# A multilayer network model of Covid-19: implications in public health policy
in Costa Rica

Fabio Sanchez  Universidad de Costa Rica, Centro de Investigación en
Matemática Pura y Aplicada - Escuela de Matemática, San José, Costa Rica
these authors contributed equally to this work  PhD  Universidad de Costa
Rica, Centro de Investigación en Matemática Pura y Aplicada - Escuela de
Matemática, San José, Costa Rica  these authors contributed equally to this
work  Juan G. Calvo  Universidad de Costa Rica, Centro de Investigación en
Matemática Pura y Aplicada - Escuela de Matemática, San José, Costa Rica
these authors contributed equally to this work  PhD  Universidad de Costa
Rica, Centro de Investigación en Matemática Pura y Aplicada - Escuela de
Matemática, San José, Costa Rica  these authors contributed equally to this
work  Gustavo Mery  Pan America Health Organization, World Health
Organization, San José 10102, Costa Rica.  these authors contributed equally
to this work  PhD  Pan America Health Organization, World Health Organization,
San José 10102, Costa Rica.  these authors contributed equally to this work
Yury E. García  Universidad de Costa Rica, Centro de Investigación en
Matemática Pura y Aplicada, San José, Costa Rica  Department of Public Health
Sciences, University of California Davis, CA, USA  these authors contributed
equally to this work  PhD  Universidad de Costa Rica, Centro de Investigación
en Matemática Pura y Aplicada, San José, Costa Rica  Department of Public
Health Sciences, University of California Davis, CA, USA  these authors
contributed equally to this work  Paola Vásquez  Universidad de Costa Rica,
Centro de Investigación en Matemática Pura y Aplicada, San José, Costa Rica
these authors contributed equally to this work  MD  Universidad de Costa Rica,
Centro de Investigación en Matemática Pura y Aplicada, San José, Costa Rica
these authors contributed equally to this work  Luis A. Barboza  Universidad
de Costa Rica, Centro de Investigación en Matemática Pura y Aplicada - Escuela
de Matemática, San José, Costa Rica  these authors contributed equally to this
work  PhD  Universidad de Costa Rica, Centro de Investigación en Matemática
Pura y Aplicada - Escuela de Matemática, San José, Costa Rica  these authors
contributed equally to this work  María Dolores Pérez  Universidad de Costa
Rica, Centro de Investigación en Matemática Pura y Aplicada, San José, Costa
Rica  these authors contributed equally to this work  MD  Universidad de Costa
Rica, Centro de Investigación en Matemática Pura y Aplicada, San José, Costa
Rica  these authors contributed equally to this work  Tania Rivas  Ministry of
Health, San José 10102, Costa Rica  these authors contributed equally to this
work  MD  Ministry of Health, San José 10102, Costa Rica  these authors
contributed equally to this work

###### Abstract

Successful partnerships between researchers, experts, and public health
authorities have been critical to navigate the challenges of the Covid-19
pandemic worldwide. In this collaboration, mathematical models have played a
decisive role in informing public policy, with findings effectively translated
into public health measures that have shaped the pandemic in Costa Rica. As a
result of interdisciplinary and cross-institutional collaboration, we
constructed a multilayer network model that incorporates a diverse contact
structure for each individual. In July 2020, we used this model to test the
effect of lifting restrictions on population mobility after a so-called
“epidemiological fence” imposed to contain the country’s first big wave of
cases. Later, in August 2020, we used it to predict the effects of an open and
close strategy (the Hammer and Dance). Scenarios constructed in July 2020
showed that lifting restrictions on population mobility after less than three
weeks of epidemiological fence would produce a sharp increase in cases.
Results from scenarios in August 2020 indicated that the Hammer and Dance
strategy would only work with $50\%$ of the population adhering to mobility
restrictions. The development, evolution, and applications of a multilayer
network model of Covid-19 in Costa Rica has guided decision-makers to
anticipate implementing sanitary measures and contributed to gain valuable
time to increase hospital capacity.

