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Article accepted for publication in Cadernos de Saúde Pública

(Reports in Public Health) on April 28, 2020.

Vertical social distancing policy is ineffective

to contain the coronavirus COVID-19 pandemic

Luiz Henrique Duczmal1

Alexandre Celestino Leite Almeida2
Denise Bulgarelli Duczmal3
Claudia Regina Lindgren Alves4,*
Flávia Costa Oliveira Magalhães5
Max Sousa de Lima6
Ivair Ramos Silva7
Ricardo Hiroshi Caldeira Takahashi8

1
Departamento de Estatística, UFMG, duczmal@ufmg.br
2
Departamento de Estatística, Física e Matemática, UFSJ, celestino@ufsj.edu.br
3
Departamento de Matemática, UFMG, bulgarelli@ufmg.br
4
Departamento de Pediatria, UFMG, lindgren@medicina.ufmg.br
5
Polícia Civil de Minas Gerais, draflaviamagalhaes@gmail.com
6
Departamento de Estatística, UFAM, maxlima@ufam.edu.br
7
Departamento de Estatística, UFOP, ivairest@gmail.com
8
Departamento de Matemática, UFMG, taka@mat.ufmg.br
All authors are members of the UFMG - COVID-19 Task Group
*
Corresponding author

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Abstract
We show, through numerical simulations, that the so-called Vertical Social Distancing health policy
(‘Isolamento Vertical’) is ineffective to contain the COVID-19 pandemic. We present the SEIR-Net model,
for a network of social group interactions, as a development of the classic mathematical model of SEIR
epidemics (Susceptible - Exposed - Infected (symptomatic and asymptomatic) - Removed). In the SEIR-
Net model, we can simulate social contacts between groups divided by age groups and analyze different
strategies of social distancing. In the vertical distancing policy, only the elderly are distanced, against the
horizontal distancing policy, where all age groups adhere to social distancing. These two scenarios are
compared to a control scenario in which no intervention is made to distance people. The vertical distancing
scenario is almost as bad as the scenario where no distancing is done, both in terms of the number of infected
and in the acceleration of the number of cases. On the other hand, horizontal distancing, if applied with the
same intensity in all age groups, significantly reduces the total number of infected and "flattening the
disease growth curve." Our analysis is done for the municipality of Belo Horizonte, but similar conclusions
apply to other cities as well. An R language program is provided.

Resumo
Mostramos, através de simulações numéricas, que o chamado Isolamento Social Vertical é ineficaz para
conter a pandemia COVID-19. Apresentamos o modelo SEIR-Net, para uma rede de interações de grupos
sociais, como um desenvolvimento do modelo matemático clássico de epidemias SEIR (Suscetível -
Exposto - Infectado (sintomático e assintomático) - Recuperado). No modelo SEIR-Net podemos simular
contatos sociais entre grupos divididos por faixas etárias e fazer análise de diferentes estratégias de
isolamento (ou mais corretamente, distanciamento) social. No distanciamento vertical apenas as pessoas
idosas são distanciadas socialmente, e no horizontal pessoas de todas as faixas etárias aderem ao
distanciamento. Esses dois cenários são comparados ainda a outro cenário de controle em que nenhuma
intervenção é feita para se distanciar socialmente as pessoas. Concluímos que o cenário de distanciamento
vertical é quase tão ruim quanto o cenário em que nenhum distanciamento é feito, tanto em termos do
número de infectados como na aceleração do número de casos. Por outro lado o distanciamento horizontal,
se aplicado com a mesma intensidade em todos os grupos etários, reduz significativamente o número total
de infectados e a aceleração do número de casos, "achatando a curva" de crescimento da doença. Nossa
análise foi feita para o município de Belo Horizonte, mas conclusões similares valem também para outras
cidades. Um programa em linguagem R é disponibilizado.

Introduction

In Brazil, there is a widespread belief that the so-called vertical social distancing health policy,
just restricting social contact with the elderly (and higher risk individuals), would be enough to
contain the propagation of the SARS-CoV-2 coronavirus disease (COVID-19). This notion is
based on the premise that people under the age of 60 would suffer only mild symptoms and could
leave their houses to work and study during the epidemic. However, we have observed a high
number of hospitalizations, with severe cases and deaths also of people under the age of 60 years
old and without underlying diseases. Besides, social distancing is not as a rule 100 percent strict,

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and the elderly tend to make social contacts during the period, thus increasing the likelihood of
being infected. Through this work, we shall use the terms 'social distance' and 'social isolation'
interchangeably to indicate a reduction in the intensity of social contact.

