over a period of 15 weeks with a timestamp of 1 hour show which
places become the hubs of infection. The paper also illustrates how hospitals
get overwhelmed as the outbreak reaches its pick.

_Keywords_ Agent-based Model $\cdot$ SARS Coronavirus $\cdot$ SEIR Epidemic
Model $\cdot$ Stochastic Epidemic Model $\cdot$ GIS Application $\cdot$
Discrete-Event Simulation Model

##  1 Introduction

The COVID-19 pandemic is growing every day across the planet and, apart from
its impact on the global economy, it is causing a dramatic loss of lives.
Since its first report on December 31st, 2019 in Wuhan, China, it has speedily
reached 213 countries. As of today, over 7 million people have been diagnosed
positive with over 400.000 casualties [42]. Policymakers and scholars all over
the globe are under pressure to make effective decisions with the purpose of
controlling and mitigating the outbreak in the short run, and of designing
valid actions to contrast future possible upsurges.
Some non-pharmaceutical interventions have being carried out by governments
and people aiming at “flattening the curve”, e.g. home quarantine, intensive
hand hygiene, closure of schools and shops, prohibition of mass events [26].
However, these counter-measures caused controversies in many countries; as an
example, several protests arose in the US against the closure of businesses
and stay-at-home impositions [15]. One of the aims of the present work is to
illustrate how these provisions can effectively flatten the curve of infected
people and save lives.
Since 1920s many stochastic epidemic models have been proposed using
differential equations of SIR (Susceptible-Infectious-Recovered) and SEIR
(Susceptible-Exposed-Infectious-Recovered) to study the dynamics of diseases
on populations [28]. Due to their generalized nature, however, differential
equations cannot explain the effect on the outbreaks of different scenarios,
geographical features or population age distribution [19]. Hence, during the
past two decades computer simulations and agent-based models were proposed to
address the structure and dynamics of transmission networks [32]. Some studies
considered static graphs as the main platforms of the disease diffusion [37],
and employed graph features and algorithms to identify the influential nodes
[22]. Other studies resorted, instead, to spatial models for investigating
such phenomena. As an example, Balcan et al. [4] developed a discrete
stochastic epidemic model based on gridded population at the worldwide scale,
whereas Cliff et al. [9] built a spatiotemporal simulator to analyse the
spread of influenza in Australia using census data.
Some works addressing the new outbreak, SARS-CoV2, have been recently
proposed. Kai et al. [24] developed two SEIR models to predict the effect on
the spread of the virus of wearing face masks in the society. The first model
uses stochastic dynamic networks to simulate the transmission among the
individuals, represented by the nodes of the network. The second model,
instead, employs an agent-based framework which indicates that “the early
universal masking” policy can effectively mitigate the pandemic. Cuevas [11]
proposed an agent-based simulator to mimic the spatiotemporal transmission
process of COVID-19. In this work, each agent has a health profile and social
characteristics, and makes decisions about its movements and interactions with
other agents inside a square, which represents a generic geographical area.
The results of this study show that simulations provide useful information to
base the decisions aimed at mitigating the transmission risks of COVID-19
within the facilities. In [21] Ivorra et al. developed a mathematical model to
study the importance of undetected infections. They proposed a modified SEIR
model, called $\theta$-SEIHRD, where $\theta$ implies the fraction of detected
cases over the actual total number of infectious cases, and H denotes the
state of the people who are hospitalized or self-quarantined. The authors
compared their differential equation results with the adjusted number of
COVID-19 cases in China, and analysed some different scenarios consequently.
In another study [14], the authors developed an agent-based model to simulate
the spread of COVID-19 based on the movements of the agents in closed built
environments, such as buildings, and studied some different social-distancing
scenarios. The model there proposed is calibrated on experimental data
collected from the Diamond Princess cruise, and relies on two factors to
determining if the virus transmits:␣the distance between the virus carrier
and the susceptible agents, and the duration of the exposure.
In this paper, we propose an agent-based model which simulates agents capable
of working, resting, taking part in social events, meeting friends and
visiting relatives stochastically. The model uses census data referred to the
working context and to the health risk factors in a given geographical area,
to create a realistic society. Moreover, it simulates the placement of
facilities and buildings in that area by using Geographic Information System
(GIS) tools to retrieve real-world geographical data, and associates them to
agents who live, work or simply visit that locations. In our work distinctive
traits of SARS-CoV2 are also considered; in particular, our model assumes that
the virus persists inside the buildings or in outdoor areas for rather long
time periods, and stochastically infects the agents in these places.
The main aim of the proposed framework is threefold. First, to enable a
detailed description of the dynamics of the SARS-CoV2 spread. Second, to
identify the high-risk places that accelerate the outbreak. Third, to provide
a decision-support system to design efficient interventions, in order to fight
against the current pandemic or similar worldwide epidemics that may occur in
the future.
The rest of the paper is organized as follows. Section 2 describes the
proposed framework, the characteristics of the agents, their set of actions,
and the rules to generate the communities in which they are organized. The
infection dynamics and the virus transmission mechanisms are explained in
detail in the same section. The simulation assumptions, the investigated
scenarios, the data collection process, the model’s calibration and the
results achieved for a given geography, represented by the State of Delaware,
are illustrated in Section 3. The research findings are discussed in Section
4, while Section 5 provides some conclusions and opportunities for future
studies.

##  2 CoV-ABM: the simulation model

The paper proposes a novel Agent-Based Model, called CoV-ABM, as a simulator
of the SARS-CoV2 infectious disease spread. An overview of the proposed
framework is depicted in Figure 1. Once the corresponding census and health
data are collected, the simulator first resorts to the Open Street Map
Networkx (OSMnx) library [6] for retrieving the GIS data of the geographical
areas under investigation. Then, it proceeds with the generation of the
buildings, the street maps, and the agents. Based on the health and census
data, each agent is associated to a health profile, the households are created
and placed in vacant houses. Different workplaces are generated based on the
buildings’ information and census data. Agents are then assigned to
professional tasks or scholar activities inside these workplaces, based on
their profiles and geographical locations. Afterwards, CoV-ABM proceeds with
making social communities as described in Section 2.2. Each of the
aforementioned steps is described in detail in the following subsections.

Figure 1: The proposed CoV-ABM framework

###  2.1 Generating the agents

Based on census data, CoV-ABM stochastically generates agents representing
citizens living, working, pursuing their needs and travelling inside a
geographical environment. To rely on a detailed and realistic simulator, each
agent is described in terms of a set of variables, defining its condition at
each timestamp. These variables are briefly depicted in Table 1. It should be
emphasized that, to the best of our knowledge, this is the first time that
such a detailed agent profile is proposed in the literature.