Keywords: Covid-19, Network Model, Public Health, Non-pharmaceutical
interventions, Computational Model

## Introduction

The global health crisis that began with the emergence of the SARS-CoV-2 virus
has triggered unprecedented collaborations and research activity among the
scientific community to better understand the transmission mechanisms of this
novel pathogen, project the burden to health care systems, and evaluate the
potential effects of the exceptional interventions that had to be implemented
to slow down the spread of the virus. [1] In this effort, mathematical and
statistical models have proven to be valuable tools in forecasting possible
scenarios. These models can provide health authorities and decision-makers a
scientifically-based tool to provide insight into the adoption of public
health measures, such as restrictions on human mobility and guidance for
resource allocation. [2] Amidst this pandemic, a variety of traditional
mathematical models have been developed. However, modeling teams have
struggled in the effort to capture the transmission dynamics of Covid-19 in
different parts of the world, given the live learning approach of the disease
behavior. [3, 2]

Some highly specialized interdisciplinary research teams have developed
sophisticated mathematical, statistical, and computational models using
publicly available data. [3, 4, 5] However, these experiments must be tailored
specifically to the circumstances of each geographic location using each
location’s available information. Modelling efforts have also faced the
challenges of rapidly adapting to the constant changes in social behavior
based on the various restrictive measures taken worldwide. In this sense,
quantitative methods incorporating a more realistic population structure and
associated contact networks provide a more flexible tool to explicitly model
heterogeneity in host contact patterns. However, this data is not available to
modelers promptly.

As shown in this article, Costa Rica has provided the necessary conditions for
modelling teams to collaborate with health authorities and develop evidence-
based tools to analyze and project possible epidemic scenarios. This central
american country of 5,163,038 inhabitants reported its first Covid-19 patient
on March 6, 2020. [6]. In the days to come ass more cases were confirmed,
public health officials began to implement interventions to promote and
facilitate physical distancing in the seven provinces and 82 cantons the
country is divided in. During the first weeks, the government announced the
cancellation of all massive events, the closing of all schools and
universities, borders, and other social gatherings sites, the implementation
of teleworking for the public and private sector, and restriction to vehicular
circulation, however by March 22,2020 the virus had been detected in all seven
provinces and by August 12, 2020 all of the 82 cantons of the county had
detected Covid-19 cases. In these early stages of the pandemic, a group of
interdisciplinary researchers, experts in mathematical modeling and other
areas, and public health officials started gathering in an attempt to use the
best available mathematical and statistical tools to inform health authorities
in preparation for the inevitable surge in cases to come, with all its medical
and social implications. Our research group started developing a network model
with different contact layers of Covid-19 in Costa Rica and gained progressive
practice to inform public health authorities.

In the months following March 2020, a reduction in cases resulted in the
government lifting several containment measures to balance public health
safety with the impact on the country’s economy. At the same time, the Costa
Rican Social Security Fund (CCSS), the country’s main public health care
provider, and other public and private institutions were working to increment
hospital capacity gradually. Early in the pandemic, CCSS had at their disposal
less than 100 intensive care units (ICU) beds and was in urgent need to
increase ICU capacity and several other competing health priorities.

In July 2020, the Ministry of Health announced the inability to trace the
source of infection of $65\%$ of the positive cases. [6] During that time,
health authorities were working diligently to contact trace confirmed cases;
this became increasingly difficult as the number of patients rapidly
increased. The government’s primary objective at the time was to prevent the
collapse of the hospital care system. The country established a four-color
alert status for every canton (green, yellow, orange, and red) to modify
restrictions according to local infections. [6] On July 11-19, 2020, a
reduction in mobility and closure of businesses was implemented in the cantons
with orange alert status, located mainly in the Greater Metropolitan Area.
This strategy was called ”epidemiological fence” by the Costa Rican
government. [6] During August 2020, the government implemented an open/close
strategy called “dance and hammer.” [7] This strategy involved alternating
between 9 days of less restriction in commercial activity and vehicular
circulation in cantons with an orange alert and 12 days of closure in
commercial activity and limiting vehicular circulation to two days a week for
people living in these cantons.

This article describes the multilayer, temporal and stochastic model developed
for Costa Rica, tailored to project and inform decision-makers of probable
transmission scenarios and the impact of the health care services under
diverse public health interventions. To depict this collaboration, we present
the simulations obtained using this model during July and August 2020, and
analyzed its role in providing insights to the Costa Rican health authorities.