Measures of social distance in the COVID-19 pandemic are already proving its effectiveness,
favoring the reduction of the number of infected people (1-7). Why is it important? Mainly because
we want the maximum peak of the epidemic to be minimized, meaning that hospitals are not
overcrowded with a large number of people with severe manifestations of COVID-19 requiring
intensive care simultaneously. This goal of public health services is popularly known as "flattening
the curve" of cases and hospitalizations. If there are not enough hospital beds to serve everyone,
many people may die simply from lack of care. Postponing the peak of cases would be potentially
beneficial so that health managers could be better prepared, and researchers would find more
effective treatments. Therefore, if social distancing can reduce the peak of cases of infected people,
and at the same time postponing its occurrence, many lives can be saved.

We will analyze these problems with a mathematical technique for simulating the evolution of
epidemics, the SEIR-Net model, obtained by us from a modification of the traditional SEIR model
(Susceptible - Exposed - Infected- Removed). In the SEIR model, people susceptible to infection
randomly come into contact with the SARS-CoV-2 virus becoming exposed. After the incubation
period, they become infected and become able to pass this virus at random to other susceptible
people. Infected people can be asymptomatic (have few or no symptoms) or symptomatic (develop
typical symptoms of COVID-19 infection). Infected people become, over time, removed (a
technical term to say that they cannot infect other people and may survive or die). In this model,
we use an estimate of the number of unreported cases, based on the number of reported (confirmed)
cases. This extrapolation/estimation is due to the fact that, in practice, it is not possible to test the
entire population at all times. In Brazil, it was estimated that there are at least 20 times more
unreported than reported cases. This estimation was based on model fitting using a mathematical
SEIR model for the disease spread in Minas Gerais (8). Still, this number now is probably much
higher, mainly due to the scarcity of available test kits. It is important to note that our model
assumes that the contacts among persons follow a uniformly random pattern of interaction, and
apart from the age groups, there is no spatial (geographic) restriction to social contact – that is

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embedded into the B transmission parameter, which is determined empirically from the observed
data at the beginning of the epidemic.

Some parameters are of interest in this simulation, such as the average incubation time Z, the
average infectious period D (for how many days the infected individual can infect others), and the
fraction of asymptomatic infected individuals which are still capable of infecting others (albeit
with less intensity). An important parameter, which does not depend only on the virus, is the
transmission rate B; it depends on the country's health system and the environment in which people
live. If the value of B is high, it means that the virus tends to spread more quickly in the population.
All COVID-19 parameters used in this work were obtained from the article (9) and adapted to the
observed case data from Belo Horizonte (8). Publicly available population data was used (10).

In the next section, we will build our new Model SEIR-Net with social distancing and network
interaction. In the following section, we will present several scenarios simulating different
conditions of vertical and horizontal social distancing in Belo Horizonte and study their impact on
reducing the number of simultaneously infected people. Code implementation of the model in R-
language is provided in the supplementary material.

The SEIR-Net model

The model proposed in this work is a development of the model used in (8), and generalizes the
SEIRis model proposed in (11). The recent model by (3) also uses the SEIR model with a
partitioning of the population into groups and considers their interaction. For a detailed description
of these models, see the links in the References section.

The SEIR-Net model divides the population between n social distancing groups and uses the F
Contact Fraction Matrix, given by

𝐹11 ⋯ 𝐹1𝑛
𝐹=( ⋮ ⋱ ⋮ )
𝐹𝑛1 ⋯ 𝐹𝑛𝑛

where the entry Fij indicates the contact intensity of virus transmission from an individual in the
group i to an individual in the group j, where 0 ≤ Fij≤1. If Fij= 1, then the contact is not restricted,
and if Fij = 0 no individual in the group i can transmit the virus to any individual in the group j.
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This system forms a connection network between the n groups. In our work, we will use groups
formed by age groups in the city of Belo Horizonte. In future work, we will extend this idea to
groups divided by income level, place of residence or work, occupation, etc.