Table 1: Variables defining each agent Variable |  Notation |  Type |  Description
---|---|---|---
Name |  $u$ |  string |  A 4-digit string as the agent’s first name
Family |  $v$ |  string |  An 8-digit string as the agent’s family ID
Home |  $m$ |  string |  A GIS location + 3-digit string indictating the building unit
|  |  |  where the agent dwells
Age |  $\zeta$ |  integer |  An integer between 1 land 100
Health vulnerability |  $h$ |  float |  A real number between 0 and 0.9999
Risk factor |  $\varphi$ |  array |  A vector of binary values with respect to different risk factors 1: if the agent has the risk factor, 0: otherwise
Disability |  $d$ |  binary |  1: severe disability, 0: without disability
SEIHRD |  $s$ |  string |  The epidemiological state of the agent
Job |  $j$ |  string |  The agent’s job name
Workplace |  $w$ |  string |  A GIS location + 3-digit string to indicate the building unit
|  |  |  where the agent works
Working table |  $W^{d,h}$ |  array |  A matrix $W=(w_{d,h})\in\mathbb{N}^{7\times 24}$
Location |  $\varrho$ |  string |  The location where the agent is at time $t$
Free |  $\phi$ |  binary |  1: if the agent is free to perform non-emergency tasks at time, 0: otherwise

When the agents are created, the model associates one with each other to
generate families based on the household size, and set their arrangement so to
be proportionate to the census data. As an example, if a family includes a kid
there must be at least an adult in the same household. In what follows, each
agent will be indicated as $a^{u,v}$, whereas $f_{i}=\\{a^{v,u}|v=i\\}$ will
represent the ith family. To reduce unnecessary complexity, the agents do not
have a gender. Moreover, their disability values, defined in Table 1, are
stochastically chosen based on a Bernoulli distribution as follows:

$a^{v,u}(d)\sim Ber(p^{g}_{d})$ (1)

where $p^{g}_{d}$ is the percentage of people with severe disabilities within
the age-group g, based on the census data. These people seldom do outdoor
daily activities, like going for grocery shopping, and mostly rely on someone
else for assistance.
CoV-ABM defines the “health vulnerability” variable for each agent, which
determines how vulnerable or strong an agent can persist against the SARS-
CoV2. A strong agent has a higher probability to get recovered after inhaling
the virus. Despite some health factors seem to play a key role in the
pathogenesis of COVID-19 infection, their relative importance is however still
unclear [23]. So far, age showed a significant correlation with the fatality
rate among infectious people. As of today, more than 70% of death cases
involved individuals older than 65, while the share of fatalities for people
under 18 is just 0.04%. At the same time, recent research findings showed that
Body Mass Index (BMI) impacts on the condition of the patients with SARS-CoV2,
since obesity compromises pulmonary function and makes the ventilation more
difficult. As a consequence, SARS mortality rate significantly increases among
patients with BMI values over than 27 [13, 27]. Moreover, Ashokka et al. [2]
demonstrated that, although the risk of severe illness for pregnant patients
in the COVID-19 pandemic is less than that of similar cases observed during
the H1N1 counterpart, the virus seems to exert significant adverse effects in
case of childbearing. Indeed, the data provided in their study showed that
roughly 9% of the patients had serious morbidities with respiratory failure or
other organs shock. There are also evidences illustrating that the history of
smoking is correlated with the deterioration of the patients’ conditions [36].
Finally, recent studies showed that, despite acute respiratory failure is
associated with higher mortality, lungs are not the only organs involved since
kidney and cardiovascular system health conditions play prominent roles as
well [8, 45].
In light of these remarks, therefore, we supposed that every agent can have
each of these risk factors with a Bernoulli distribution as follows:

$a^{v,u}(\varphi_{f})\sim Ber(p^{g}_{f})$ (2)

where $p^{g}_{f}$ represents the prevalence of having the factor f within the
age-group g. Moreover, since correlations might exist among these different
determinants, we introduced an upper and a lower health bound for each agent,
from which the health vulnerability value is stochastically chosen by using a
uniform distribution:

$b_{upper}(a^{v,u})=min((1-q_{f})^{a^{v,u}(\varphi_{f})}|f\in riskfactors)$
(3) $b_{lower}(a^{v,u})=\prod\limits_{f\in
riskfactors}(1-q_{f})^{a^{v,u}(\varphi_{f})}$ (4) $a^{v,u}(h)\sim
unif(b_{lower}(a^{v,u}),b_{upper}(a^{v,u}))$ (5)

where $q_{f}$ represents the percentage of severe morbidity cases with the
risk factor f among SARS-CoV2 patients in the age-group g. People with health
vulnerability values lower than a specific threshold were considered as in
need of hospital care and treatment. Since they barely perform shopping or
similar tasks, we considered them like agents with a serious disability, i.e.
$a^{v,u}(d)=1$. Notice that, in an unrealistic situation where no correlations
exist, so that risk factors can be deemed independent one from each other,
based on the Bayes’ theorem the probability of their coincidences is the
product of the probability of their solely occurrences, which is equal to the
lower health bound. At the other extreme, if all the events’ probability
spaces are subsets of each other, the probability of their coincidences is
given by the probability of the rarest event, which in its turn is equal to
the upper health bound. Table 2 shows the $p^{g}_{f}$ and $q_{f}$ values for
the group composed of people between 25 and 29 years old for a city in the US,
based on census data.

Table 2: Risk factors for the age-group 25-29 in the US Risk factors | Census-based prevalence $p^{g}_{f}$ | Severe morbidity probability $q_{f}$
---|---|---
Value | References | Value | References
Age | 1 | - | 0.02 | [41]
Obesity | 0.35 | [17] | 0.1 | [27, 13]
Pregnancy | 0.08 | Statista111https://www.statista.com/statistics/295731/us-pregnancy-rates-by-age/ | 0.09 | [2]
Smoking | 0.16 |  [10], CDC222https://www.cdc.gov/tobacco/data_statistics/fact_sheets/adult_data/cig_smoking/index.htm | 0.15 | [36]
Chronic respiratory diseases | 0.001 | [43] | 0.08 | [41]
Acute renal disease | 0.001 | [33] | 0.91 | [7, 8]
Heart disease | 0.05 | [35] | 0.13 | [41]
Diabetes | 0.03 | [34] | 0.09 | [41]

In the CoV-ABM framework two types of agents are generated: those who have a
job, in the form of a working or a school activity, which has been assigned to
them with a random job-filler algorithm which considers their profiles, and
those who aren’t associated to any job. This second group includes job-
seekers, whose population size has been set according to the most recent
observed unemployment rate, children at preschool ($a^{v,u}(\zeta)<7$) or
people older than $x_{retired}$, where this last threshold is given by a
stochastic number having a normal distribution as follows:

$x_{retired}\sim N(\mu_{retired},\sigma_{retired}^{2})$ (6)

where and represent, respectively, the average and the standard deviation of
the retirement age recorded in the geographical area under investigation.
Finally, at each timestamp every working agent decides whether to travel to
the workplace or not. To rely on a realistic simulator, agents are not always
on-time and they might be up to two timestamps late or one timestamp in
advance when they go to work or come back home.

###  2.2 Generating communities

In the proposed model, agents belong to different friendship groups or
communities of relatives. Friendship groups are composed on average by four
individuals, whose membership is not exclusive. This means that agents
$a^{v_{1},u_{1}}$ and $a^{v_{2},u_{2}}$ can be friends in more than one group
and can share different sets of common friends. At every timestamp, each
friendship group stochastically searches among its members with the aim of
finding those willing to organize a meetup, where the probability of this
event differs according to the specific day in the week and time in the day.
The likelihood of successfully arranging a meeting, for example, is much
higher for a weekend evening compared to any other evening during the week. If
such a meetup is made possible, the available members decide where to go and
the place is chosen randomly either from the members’ homes or from open
public places in their city, e.g. parks, cafes, shopping centres or stadiums.
The model assumes that if an agent attends a meetup it is engaged for a given
time interval and, therefore, it is not able to take part to any other meeting
for that period. In what follows, $W^{t}$ will denote the set of willing
members at time $t$, and $y$, the place where they choose to go from timestamp
$t+1$ to $t+l$:

$\forall(v,u)\in W^{t}:a^{v,u}(\phi^{i}|i\in\\{t+1,\ldots,t+l\\})=0$ (7)
$\forall(v,u)\in W^{t}:a^{v,u}(\varrho^{i}|i\in\\{t+1,\ldots,t+l\\})=y$ (8)

It is worth mentioning that the friendship groups comply with the following
rules:

  1. 1.