## Methods

### Network Model

A network is a mathematical object composed of nodes and edges (links between
vertices), that in our case, represent individuals in the population and
interactions among them, respectively. In our model, the number of nodes in
the graph is equal to the total population in Costa Rica. An edge connects two
nodes on a specific day if they have contact. We try to mimic how the contacts
of a node (individual) change over time by changing its number of edges
(connections with neighbors) daily. The degree of a node refers to the number
of edges at a particular time.

To differentiate types of contacts in the population, we consider a temporal
multilayer network; see. [8] Moreover, we consider three layers as depicted in
Fig 1. Each layer represents a different type of contact between individuals.
These layers are: (1) a household network (individuals that live in the same
house), (2) a social network (known contacts such as friends and colleagues),
and (3) a sporadic network (strangers that you may encounter in short periods
when you visit random locations). Networks are connected by intra-layer edges
(links in one layer) and inter-layer edges (links between layers). Layers one
and two are fixed for each simulation, while layer three can change each day
since an individual has no control over sporadic encounters.

Figure 1: Representation of contact layers in the network. Three different
layers are included in the network: household, social, and sporadic contacts.

Each node has two types of attributes: (1) fixed parameters, such as canton of
residence, household members, degree and graph connectivity for layer two, and
(2) variable attributes, such as connectivity for layer three, epidemiological
state, number of days at current epidemiological state, number of interactions
with social contacts per day, use of personal protective measures, and self-
care behavior. The former is defined at the beginning of each simulation, and
the latter can take random values every day accordingly. These attributes such
as canton of residence is used for connectivity between individuals; household
members, interactions, degree and graph connectivity for the different layers,
are used to construct the multilayer network. Epidemiological state, number of
days at current epidemiological state, use of personal protective measures,
and self-care behavior are used for the epidemiological model depicted in
Figure 2. For implementation details, we refer to. [9]

### SARS-CoV-2 transmission model

The classical SEIR type dynamics is incorporated into the multilayer network
model to describe the SARS-CoV-2 transmission process. The population is
divided according to the epidemiological status. Thus, the first compartment
corresponds to susceptible individuals. When susceptible individuals become
infected, they move to a latency stage where eventually they become infectious
(undiagnosed or diagnosed). Individuals could potentially recover, require a
hospital bed, move into an ICU or die; see Fig 2. We use a discrete time step
(one day) for transitions between states.

Figure 2: Transmission model. Individual’s epidemiological states for SARS-
CoV-2.

A susceptible individual can become infected by computing its probability of
infection, which depends on its daily interactions with its infectious
contacts (nodes). If the probability that node $j$ infects a susceptible node
$i$ at a given day is $\beta_{ij}$, then the probability of infection $p_{i}$
for node $i$ is given by

$\displaystyle p_{i}=1-\prod_{j\neq i}(1-\beta_{ij}),$

where $j$ includes all the indices of nodes that can infect node $i$ that
particular day.

The value $\beta_{ij}$ may depend on different attributes (such as layer,
epidemiological state, human mobility, use of personal protective measures and
self-care behavior, time of exposure, environment, and other factors).
Depending on the interaction, the value of $\beta_{ij}$ can be reduced
according to contact type (layer), personal protective measures, human
mobility, and sneezing/coughing protocols, among others. In this way, we mimic
the different health measures by setting

$\displaystyle p_{i}=1-\prod_{j\neq i}(1-c_{j}\beta_{ij})^{\gamma}$ (1)

where $c_{j}$ is a reduction percentage that depends on the layer, use of
personal protective measures, and human mobility in the interaction between
nodes $i$ and $j$, respectively (see Table 1), and $\gamma\in(0,1]$ is a
calibration parameter estimated for each simulation. For approximating
$\gamma$, we consider a period of seven days where we calibrate the model by
using a standard bisection method for finding an optimum value for $\gamma$,
such that the square error between observed and predicted cases is minimum.
The value of $\gamma$ is then fixed for all the simulations.