How will the COVID-19 epidemic evolve in this case? In the next section we will analyze
scenarios with different structures for the F matrix.

Case Studies

In the SEIR-Net model, we can simulate social contacts between groups divided by age groups
and analyze different strategies of social distancing.

The population of the municipality of Belo Horizonte, MG, with approximately 2.5 million
inhabitants, has the following age distribution interpolated for 2015 (10):

0-9 years old: 11.7 %

10-24 years old: 21.8 %

25-59 years old: 52.3 %

60+ years old: 14.2 %

Initially, we present a control scenario in which no distancing intervention is performed. In this

case, all elements Fij of the F matrix are equal to 1.

In vertical distancing, only people aged 60+ years old are socially distanced. The F matrix is given
by

1 1 1 𝑐
1 1 1 𝑐
𝐹=
1 1 1 𝑐
𝑐 𝑐 𝑐 𝑐

where c = 1/k means that contacts between individuals aged 60+ years old have a k-fold reduced
social contact with individuals of all age groups and vice versa.

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Finally, in the scenario of horizontal distancing, individuals of all age groups adhere to distancing.
The F matrix becomes

𝑐 𝑐 𝑐 𝑐
𝑐 𝑐 𝑐 𝑐
𝐹=
𝑐 𝑐 𝑐 𝑐
𝑐 𝑐 𝑐 𝑐

where the value of c depends on the social contact reduction factor. A value of c = 1∕15 corresponds
to the estimated social contact reduction factor for New York City during the end of March, with
a 15-fold social contact reduction, meaning that the social contacts where reduced by 1 - 1∕15 =
93% (2). When c=1, we go back to the control scenario, with 0% social contact reduction. As we
will see, a value of c= 0.55 (1.8 fold, or 45% reduction of social contact) for the above matrix is
consistent with the observed data in Belo Horizonte city.

Below we present the results of the simulations using the SEIR-Net model.

The four dashed curves measure the cumulative number of individuals who have been infected
over time for each age group, according to the legend. The solid curve indicates the total number
of individuals from all age groups which are currently infected. The thin dotted horizontal lines
indicate the number of persons for each age group, thus showing the ceiling for the accumulated
number of possible infected persons in each group.

The numbers in parentheses indicate the approximate percentage of each age group within the
population as a whole.

In the control scenario (without distancing) of Figure 1, a maximum of about 500 thousand
simultaneously infected people is reached about 55 days after the beginning of the epidemic. The
number of infected people is very high for all age groups. In particular, within the age group of
60+ years old, we would have more than 350 thousand infected people accumulated over the
period.

[Figure 1 about here]

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In the scenario of vertical distancing, with a 4-fold (75%) reduction of social contact intensity for
only the 60+ years old age group (Figure 2), a maximum of about 400 thousand simultaneously
infected people is reached approximately 65 days after the beginning of the epidemic. The number
of infected people is very high for all age groups. In the age group of 60+ years old, we would
have more than 200 thousand infected people accumulated over the period. In other age groups,
virtually everyone has been infected.

[Figure 2 about here]

The scenario of horizontal distancing with the same 4-fold (75%) reduction of social contact
intensity, for all age groups, is shown in Figure 3. The epidemic does not reach significant
dimensions in the first 180 days of simulation. As can be seen in Figure 4, the number of
simultaneously infected people only becomes significant about 18 months later, with a relatively
small number of simultaneously infected people (less than 10 thousand).

[Figure 3 about here]

[Figure 4 about here]

By the end of March, it was estimated that social contacts decreased between 30% to 50%
(corresponding to contact intensity between 0.50 and 0.70) in Belo Horizonte (12). Using an
intensity of c= 0.55, corresponding to a value of a (1/0.55) = 1.8-fold (or 45%) reduction in social
contact, we obtain the graph of Figure 5. It is immediately apparent that this level of 1.8-fold
reduction is not sufficient to deter the epidemic outbreak, as could be observed by the accumulated
case's curves reaching more than 85% of the ceiling limit of the number of infected persons for all
groups. The peak of simultaneous infections (more than 200 thousand) is reached after about 105
days.

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 [Figure 5 about here]

Conclusions

The vertical distancing scenario is only marginally better than the situation in which there is no
social distancing at all, and much worse than the horizontal distancing scenario, with an equivalent
level of reduction in social contact.