Let $a^{v,u}$ denote the first agent included in a group. The other members
are selected so that they are approximately at its own age. Specifically, the
age of the other agents is taken from the range
$a^{v,u}(\zeta)\mp(2\times\sigma_{age})$, where $\sigma_{age}\propto
a^{v,u}(\zeta)$. As a consequence, in our framework adolescents or young
people have only other adolescents or young people as friends. For older
individuals friends are chosen similarly but considering a wider range for the
age.

  2. 2.

Agents younger than a certain age do not join any friendship group. The
rationale behind this rule is that children are supposed to meet their friends
only at school or preschool and aren’t able to organize meetups in public
places.

Beside friendship groups, the model includes also communities of relatives
involving different families. In this case the membership is exclusive so that
each family belongs to at most one of these groups. At every timestamp, each
community stochastically searches among its families to find those available
to organize a meetup. A family is declared available when at least half of its
members are willing to attend the meeting. Once a meeting is set up and its
location is chosen, the available family members will attend it and won’t be
free for a certain time.
Notice that, unlike other agent-based models proposed in the literature which
resort to random graphs to represent interactions among the agents, our model
relies on a more detailed relationship framework to better reflect the real-
world interpersonal interactions. This feature plays an important role since
it allows to study quarantine scenarios and social distancing impositions with
higher realism and precision. For instance, if people are forced to forego the
meetups with their friends due to lockdown or other social restrictions, they
are still in contact with their own family members, they might still visit
some of their relatives, like grandparents, or they might want to limit the
meeting locations to their own houses. As far as we know, the models so far
proposed lack such representation capabilities.

###  2.3 Agents’ needs and actions

As the discrete-event model starts, each family is stochastically associated
to a variety of needs which have to be fulfilled. These needs can take the
form of emergency needs, which are strictly connected to the family members’
health conditions, and of daily needs, which are instead referred to every-day
activities such as food provisioning, shopping, refueling and so on. The
probability of having daily needs depends on the frequency of each need in the
real society and it is here assumed to be equal for all families. The
likelihood for an emergency need to arise is, instead, dependent on each
member’s health conditions, $a^{v,u}(h)$ and $a^{v,u}(s)$, as follows:

$v_{emergency}^{regular}(u)^{t}\sim Ber(p_{emergency}^{regular}a^{v,u}(h))$
(9) $v_{emergency}^{COVID}(u)^{t}\sim Ber((p_{i\rightarrow
h}^{t})^{v,u});\\{(v,u)|a^{v,u}(s)=infected\\}$ (10)

where $p_{emergency}^{regular}$ is the likelihood of being hospitalized for a
“highly strong agent”, defined in this study as an agent with a health
vulnerability value near to 1, as indicated in Table 1. Whereas
$p_{i\rightarrow h}^{t}$ represents the likelihood of being hospitalized due
to COVID-19. This second term will be explained in more details in Section
2.4.
Emergency needs involve a given family member and can be satisfied at a
specific hospital located in the county where he/she lives. When an emergency
need arises, the affected agent immediately plans to go to the hospital with
the aim of fulfilling that need. In case of disability or precarious state, a
good health adult belonging to the same family is randomly selected to help
the member in need.
In our model, the length of staying (LOS) at the hospital for patients
affected by COVID-19 is defined as the number of days elapsed between
hospitalization and discharge (in case of both recovery and death), while for
the other patients it is set to a stochastic number. Notice that, the average
LOS value may vary from city to city and for different diseases. However, for
the sake of simplicity its value was here set to 4.5 days, which corresponds
to the average hospital LOS recorded in the US [39].
Harini et al. observed that non-transformed distributions cannot fit the LOS
data appropriately, and suggested to use the gamma-pareto distribution for
hospitals LOS fitting purposes [18]. In our model, therefore, each non-COVID
patient stays at the hospital for a period of days, which follows a gamma-
pareto distribution [1]

$L\sim Gamma-Pareto(\alpha_{LOS},c_{LOS},\theta_{LOS})$ (11)

with parameter values $\alpha_{LOS}=7$, $c_{LOS}=0.2$ and $\theta_{LOS}=1$, as
suggested in [18].
Daily needs are by their nature less urgent and can ideally be satisfied by
any eligible member of the family. The eligibility is given to not-disabled
members showing health vulnerability values greater than a given threshold. In
particular, as soon as an eligible member manages to find the time and the
right place is open, he/she goes there to fulfill the need. In the event that
several places are available, the priority is given to the closer destination.
To this aim, we resorted to a graph-based shortest path algorithm proposed in
NetworkX [16]. Furthermore, to govern the complexity, we assumed that on
average each family daily need requires one hour to be satisfied.
The model uses a Bernoulli trial to generate new needs for each family, the
probability of which is given by:

$v_{daily}^{n,t}\sim Ber(\vartheta_{n}dt/24);n\in N$ (12)

where $\vartheta_{n}$ is the frequency of expressing the daily need $n$. $dt$
denotes the timestep and $N$ is the set of different daily needs.
When an agent is selected to satisfy a need, either an emergency or a daily
one, it won’t be free to accomplish other tasks until the need is fulfilled.
For a family, if a daily need is not yet satisfied and a new similar need is
generated, these two needs are merged into a single one.

###  2.4 Infection dynamics

In this paper we propose a novel way of disease transmission among the agents.
Specifically, we assume that the individual suffering from the disease can
infect other agents indirectly by spreading the virus in the environment. As
of today, it is believed that COVID-19 is often transmitted through
respiratory droplets produced by coughing and sneezing. This virus can however
remain suspended in the air for a significant period of time [5] and may also
persist on surfaces like metal, plastic or glass for up to 9 days [25].
Developing a model which accounts for such an important characteristic is
fundamental to properly represent the spread of SARS-CoV2. Hence, unlike other
agent-based models which assume directly contacts as the only mean of
transmission, we suppose that the environment itself (the air and/or a
surface) can convey the virus from one agent to the other.
In the CoV-ABM framework each location, either indoor or in the form of an
outdoor area, can contain a given amount of virus. The volume of the virus
persisting in the given location divided by the volume of the location itself
determines the probability of infection. The persistence of the virus is not
yet exactly known and is estimated to range between two hours and several
days, depending on the environmental conditions [25]. However, we can suppose
that in a given venue COVID-19 gets inactivated in the environment after $k$
hours on average. Thus, after each timestamp a portion of virus equal to
$\delta=dt/k$ at each location disappears. Note that the value of $k$ can
change by applying different mitigation measures, such as the use of
detergents and UV lamps. According to this model, an agent is be able to
transmit the virus to any other agent residing or visiting the locations it
stayed or passed by. To the best of our knowledge, none of the existent agent-
based models in the literature relies on such an assumption.