### Data

We describe each set of data that we used in our analysis as follows:

  * •

Contact tracing. The initial conditions of the model are based on the contact
tracing provided by the Ministry of Health. [6] This data allowed us to create
a more realistic contact network for the early stages of the pandemic.
However, starting in July 2020, the number of cases increased, and contact
tracing became a problem for public health authorities. From July 2020, as
community transmission was declared, we only incorporate the number of daily
cases per canton provided by the Ministry of Health. [6]

  * •

Household layer. The number of households per canton and the average number of
individuals per household was taken from the “National Household Survey”, [10]
made by the National Institute of Statistics and Census (INEC) in 2018. In our
model, the number of inhabitants of each household is determined by a Poisson
distribution with a mean equal to the average number of inhabitants per
household per canton. Total connectivity is assumed in each household with no
reduction in the probability of infection ($c_{j}=1$).

  * •

Mobility matrix. Possible connections between nodes in different cantons are
selected according to a connectivity matrix that provides the place where
individuals reside and work, according to the “National Household Survey”.
[10] Connections (edges) between individuals of two different cantons may
exist if there is mobility between them. The degree of a node in layers two
and three is calculated using a uniform distribution on a given interval; [11]
see values in Table 1. We assume a fixed set of social contacts in layer two,
but daily interactions are limited by choosing a subset of its edges.

Table 1: Network parameters. (Left) Degree node distribution for layers two
and three; GMA stands for Greater Metropolitan Area. (Right) Infection
parameters for Eq (1). PM and HM stand for protective measures and human
mobility, respectively.

| Interval
---|---
Contacts in sporadic network | $[0,20]$
Contacts in social network | $[5,30]$
Social contacts per day, GMA | $[5,25]$
Social contacts per day, off GMA | $[5,15]$

| $c_{j}$ | Ref.
---|---|---
Sporadic contacts | 0.5000 | Est.
PM | 0.1476 | [20]
HM | 0.1238 | [20]
PM and HM | 0.0182 | [20]

  * •

Proportion of undetected individuals. Asymptomatic and pre-symptomatic
individuals play an essential role in the spread of the virus. [12, 13]
However, the proportion of infected individuals who remain asymptomatic and
the transmission rate were not evident during the period when the first
simulations were conducted. [14, 15] The scientific literature reported
different percentages of asymptomatic individuals worldwide. [16, 17] Based on
this information and the testing capacity in the country, we set the
percentage of undetected persons at $25\%$. This percentage accounts for
asymptomatic individuals and those who were not captured by the health system
even with symptoms, either because they did not consult a health care center
or due to the country’s testing capacities. In mid-2020, Costa Rica conducted
3·3 tests per 10,000 inhabitants on average, with a positivity rate of $25\%$.
[18, 6] To date, the World Health Organization (WHO) recommendation was that
positivity rates should remain at $5\%$ or lower for reopening. [19]

  * •

Protective measures. Previous studies [20, 21, 22] have shown that
incorporating appropriate personal protective measures and restricting human
mobility reduce the probability of infection. If a particular interaction
between two nodes involves using personal protective measures, then we reduce
the probability $\beta_{ij}$ accordingly; see Table 1. We assumed that the
percentage of individuals who effectively used personal protective measures in
July and August 2020 was $50\%$ (the use of masks was recommended but not
mandatory). We also included a parameter that accounts for the percentage of
individuals circulating and its impact on human mobility (represented as HM in
Table 1), which is affected by public health restrictions and the
consequential social behavior changes. To simulate the different levels of
restrictions or the lifting of measures, we reduce or increase the value of
this parameter, respectively. Furthermore, we decrease the number of
connections (edges) between nodes in different cantons to simulate the
mobility restriction between specific locations.

  * •

Epidemiological parameters. Parameters related to hospitalization and ICUs
were provided by the Costa Rican Social Security Fund (CCSS), and parameters
related to the transmission are from the literature; see Table 3 and
hospitalization parameters are provided by the CCSS (see Table 2). We have
used the exact number of days for transitions between states.