Figure 2: Schematic diagram of the proposed SEIHRD model, where arrows
indicate possible transitions with the corresponding probability notations

In this study an extension of the SEIR model, called SEIHRD, is used to shape
the epidemiological phases of the disease. As discussed earlier in Section 1.,
some SEIR extensions have been proposed in the literature addressing the
spread of COVID-19 [21, 44] and [26]. As it is shown in Figure 2, our SEIHRD
model comprises 6 states: Susceptible (S), Exposed or Asymptomatic Infectious
(E), (Symptomatic) Infectious (I), Hospitalized (H), Recovered (R) and Dead
(D). For the sake of simplicity, unlike some recent works [30], we do not
separate ICU beds and regular beds.
By default, all the agents are susceptible except for the initial spreaders,
called seeds:

$a^{v,u}(s^{t=0})=\begin{cases}susceptible;\forall(v,u)\notin seeds\\\
exposed;\forall(v,u)\in seeds\end{cases}$ (13)

At each timestamp, exposed and infectious agents spread the virus in the
locations where they resided or passed by. For each location $y$, the
infection is calculated as:

$y(\Delta^{t})=y(\Delta^{t-1})(1-\delta)+\sum\limits_{(v,u)\in
A_{y}^{t}}\epsilon^{a^{v,u}(s^{t})};A_{y}^{t}=\\{(v,u)|a^{v,u}(\varrho^{t})=y\\}$
(14)

where $y(\Delta^{t})$ is the volume of active viruses inside $y$, and
$\epsilon^{a^{v,u}(s^{t})}$ represents the average quantity of virus being
spread in the environment, at a given timestamp, by an agent who is in the
epidemiological state $s^{t}$. $A_{y}^{t}$ is the set of agents located in $y$
at time $t$.
The probability of being infected for any susceptible agent inside the given
area is then determined by a Bernoulli trial with transmission probability
$p_{s\rightarrow e}^{t}$. Specifically, for agent $a^{v,u}$ at time $t$ this
probability can be computed as:

$p_{s\rightarrow
e}^{t}=\alpha(a^{v,u}(\varrho^{t})(\Delta^{t})/a^{v,u}(\varrho^{t})(\Omega^{t}))$
(15)

where $\alpha$ is the infectiousness rate of the disease, which is a real
number between 0 and 1, and $y(\Omega^{t})$ denotes the volume of location
$y$.
If the agent receives the virus, during the “exposed” phase it won’t show any
clinical symptom. Therefore, it will work and live like a susceptible agent
while spreading the virus over the places it visits or resides in. After a
stochastic period, whose average is $L_{e}$(hours), the agent is either
“infectious” or “recovered”:

$p_{e\rightarrow i}^{t}=\beta(1-\rho_{er}(a^{v,u}(h^{t})))$ (16)
$p_{e\rightarrow r}^{t}=\beta\rho_{er}(a^{v,u}(h^{t}))$ (17)

where $\beta=dt/L$, and $\rho_{er}$ is the rate of being directly recovered
after being asymptomatic infectious for a highly strong agent. Thus, as
$a^{v,u}(h^{t})$ decreases, the probability of being “infectious” raises.
During the “infected” phase the agent shows symptoms; it still spreads the
virus but with a different rate $\epsilon^{infectious}$. The average time
length of being infectious for a highly strong agent is given by
$L_{i}$(hours). Let $\gamma=dt/L_{i}$ be the rate of being “recovered” for a
highly strong agent. The probability of becoming “hospitalized”, “recovered”
or “dead” at each timestamp is:

$p_{i\rightarrow r}^{t}=\gamma(a^{v,u}(h^{t}))$ (18) $p_{i\rightarrow
h}^{t}=\rho_{ih}(a^{v,u}(h^{t}))$ (19) $p_{i\rightarrow
d}^{t}=\rho_{id}(1-\gamma(a^{v,u}(h^{t})))$ (20)

where $\rho_{ih}$ and $\rho_{id}$ represent the hospitalization and the death
rate, respectively, among the strong individuals who are not hospitalized. For
hospitalized patients, instead, the probability of being “recovered” is still
the same, since there is no evidence illustrating that hospitalized people
have shorter period of infection: $p_{i\rightarrow r}^{t}=p_{h\rightarrow
r}^{t}$ . However, due to the presence of dedicated facilities such as
Intensive Care Units and of special medical equipment, hospitalization reduces
the death rate. Consequently, we propose to compute this probability as:

$p_{h\rightarrow d}^{t}=p_{i\rightarrow d}^{t}/H_{q}^{t}$ (21)

where $H_{q}^{t}$ represents the efficiency of hospital $q$ at time $t$,
called “hospital factor”. If hospital $q$ has a treatment capacity equal to
$C_{q}^{t}$, the hospital reaches its maximum efficiency if there are less
than $C_{q}^{t}$ agents being hospitalized. If the number of hospitalized
individuals at $q$ increases, the hospital factor plunges due to the
saturation of the structures and the scarcity of available care beds. Let
$n^{t}$ be the number of patients in hospital $q$ at time $t$; we propose a
modified pareto curve to calculate the efficiency values as follows:

$H_{q}^{t}=1+\tau_{0}C_{q}^{t}/(C_{q}^{t}+\tau_{1}n^{t})$ (22)

Figure 3: Hospital factor with different parameter values

Figure 3 illustrates the hospital factors with different parameter values.
These values may change from city to city and hospital to hospital, so a
parameter tuning needs to be accomplished. The pseudocodes of the proposed
SEIHRD diffusion model and the main discrete-event of CoV-ABM are provided in
Algorithm 1 and Algorithm 2 (see appendix A).

##  3 Simulation results

In this section we illustrate how the model can simulate the spread of
COVID-19 in the State of Delaware, USA, which has a population around one
million people. This state is divided into three counties, New Castle, Sussex,
and Kent, with populations over 558, 234 and 180 thousand, respectively.
Delaware is one of the smallest states in the US and ranked 46th out of the
fifty US states in terms of population. However, the virus there has been
spreading widely in the recent weeks. As of today, Delaware is ranked 6th in
the US in terms of total COVID-19 positive cases per one million population,
with near 10,000 approved cases [34]. Figure 4 depicts the network of streets
of the three counties, separately.

Figure 4: Street networks of New Castle County (left), Sussex County (middle)
and Kent County (right)

###  3.1 Simulation assumptions and different scenarios

In order to govern the complexity of the simulator we assumed the environment
to be isolated. Consequently, the model doesn’t account for incoming or
outgoing travels from the region at hand. Notice that, this hypothesis well
fits with our simulation environment since the State of Delaware occupies the
north-eastern portion of the Delmarva Peninsula, besides some islands and
territory within the Delaware River, and is therefore less connected to other
states compared to inland states.
Further assumptions were then introduced to cope with the lack of complete
information, preserving however the compatibility and the adherence to
detailed census data. From one side, some buildings in the state were randomly
assigned to different uses, being unaware of their actual usage. From the
other, the unavailability of census data reporting an exhaustive list of job
titles and workplace types, required to synch the environment generator,
forced us to simplify jobs, workplaces and the corresponding working staffs.
Table 3 shows the different working and living spaces investigated in the
present study. The amount of working hours was randomly assigned to each
worker, by assuming that no healthy individual works less than 6 hours or more
than 10 hours, daily. Moreover, some workplaces such as hospitals, public
transportations or police stations were supposed to be always up and running.