Table 2: Hospitalization parameters. Values were provided by CCSS. Parameter |  July 12-16, 2020 |  July 22, 2020 |  August, 2020
---|---|---|---
Diagnosed people requiring hospitalization (%) |  6.0 |  8.0 |  8.0
Hospitalized people requiring ICU (%) |  17 |  17 |  18
Average days a person stays in ICU |  17.4 |  17.4 |  18.1
Average days a person stays in ward |  9.6 |  10.2 |  10.2
Mortality in ward (%) |  2 |  4 |  5
Mortality in ICU (%) |  60 |  46 |  60
Table 3: Epidemiological parameters. Values are updated throughout the pandemic as new information is available. The probability of infection was estimated using an Approximate Bayesian Computation [23] method at the beginning of the pandemic. Parameter |  Value |  Ref.
---|---|---
Average days from onset of symptoms to hospitalization |  7 |  [24]
Average days from onset of symptoms to ICU | 11 |  [17, 25]
Recovery time (days) |  14 |  [26]
Percentage of diagnosed people |  75% |  [16, 17]
Percentage of diagnosed people who do not isolate themselves | 10% |  [14]
Incubation period (days) |  6 |  [27, 17]
Probability of infection $\beta_{ij}$ |  0.21 |  Est.
Percentage of undiagnosed people requiring hospitalization |  1% |  Est.
Mortality out of hospitalization |  0.008% |  Est.

### Scenarios performed in July 2020

In response to the requests that arose from the health authorities during
different periods, we proposed three scenarios to evaluate and observe the
impact of interventions on reducing cases and hospital admissions. Each
scenario was designed and implemented on different dates. Differences were
related to the reopening assumption after the “epidemiological fence” and the
percentage of people adhering to the restrictions.

  1. 1.

Scenario 1: July 12, 2020.

     * –

Restrictions based on canton alert status are removed from the simulation
after July 19, 2020.

     * –

It is assumed that the $70\%$ of the country maintained human mobility during
that period.

  2. 2.

Scenario 2: July 16, 2020.

     * –

Restrictions based on canton alert status are maintained until August 2, 2020
only in cantons that remain in orange alert status.

     * –

It is assumed that the $70\%$ of the country maintained human mobility during
that period.

  3. 3.

Scenario 3: Jul 22, 2020.

     * –

Restrictions based on canton alert status are removed from the simulation
after July 19, 2020.

     * –

Percentage of individuals that maintained human mobility changed gradually in
places with an orange alert status: $70\%$ from July 11 to 19, 2020, $60\%$
from July 20 to 31, 2020, and $50\%$ after July 31, 2020. In cantons with a
yellow alert status, $50\%$ of the population maintained human mobility.

In all the scenarios, we assumed that (i) $50\%$ of the population used
protective measures and the virus transmission probability is reduced
accordingly, and (ii) the number of contacts is sampled from a uniform
distribution; see Table 1. We also assumed that from July 11-19, 2020,
individuals living in areas where the “epidemiological fence” was implemented
could only contact people from areas with the same alert status.

### Scenarios performed in August 2020

In August 2020, public health authorities decided to implement a strategy of
closing and opening periods called the hammer and dance; [7] details about the
interventions are on the official website of the Ministry of Health of Costa
Rica. [6]

To help inform the potential effects of this new strategy, we simulated a set
of scenarios considering several closing periods of 12 days and opening
periods of nine days for two and a half months. Human mobility restriction was
simulated considering two assumptions: the first related to the percentages of
individuals who contributed to new infections (human mobility parameter (HM)).
The canton alert differentiated the parameter value. The second was related to
the connections of the individuals (nodes) with people of different cantons
(mobility restriction (MR)). We restricted the interaction of individuals in
cantons in orange alert to only contact individuals within the same canton. On
the other hand, people living in cantons with a yellow alert could contact
those with the same yellow alert. Scenarios are summarized in Table 4.

Table 4: Scenario assumptions performed in August, 2020. MR stands for
mobility restrictions between cantons with different levels of alert. In the
model, people living in cantons with an orange alert only had interactions
with people living in the same canton.

Level of alert |  Open/Close strategy |  Scenario 1 50%-70% |  Scenario 2 50%-60% |  Scenario 3 50%-65% |  Scenario 4 45%-55% |  Scenario 3 30%-40%
---|---|---|---|---|---|---
Yellow |  Dance |  50% HM |  50% HM |  50% HM |  45% HM | 30% HM
Yellow |  Hammer |  50% HM |  50% HM |  50% HM |  45% HM | 30% HM
Orange |  Dance |  9 days open
50% HM |  9 days open
50% HM |  9 days open
50% HM |  9 days open 45% HM |  9 days open
30% HM
Orange |  Hammer |  12 days closed
70% HM and MR |  12 days closed
60% HM and MR |  12 days closed
65% HM and MR |  12 days closed
55% HM and MR |  12 days closed
40% HM and MR

During the hammer period, restrictions on vehicular circulation and the
opening of commerce were applied only in the cantons with an orange alert
status. Every week, the alert status changed. As an attempt to predict which
cantons would change status, we introduced an indicator given by:

$\text{{Cantonal Hazard Rate}}=\dfrac{\dfrac{\text{new cases in the last 3
weeks in a canton}}{\text{canton population}}}{\dfrac{\text{new cases in the
last 3 weeks in the country}}{\text{country population}}}$

If the Cantonal Hazard Rate is greater than one in a canton with a yellow
alert status, it changes to an orange alert. All cantons with an orange alert
status do not change status. Finally, connections between nodes are only
allowed in places with the same alert status (yellow or orange) in the hammer
phase.

## Results

All simulations were implemented in Matlab R2020a [28] and run in two remote
servers: (i) a Dell PowerEdge R740 with 64GB of RAM and two Intel®(R) Xeon®(R)
Silver 4114 CPU @ 2.20GHz processors, and (ii) a Lenovo SR650, with two Intel®
Xeon® Plata 4214, 2.20 GHz processors with 128GB of RAM. The results presented
below correspond to the simulations and scenarios implemented in July and
August 2020, modeled in real-time. Therefore, the number of simulations per
scenario varies between 20 to 60 since it depends on the response time. We
present the average of the total simulations in each scenario.

### Scenarios conducted in July 2020

Our results showed that infections and hospitalizations were expected to
increase abruptly if, after only a week of restrictions, measures were lifted
to levels similar to those before the epidemiological fence (July 12, 2020,
scenario). This rapid growth could have been controlled partially if the
reopening occurred gradually, as shown in the July 22, 2020, scenario.
However, if restrictions were maintained for three weeks in cantons under
orange alert (July 16, 2020, scenario), the model showed a significant
reduction in infections, hospitalizations, and ICU admissions; see Fig 3.

Figure 3: Scenario in July: Cumulative confirmed Covid-19 cases (Left),
hospitalizations (Center), and ICU (Right) admissions.

### Scenarios conducted in August 2020

Results showed that establishing the three closing phases of the “Hammer and
Dance” period was expected to curtail the rise in cases successfully, but only
if they could restrict population mobility in ranges over $50\%$. Furthermore,
ICU admissions were only expected to remain within the 159 threshold with
$50-65\%$ population adherence.

There were also three abrupt decreases in hospitalization as a consequence of
the three closing periods and the assumptions related to the connectivity of
individuals since we assumed that each node only interacted with contacts in
cantons with the same yellow alert and restricted the connection to people in
the same canton for those living in cantons with orange alert see Fig 4.

Figure 4: Scenario in August: Cumulative confirmed Covid-19 cases (Left),
hospitalizations (Center), and ICU (Right) admissions. Percentages in the
legend correspond to changes in the human mobility (HM) parameter. The first
values are for cantons with a yellow alert, and the second values are for
cantons with an orange alert.

## Discussion

A multilayer network model was developed in a joint effort between members
from the Ministry of Health, the Pan American Health Organization (PAHO), and
scientists from academia. The model’s flexibility allowed us to incorporate a
socio-demographic component, public health interventions, and the basic
epidemiological characteristics of the virus transmission. The primary use of
the model focused on three aspects: short and mid-term projection of the
number of cases and hospital demand, simulation of hypothetical scenarios that
helped health authorities and government officials in the design of preventive
control measuresVaccine administrated by locationVaccine administrated by
location, and finally, on simulating the impact of interventions (results were
reported by PAHO in the monthly reports on the Covid-19 pandemic in Costa
Rica). [29]

The findings presented here show the experience of real-time modelling
scenarios developed according to the necessities of Costa Rican public health
officials during July and August 2020. After lifting most restrictions due to
a low case count, the country started experiencing increased cases and
hospitalizations. This increase led decision-makers to implement an
“epidemiological fence” from 11 to July 19, 2020. Scenarios projected on July
12, 2020, warned of an expected abrupt increase in cases and hospitalizations
if restrictions were lifted after only a week. Instead, maintaining them for
three weeks was expected to contain the growth in disease transmission.