Table 3: Categories of workplaces and living spaces Categories |  Working and living spaces
---|---
Education |  University, College, School, Kindergarten
Emergency |  Hospitals, Fire stations, Police stations
Amenity |  Parks, Cafés, Restaurants, Gyms, Casinos
Event |  Pubs, Stadiums
Crucial |  Banks, Post Offices, Courts, Supermarkets, Drugstores, Gas stations, Public transportations, Crucial factories, Power plants
Cultural |  Museums, Churches, Mosques, Zoos
Private |  Shops, Industrial plants, Offices
Residential |  Apartments, Detached houses, Villas

In our framework, friends and relatives can get together in one of the places
comprised in the categories Residential, Amenity, Event or Cultural, with
different probabilities, which take into account the specificity of both
location and meeting attendees. For example, the likelihood for a reunion to
take place in a café is higher than in a museum; at the same time, friends
tend to meet in Stadiums, Cafés, Gyms, Parks, while relatives are more
inclined to visit each other at their own houses.
According to the hypothesis made in [9], we further assumed that schools and
kindergartens admit people from the closer districts. Hence, the school
located closest to a child’s home had a higher probability to be selected.
Finally, to limit the computational effort the model was run on each county
separately, and the results were then summed together. Notice that, the same
framework can be run on the whole country with few minor modifications at the
expense of an increase of the simulation time.
With the aim of studying the usefulness of some virus counter-measures and
investigate at what extent they can affect the curves of infected people and
cause different spread outcomes, two alternative scenarios were defined as
follows:

  * •

No interventions: All places remain open and practicable full-time. The
probability of visiting friends is $1/14(day^{-1})$ and relatives is
$1/30(day^{-1})$.

  * •

Quarantine: All the places in the Education, Amenity, Cultural and Event
categories are closed with their staff staying at home. The probability of
visiting friends and relatives drops to $1/30(day^{-1})$ and $1/60(day^{-1})$,
respectively. In this case, therefore, people still have contacts with friends
and relatives; however, meeting destinations are no longer represented by
public places like parks or cafés, closed by the restrictions, but are
replaced by the individuals’ own houses.

It is worthwhile to observe that the previous scenarios were conceived for the
purpose of analyzing the effectiveness on the mitigation of the outbreak of
two specific types of interventions, represented by the drastically reduction
of social meetings and the closure of non-critical places.

###  3.2 Simulation of agents on census data

In order to generate the simulated environment and the agents’ profiles all
the necessary demographic and health information about Delaware’s counties
were gathered from US census and US health reports. In particular, data on
each county’s hospitals and relative capacities were collected from the
American Hospital Directory
website333https://www.ahd.com/states/hospital_DE.html, whereas the Open Street
Map tools444https://wiki.openstreetmap.org/wiki/Main_Page, e.g. OSMnx, were
used to retrieve information about locations, building usage and street
graphs.
Once the agents were generated based on the census and health data, their
profiles were analysed to verify the validity of the achieved population.
Table 4 compares the population of the simulated agents, divided into 5 age-
groups, with the real-world census, and shows the close correspondence between
the created and the actual population’s demography. Figure 5, instead,
illustrates the number of generated agents affected by the different risk
factors in each county.

Table 4: Population of agents in different age-groups Risk factors | New Castle County | Sussex County | Kent County
---|---|---|---
Simulated | Real-world | Simulated | Real-world | Simulated | Real-world
< 5 | 5.5% | 5.6% | 5.2% | 5.2% | 5.5% | 6.3%
5 - 17 | 16.3% | 15.7% | 14.8% | 14.0% | 16.3% | 17%
18 - 44 | 37.1% | 36.1% | 30.3% | 27.1% | 36.9% | 35.6%
45 - 64 | 25.3% | 26.7% | 25.1% | 28.4% | 25.1% | 25.3%
65 + | 15.9% | 15.6% | 24.6% | 25% | 16.2% | 15.8%
Figure 5: Risk factors among the simulated agents

###  3.3 Simulating the spread of SARS-CoV2

Simulations were performed by using a mid-range personal laptop with an i7
processor and 8GB of memory on an Ubuntu 18 operating system. The simulator
was coded in Python 3.8 and the model was run on each scenario 5 times
iteratively, to average the results. The runtime of each iteration for the
counties Kent, Sussex and New Castle was roughly 2, 2.5 and 7 hours,
respectively.

####  3.3.1 Simulator calibration

In order to run the simulator on the State of Delaware, we first set some
initial parameters, described in Section 2., as shown in Table 5.

Table 5: Parameters and their values for simulating on the State of Delaware Parameter |  Description |  Estimated value |  References
---|---|---|---
$dt$ |  Timestamp |  1 (hour) |  -
$hr$ |  Average time length of virus persistence |  20 (hours) |  [25]
|  in the environment. |  |
$\alpha$ |  Infectiousness rate |  0 to 0.25 (1/hour) |  Estimated
$L_{e}$ |  Average time length of being exposed |  120 (hours) |  [29, 38]
$L_{i}$ |  Average time length of being infectious |  110 (hours) |  [31]
$\rho_{er}$ |  Proportion of cases having no symptoms |  80% |  [40, 3]
$\rho_{ih}$ |  Hospitalization rate for an infectious case without risk factors |  10% |  Assumed
$\rho_{id}$ |  Death rate for an infectious case without risk factors |  5% |  Assumed
$\epsilon^{exposed}$ |  Average quantity of virus being spread hourly in the environment by an asymptomatic agent |  0.5 (1/hour) |  Assumed
$\epsilon^{infectious}$ |  Average quantity of virus being spread hourly in the environment by a symptomatic agent |  1 (1/hour) |  Assumed

As it is defined in Section 2.4, the infectiousness rate $\alpha$ determines
how easily the virus transmits to a susceptible individual. The exact value of
the infectiousness rate is unknown and depends on several factors, such as the
virus characteristics, the environment (e.g. humidity, temperature, wind
speed) and the personal protective behaviour. As an example, facial masking
and careful hand hygiene can reduce the transmission, so that the
infectiousness rate in societies with rigorous protective behaviours is
usually lower.
In order to figure out how the infectiousness rate affects the viral spread
dynamics and to properly set its value, we performed several simulations on
the Kent county with different values, ranging from 0 to 0.25 with a step size
equal to 0.05. Figures 6 to 10 depict the impact of the infectiousness rate on
the SEIHRD phases’ surfaces. As it can be seen, high values result in more
infected people and more deaths. Moreover, from the same charts it can be
inferred that quarantine and lower infectiousness rates make the whole spread
process slower and longer, eventually leading to fewer casualties.

Figure 6: The impact of $\alpha$ on the number of susceptible agents in Kent
County with No intervention (left) vs. Quarantine (right) Figure 7: The impact
of $\alpha$ on the number of exposed agents in Kent County with No
intervention (left) vs. Quarantine (right) Figure 8: The impact of $\alpha$ on
the number of infectious agents in Kent County with No intervention (left) vs.
Quarantine (right) Figure 9: The impact of $\alpha$ on the number of recovered
agents in Kent County with No intervention (left) vs. Quarantine (right)
Figure 10: The impact of $\alpha$ on the number of dead agents in Kent County
with No intervention (left) vs. Quarantine (right) Figure 11: Infected places
in Kent County after 7 weeks:$\alpha=0.1$(left), $\alpha=0.15$(middle),
$\alpha=0.2$(right)

Figure 11 depicts the infected places in Kent County, with different
infectiousness rate values, at the corresponding infection peaks. Notice that,
due to the virus inactivation period, the highlighted locations only represent
the places where the virus still persists and is not disabled. The blue
circles are proportional to the virus densities inside the corresponding
areas.
Once observed the effect of the infectiousness rate at different intensity
levels and compared it with the actual curves derived from the available
reports, we consequently set it at $\alpha=0.15$ to proceed with the remaining
simulations.

###  3.4 Simulation results for Delaware

Figure 12: SEIHRD dynamics in the State of Delaware when $\alpha=0.15$ \- No
intervention (left) vs. Quarantine (right) Figure 13: SEIHRD dynamics in each
county when $\alpha=0.15$ \- No intervention (left) vs. Quarantine (right)

As it is shown in Figure 12 and Figure 13, the quarantine measures can
drastically flatten the curves. Notice that, our quarantine scenario doesn’t
impose extreme restrictions, since people can still interact with their
relatives and friends and private offices, factories and crucial workplaces
are still open with regular opening hours. Therefore, the results achieved
emphasize that even small actions can play a prominent role.
Table 6 shows the number of deaths in different age groups after 15 weeks of
simulation and demonstrates how vulnerable old people are and how the
interventions can save their lives.

Table 6: Average number of dead agents after 15 weeks with different measurements Age Group | New Castle County
---|---
No intervention | Quarantine
Deaths | %of the age group | Deaths | %of the age group
< 5 | 0.0 | 0.000% | 0.0 | 0.000%
5 - 17 | 20.4 | 0.022% | 0.0 | 0.000%
18 - 44 | 273.4 | 0.132% | 13.4 | 0.006%
45 - 64 | 911.0 | 0.645% | 27.2 | 0.019%
65 + | 2085.8 | 2.350% | 119.2 | 0.134%
Total | 3290.6 | 0.5889% | 159.8 | 0.0286%
Age Group | Sussex County
No intervention | Quarantine
Deaths | %of the age group | Deaths | %of the age group
< 5 | 0.0 | 0.000% | 0.0 | 0.000%
5 - 17 | 0.0 | 0.000% | 0.0 | 0.000%
18 - 44 | 49.4 | 0.070% | 0.0 | 0.000%
45 - 64 | 330.2 | 0.562% | 2.8 | 0.005%
65 + | 1401.6 | 2.433% | 50.6 | 0.088%
Total | 1781.2 | 0.7604% | 53.4 | 0.0228%
Age Group | Kent County
No intervention | Quarantine
Deaths | %of the age group | Deaths | %of the age group
< 5 | 0.0 | 0.000% | 0.0 | 0.000%
5 - 17 | 6.6 | 0.022% | 0.0 | 0.000%
18 - 44 | 52.2 | 0.078% | 4.2 | 0.006%
45 - 64 | 303.0 | 0.668% | 3.2 | 0.007%
65 + | 916.8 | 3.130% | 9.8 | 0.033%
Total | 1278.6 | 0.7072% | 17.2 | 0.0096%

The reports from Delaware, Division of Public Health [12], state that the
first positive case has been identified in March 10th, 2020. As of today,
after 12 weeks of spread, COVID-19 casualties raised to 149, 126 and 56 cases,
for the New Castle, Sussex and Kent counties, respectively. These reports show
that Sussex has a higher percentage of deaths, which matches our “no
intervention” scenario. It is worth mentioning that, the infection process can
be highly path-dependent when the ratio between the number of seeds and the
overall population is infinitesimal. In such cases, the number of deaths can
significantly vary in each iteration, and relies on the selection of the seed
sets and on their actions during their infectious period. For example, if
there are only few seeds with too few interactions, or they are in good health
conditions to get recovered speedily, the infection won’t sprout up and won’t
last for a long period.
Another important issue during the outbreak is the saturation of health
facilities. Figure 14 illustrates the number of beds required for the patients
in each of the hospitals in the three counties. During the quarantine scenario
this demand is less urgent since the speed of the virus spread slows down. The
plotted lines are the cumulative of both COVID-related and non-COVID
hospitalization needs.

Figure 14: Occupied or needed beds in different hospitals - No intervention
(left) vs. Quarantine (right)

###  3.5 Identifying the hubs of infection

Our simulator has the power to monitor the spread of the virus and identify
hazardous places where the probability of transmission rises. To show the
phenomenon with a closer lens, we separately ran the model on the city of
Dover, the capital of Delaware, with 38,000 agents. Figure 15 and 16 depict
the infected places through the time. The plots were taken at the end of every
2 weeks and illustrate the growth and decline of the outbreak in the city.
In order to identify the places with higher transmission potential, we
extracted the areas where each “exposed” agent received the virus and became
exposed. For the “no intervention” scenario, the place types inside which at
least 100 incidents occurred are listed in Table 7. Educational places
collectively incubated more than half of all the virus transmissions, with
approximately 7500 incidents on average. It must be pointed out that most of
these cases involved young and healthy people; therefore, only a small portion
of them became symptomatic infectious. On the contrary, these exposed young
people conveyed the virus to their houses and made their family members
exposed as well. This is one of the reasons why the number of virus
transmissions inside the houses was high on average (with almost 3800 cases).
For the “quarantine” scenario, on the other hand, the city witnessed a
dramatic decrease in the number of transmissions inside the houses due to the
lock down of the educational and cultural places. Despite houses remain among
the most risky locations, the average number of transmissions therein plunged
to one-fifth.
According to Table 8, in the quarantine scenario hospitals were the most
hazardous places in terms of virus transmission. It should be emphasized that,
an agent with poor health conditions or disabilities needed an eligible family
member to accompany it to the hospital. On such occasion, it is very likely
that the attendant gets exposed in the hospital, if he/she did not inhale the
virus at home or anywhere else. Also public transportation had an important
role in the viral spread, since in both scenarios it led to more than 800
incidents, on average, with a small standard deviation.

Table 7: The places with the most transmission incidents (No intervention) Location type | Number of incidents
---|---
Mean |  Standard deviation(%)
Schools and Kindergartens |  5914.8 & 4.2% |
Houses |  3792.0 |  6.6%
Universities and Colleges |  1596.0 |  3.7%
Churches |  1127.2 |  4.5%
Hospitals |  1012.6 |  2.5%
Public transportations |  878.2 |  3.2%
Gyms |  345.4 |  8.0%
Café and Restaurants |  269.2 |  20.4%
Museums |  201.0 |  20.3%
Private Offices |  154.8 |  1.0%
Industrial plants |  150.4 |  23.4%
Table 8: The places with the most transmission incidents (Quarantine) Location type | Number of incidents
---|---
Mean |  Standard deviation(%)
Hospitals |  934.0 |  0.8%
Public transportations |  802.2 |  8.3%
Houses |  687.0 |  3.2%
Groceries |  37.2 |  24.0%
Private Offices |  29.2 |  9.1%
Industrial plants |  28.0 |  3.6%
Shops |  13.6 |  23.5%
Banks |  13.4 |  22.9%
Gas stations |  11.0 |  39.6%
Figure 15: Simulating the spread of COVID-19 in Dover, DE with no
intervention. The size of the circles represents the locations’ virus density
Figure 16: Simulating the spread of COVID-19 in Dover, DE in quarantine. The
size of the circles represents the locations’ virus density

The comparison between Figure 15 and 16 graphically demonstrates the
effectiveness of the quarantine measures in terms of the outbreak mitigation.
During the quarantine, indeed, the prevalence and density of the virus inside
the city buildings and the outdoor areas are considerably lower with respect
to the other scenario.
Moreover, both figures show that the buildings in the city downtown (the
center of the map) can reach the highest virus densities. In particular,
Figure 15 illustrates that the areas with a higher number of churches,
colleges and schools (e.g. the downtown), the south-eastern district and the
northern district, where Delaware State University and Wilmington University
are located, have higher likelihoods to become virus hubs.