In August 2020, decision-makers needed to balance public health with economic
recovery. The Hammer and Dance Period was an attempt to achieve precisely
that. Results showed that such strategy was expected to contain the disease
only if population mobility was reduced in ranges over $50\%$ during closure
phases. We did not know for sure if and how effective these sanitary measures
were at reducing human mobility. We performed a parallel analysis [22]
describing the impact that reduction in mobility had on infections and
hospitalization. However, we also anticipated that the same intervention
imposed at different pandemic stages was unlikely to produce the same results.
A population growing tired of confinement and those with precarious employment
are likely to countermeasure the effect of restrictions. The use of masks was
not mandatory at that time but was under consideration. In our model, we set
the use of personal protection measures at $50\%$. This parameter became a
factor of adjustment for subsequent scenarios.

It is crucial to highlight the implications of the analysis in hospital
occupancy. By July 2020, Costa Rica had been working on increasing its ICU
capacity from 67 to 159 beds (0.013 to 0.03 beds per 1000 hab) for Covid-19
and had the goal to reach 359 (0.07 beds per 1000 hab) by the end of the year.
Intensive care services are offered mainly by the three leading hospitals in
the country, the Calderón Guardia Hospital, the Mexico Hospital, and the San
Juan de Dios Hospital located in the city of San José also known as the Great
Metropolitan Area. Therefore, patients must be transferred from anywhere in
the country to the Great Metropolitan Area, which implies additional
logistical considerations. Lifting restrictions after only one week of
epidemiological fence presented a scenario of rapid collapse in hospital
capacity. Our results suggested that adopting a Hammer and Dance strategy
could potentially contain this collapse and gain valuable time for the health
care system to reach its goal of 359 ICU beds. However, these measures alone
could not prevent this collapse if population adherence to mobility
restrictions was lower than $55\%$. Furthermore, with adherence lower than
$45\%$ the projected ICU admissions could surpass the threshold of 359 ICU
beds. After the public health interventions, the average ICU occupancy in July
and August was 0.0068 and 0.0022 patients per 1000 inhabitants, respectively.

One of the challenges and limitations of the analysis is how difficult it is
to know what variable is attributable to which effect. Sanitary measures were
imposed in an ever-changing social scenario where people’s adherence is as
relevant as the interventions themselves. If our model predicted that certain
restrictions, based on past data, were going to reduce cases and then cases
did not drop, it is as likely that the people responded differently to such
measures as that the model was wrong. Indeed, it would probably be a
combination of both, plus several other factors. Hence, the model’s value must
not be judged by the accuracy of its forecasts but by the value in informing
decisions and their impact in containing the disease, the collapse of health
care systems, and ultimately saving lives, highlighting the value of close
collaboration between researchers and policy-makers.

The ongoing global health crisis has evidenced the challenges of modeling a
real-time pandemic and the importance of having an interdisciplinary research
team working hand in hand with health authorities. Effective communication
between scientists, public health entities, and government officials is vital
in developing mathematical, statistical, and computational models that can
provide insight to create successful disease prevention and control tools in a
practical setting.

Counting with adequate, robust models, sophisticated enough to capture the
changing dynamics and variety of factors interacting simultaneously in a
pandemic of the scale of Covid-19 could be limited by the computational
capacity and the available data. Furthermore, what we know about the virus is
still very limited. After more than a year of this pandemic, virus
transmission dynamics are still a subject of research. Many open questions
remain, and the assumptions have to be constantly updated.

## Contributors

Y.G. and P.V. designed the study, performed simulations, analyzed data, and
wrote the initial draft. G.M. and F.S. designed the study, interpretation and
analysis of the results, and edited the manuscript. J.C. coded the model and
edited the manuscript. L.B., T.R. and M.D.P. designed the study and edited the
manuscript. All authors reviewed the manuscript. G.M. and M.D.P. are staff
members of the Pan American Health Organization. The authors alone are
responsible for the views expressed in this publication, and they do not
necessarily represent the decisions or policies of the Pan American Health
Organization.

## Acknowledgments

The authors would like to thank the Research Center in Pure and Applied
Mathematics and the School of Mathematics at Universidad de Costa Rica for
their support during the preparation of this manuscript. They also thank the
Ministry of Health and CCSS for providing data and valuable information for
this study.

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