##  4 Discussion

ABM is one of the computational approaches which can simulate individuals and
their decisions based on the rules and the set of actions that are programmed
in it. These rules are configured so that the simulator can better mimic the
behaviour of its real-world counterpart. Such simulators can be of great
importance when the interactions among the citizens and their decisions
profoundly shape the dynamics of the system. Thus, since the intensity of the
outbreaks is highly reliant upon the behaviour of the individuals, this paper
proposes an ABM framework, named CoV-ABM, considering such actions and
decisions in a higher detail compared to other existing frameworks. In
addition, due to the unique epidemiological and clinical features of the novel
coronavirus SARS-CoV2 [38], and its remarkable durability and persistence in
the environment [25], the development of a simulator being able to consider
these characteristics was needed.
During the past few months, the COVID-19 pandemic heavily affected the lives
of billions of people around the world, as well as the global economy.
Different countries and international organizations are taking unprecedented
measures to combat its spread [20]. In this scenario, CoV-ABM can be used as a
decision-support tool for policymakers and researchers to analyse the
consequences of different decisions and regulations devoted to halt the
outbreak and prevent future potential occurrences.
This paper proposes a novel extension of SEIR, called SEIHRD, which formulates
the hospitalized cases and the saturation of the facilities and considers both
symptomatic and asymptomatic cases. According to a study from China [3],
approximately 80% of SARS-CoV2 cases had no clinical symptoms, even though
they could release large amounts of viruses at the early phases of the
infection. CoV-ABM was developed to being able of tracking the active viruses
existing at every indoor or outdoor areas, and computing the transmission
probability based on the virus density at each location. In the proposed
model, indeed, a place visited by infectious people remains infectious until
all its viruses get inactivated.
This novel assumption, unlike that of studies whose diffusion models rely on
social networks or agent-agent interactions, is totally compatible with the
peculiarity of SARS-CoV2. Former studies assumed that an agent directly
transmits the virus to another agent when both are close enough for a certain
period or time, or when a graph-edge gets activated. In our model, instead, an
agent can infect any other individual indirectly through the environment. As
an example, agent $a^{v_{i},u_{j}}$ releases an amount of virus at location y,
where it passes by $y$ at time $t$. Any susceptible agent $a^{v_{m},u_{n}}$
who visits the same location at time t might get exposed regardless of having
a social link with agent $a^{v_{i},u_{j}}$, until the viruses in that area are
not disabled (i.e. $y^{\Delta^{t}}>0$). In graph-based models this can be
possible only if the graph is complete.
Another novelty of this research is how the agent communities and their
interactions were modelled, specifically the interactions among family
members, relatives and friends. In our framework, each agent is a member of a
family, has a group of relatives and a set of friendship groups. When a group
of friends or relatives decide to visit each other, they choose a destination
among the possible open locations where they can go together. Even if no open
public places exist, they still have the chance to set the destination to one
of the available members’ houses; this possibility enables the escalation of
the virus spread. Hence, an effective mitigating scenario should consider not
only limitations of the opening hours of public places but also measures
devoted to lowering as possible the probability of interactions among
individuals.
By defining two scenarios, “no intervention” and “quarantine”, this study aims
at demonstrating how lock down measures can halt the outbreak and “flatten the
curve”. In particular, the simulation results for the State of Delaware showed
that the restrictions can hugely reduce the number of infected people and,
consequently, the number of victims and the hospital bed demand. From this it
can be inferred that in the “no intervention” scenario, Delaware hits the peak
of outbreak between the 2nd and the 3rd week, and the number of active persons
remains relatively high for about 9 more weeks. In the “quarantine” scenario,
instead, the outbreak peak happens somewhere between the 6th and the 12th
weeks while the number of active agents is relatively low and lasts without
any significant change by the end of our simulation period. The results also
illustrated that the spread dynamics slightly differ in different counties,
maybe due to their different health structures and urban arrangements. This
opens up an opportunity for future studies to investigate the impact of urban
design and street graphs on the spread of COVID-19.
By tracking the locations of all transmission incidents and calculating the
virus densities for different areas, CoV-ABM highlighted the high-risk places
inside the geographical environment. For the city of Dover, the simulation
results showed how educational and cultural places can become hubs of
infection that accelerate the outbreak. In the “no intervention” scenario,
educational buildings caused roughly 48% of all the transmissions. The same
percentage for cultural places, including churches and museums, was just over
8% on average. On the other hand, our results draw attention on the protection
policies inside hospitals and public transportation, since the simulations
showed that these places, taken together, can cause more than 70% of all the
infection incidents during the quarantine. Households achieved a share of 25%
and 23% of all the incidents for the “quarantine” and the “no intervention”
scenarios, respectively.
Notice that, this paper tried to gather and use all the available real-world
data to generate a simulation environment as accurate as possible. However,
there was still a substantial portion of urban and suburban buildings for
which complete information was missing, such as their usage, their indoor
volume size and their opening hours. In such cases, CoV-ABM assigned random
values to make the environment complete. In order to enhance the precision of
the simulator, more comprehensive census and geographical information
databases are required.

##  5 Conclusion and future opportunities

In this paper we proposed a novel spatiotemporal simulation framework, called
CoV-ABM, to mimic the spread of SARS-CoV2 virus in different geographical
domains. This framework uses census data and GIS information to generate a
realistic geographical environment where people live, work, and interact with
each other. CoV-ABM employs a new extension of SEIHRD diffusion model which
considers the virus characteristics and its ability to persist in the air
and/or on surfaces for long periods of time. Our diffusion model works based
on the density of active SARS-CoV2 viruses at each place’s volume. Therefore,
unlike other simulators proposed in the literature, agents in our framework
can inhale the virus in an infectious place, even if there is no other agent
around. Moreover, the diffusion model considers also the actual number of
infectious people who can have asymptomatic periods of illness, and not only
those who have been tested, and some risk factors, such as pregnancy, smoking
record, heart diseases and kidney misfunctioning, to generate the health
profiles for each population based on the local health information.
CoV-ABM simulates the agents’ daily needs and their social interactions with a
high resolution, which enabled us to study the consequences of different
mitigation policies. Therefore, in this study we examined the impact of two
different scenarios on the dynamics of the outbreak in the State of Delaware,
with a population of one million. The simulation results illustrated that the
number of deaths drastically grew in different age-groups in case of no
interventions compared to the quarantine scenario. Our framework assumed that
the plethora of agents inside hospitals and the higher number of
hospitalizations reduced the hospital efficiency, and compared the difference
between the hospitals’ bed demands in the two scenarios.
Finally, CoV-ABM identified the high risky locations and buildings, in terms
of the virus transmission likelihood, in the geographical areas under
investigation. According to the simulation results, the top three virus hubs
were schools, houses and universities when no restrictions were imposed, while
the top-ranked places were hospitals, public transportations and houses in the
quarantine scenario. These outcomes can be of great importance for
policymakers to understand the consequences of their decisions, in particular
of laws and protective regulations focused on limiting the attendance of such
locations.
The current implementation of CoV-ABM was run by simulating millions of agents
at a country-level geography, to control the computational effort and enable
its usage on a personal computer. The proposed simulator, however, could be
applied on continental domains and/or on a world-wide scale by adding or
removing few assumptions. Several research opportunities and working
directions are identified and recommended as follows: (1) developing a variety
of different scenarios based on the current or future policy responses to
COVID-19; (2) extending the SEIHRD model to separate the ICU beds and regular
beds for patients with different morbidity conditions at hospitals and
considering the effect of self-quarantine; (3) adding airports and harbours to
the model and investigating the impact of travellers and tourists on the
spread dynamics;(4) investigating the influence of different urban designs on
the severity of outbreak.

## Acknowledgement

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.

## CRediT author statement

Masoud Jalayer: Methodology, Conceptualization, Software, Data Curation,
Investigation, Visualization, Writing- Original draft preparation. Carlotta
Orsenigo: Validation, Writing- Reviewing and Editing, Supervision. Carlo
Vercellis: Supervision

##  Appendix A Algorithms

Data: $t,a^{v,u},locations,parameters$

Result: $a^{v,u},locations$

if  _$a^{v,u}(s^{t})=infectious$_ then

infect $a^{v,u}(\varrho^{t})$ ; calculate $a^{v,u}(\phi^{t+dt})$ ;

if  _$a^{v,u}(\varrho^{t})\neq a^{v,u}(\varrho^{t+dt})$_ then

infect $a^{v,u}(\varrho_{out}^{t})$ and $a^{v,u}(\varrho_{out}^{t+dt})$;

if  _$a^{v,u}(\varrho^{t})\in hospitals$_ then

calculate $H^{t}_{q}$;

if  _random <$p_{h\rightarrow d}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow dead$; set $a^{v,u}(\varrho^{t+dt})\leftarrow$ a
cemetery nearby

else if  _random <$p_{h\rightarrow r}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow recovered$;

else

if  _random <$p_{i\rightarrow d}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow dead$; set $a^{v,u}(\varrho^{t+dt})\leftarrow$ a
cemetery nearby

else if  _random <$p_{i\rightarrow r}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow recovered$;

else if  _$a^{v,u}(s^{t})=exposed$_ then

infect $a^{v,u}(\varrho^{t})$;

if  _$a^{v,u}(\varrho^{t})\neq a^{v,u}(\varrho^{t+dt})$_ then

infect $a^{v,u}(\varrho_{out}^{t})$ and $a^{v,u}(\varrho_{out}^{t+dt})$;

if  _random <$p_{e\rightarrow r}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow recovered$;

else if  _random <$p_{e\rightarrow i}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow infectious$;

else if  _$a^{v,u}(s^{t})=susceptible$_ then

if  _random <$p_{s\rightarrow e}^{t}$_ then

set $a^{v,u}(s^{t})\leftarrow exposed$;

Algorithm 1 Pseudo-code of the SEIHRD model

Initialization: set objects to default values; for $(v,u)\in seeds$ then set
$a^{v,u}(s^{0})\leftarrow exposed$;

Main loop:

while  _$t <t^{max}$_ do

foreach  _$\varrho\in$ locations_ do

update $\varrho(\Delta^{t})$

foreach  _$v\in$ families_ do

foreach  _$n\in$ daily needs_ do

update $v_{n}^{t}$

foreach  _$u\in v(members)$_ do

update $v_{emergency}^{regular}(u)^{t}$;

if  _$v_{emergency}^{regular}(u)^{t}=1$_ then

set $a^{v,u}(\varrho^{t+dt})\leftarrow$ a related place nearby; set
$v_{emergency}^{regular}(u)(t_{0})\leftarrow t+dt$

update $a^{v,u}(\phi^{t+dt})$; update $a^{v,u},locations\leftarrow
SEIHRD(t,a^{v,u},locations,parameters)$

if  _$a^{v,u}(s^{t})=infectious$ & $a^{v,u}(\varrho^{t})\notin hospitals$_
then

update $p_{i\rightarrow h}^{t}$

if  _$random <p_{i\rightarrow h}^{t}$_ then

update $v_{emergency}^{COVID}(u)^{t}$; set $a^{v,u}(\varrho^{t+dt})\leftarrow$
a related emergency place nearby

foreach  _$n\in needs=\\{n|v_{n}^{t}\neq\varnothing\\}$_ do

if $a^{v,u}$ is eligible then set
$O_{n}^{t+dt}\leftarrow\\{o_{i}|\|shortestpath(o_{i},a^{v,u}(\varrho^{t})\|<2km\\}$;

if  _$O_{n}^{t+dt}\neq\varnothing$_ then

set $a^{v,u}(\varrho^{t+dt})\leftarrow$ random choice from $O_{n}^{t+dt}$; set
$v_{n}^{t}\leftarrow\varnothing$; set $v_{n}^{t+dt}\leftarrow\varnothing$;

if  _$\\{v_{emergency}^{COVID}(u)^{t}\bigcup
v_{emergency}^{regular}(u)^{t}\\}\neq\varnothing$_ then

if  _$a^{v,u}(\zeta) <18$ or $a^{v,u}(h)<0.3$ or $a^{v,u}(d)=1$_ then

$a^{v,q}\leftarrow$ random choice from eligible family members (if any) for
help;

set $a^{v,q}(\varrho^{t+dt})\leftarrow a^{v,u}(\varrho^{t+dt})$

if  _$v_{emergency}^{COVID}(u)^{t}\neq\varnothing$_ then

if $a^{v,u}(s^{t})\neq infectious$ then set
$v_{emergency}^{COVID}(u)^{t}\leftarrow\varnothing$

else if  _$t-v_{emergency}^{regular}(u)(t_{0})\geq L$_ then

set $v_{emergency}^{regular}(u)^{t}\leftarrow\varnothing$

foreach  _$g\in friendgroups$_ do

set $F\leftarrow\\{(v,u)|a^{v,u}(\phi^{t+dt})=1\\}$;

if  _$random <p_{friends}\times timefactor(t+dt)$ and $\|F\|>1$_ then

set $l\leftarrow$ a random location where they can get together;

foreach  _$(v,u)\in g$_ do

set $a^{v,u}(\varrho^{t+dt})\leftarrow l$; set $a^{v,u}(\phi^{t+dt})\leftarrow
0$

foreach  _$g\in relativesgroups$_ do

foreach  _$v\in g$_ do

set $\Phi^{v}\leftarrow\\{(v,u)|a^{v,u}(\phi^{t+dt})=1\\}$

set $F_{g}\leftarrow\\{v_{i}|\|\Phi^{v_{i}}\|>1\\}$;

if  _$random <p_{relatives}\times timefactor(t+dt)$ and $\|F_{g}\|>1$_ then

set $l\leftarrow$ a random location where they can get together;

foreach  _$v\in g$_ do

foreach  _$u\in\Phi^{v}$_ do

set $a^{v,u}(\varrho^{t+dt})\leftarrow l$; set $a^{v,u}(\phi^{t+dt})\leftarrow
0$

foreach  _$h\in hospitals$_ do

update $H_{h}^{t}$

set $t\leftarrow t+dt$

Algorithm 2 Pseudo-code of the CoV-ABM discrete-event model

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