Vertical distancing with a 4-fold (75%) reduction in social contact, only for the 60+ years old age
group, could not prevent a large number of infected elderly (more than 200 thousand) from
appearing, with 350 thousand individuals simultaneously infected. It would also cause a massive
flow of patients requiring immediate hospitalization. That would quickly exceed the bed capacity
in the Belo Horizonte hospital network.

The proposed horizontal distancing, with a similar 4-fold (75%) reduction for all age groups,
should slow the surge of cases, postponing the cases' peak for about two years. That should relieve
the hospital network, reducing the number of fatal victims, and still allowing future interventions
that may occur later (vaccination, new treatments, etc.).

However, this 4-fold reduction is far from being adopted by the general population. Using mobility
data for Belo Horizonte, a value of only a 1.8-fold (45%) reduction in social contact was achieved
during the last two weeks, which is clearly not sufficient to deter the epidemic outbreak. An urgent
effort is recommended to improve social contact reduction through a stricter horizontal distancing
health policy for several months.

Acknowledgments
We thank the Editor and Reviewers for their thoughtful comments, which improved the quality
of the paper. We also thank the UFMG COVID-19 Task Group and the UFMG contingency
committee.

References
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http://www.bmj.com/cgi/doi/10.1136/bmj.b3675

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2. Bakker M, Berke A, Groh M, Pentland AS, Moro E. Effect of social distancing measures
in the New York City metropolitan area Main findings. Boston; 2020. p. 17.
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6. Adam D. The simulations driving the world’s response to COVID-19 How
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“infodemics” in response to COVID-19 epidemics [Internet]. Ithaca; 2020. Available
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19 em Belo Horizonte (23/03 a 29/03/2020) [Internet]. Belo Horizonte; 2020 [cited 2020
Apr 3]. p. 22. Available from:
https://drive.google.com/file/d/1dkOfGHZBwuxiAhScRVvC2saQw4CbjhQa/view
9. Li R, Pei S, Chen B, Song Y, Zhang T, Yang W, et al. Substantial undocumented infection
facilitates the rapid dissemination of novel coronavirus (SARS-CoV2). Science (80- )
[Internet]. 2020 Mar 16;3221(March):eabb3221. Available from:
https://www.sciencemag.org/lookup/doi/10.1126/science.abb3221
10. Brasil. Ministério da Saúde. Secretaria de Vigilancia em Saúde. DATASUS. Informações
de Saúde (TABNET) demográficas e socioeconômicas. População residente - estudo de
estimativas populacionais por município, idade e sexo 2000-2015 - Brasil [Internet]. 2020
[cited 2020 Apr 3]. p. 1. Available from:
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11. Duczmal LH, Almeida ACL, Duczmal DB. Avaliação de cenários de isolamento social
para a pandemia COVID-19 no Município de Belo Horizonte [Internet]. Belo Horizonte;
2020 [cited 2020 Apr 3]. p. 12. Available from:
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Figure 1: Control scenario without any social distancing, presented for comparison purposes.

10
Figure 2: Vertical distancing, only with the 60+ years old age group socially distanced (4-fold, or
75% reduction). This scenario is almost as unfavorable as the scenario in which there is no
distancing at all.

11
Figure 3: Horizontal distancing, with a 4-fold social contact intensity factor (75% reduction) for
all age groups. The epidemic does not reach significant levels in the first 180 days of simulation.

12
Figure 4: Same as the previous scenario, here displayed for five years, for the horizontal distancing,
with a 4-fold social contact intensity factor (75% reduction) for all age groups. The epidemic only
manifests itself in a reduced way, about 18 months later.

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Figure 5: Scenario using a 1.8-fold (45%) social contact reduction. This is not enough to deter the
epidemic.

14
Appendix

SEIR-Net model description

The SEIR-Net Model is an extension of the SEIR epidemiological model. With the population
partitioned into n groups, the SEIR-Net model consists of a system 4n ordinary differential
equations and a set of initial conditions. The system solution consists of 4n time functions in days,
which show the evolution of the epidemic's variables (number of susceptible, exposed, infected
(reported and unreported) and removed over time.

SEIR-Net Model Parameters

The following variables and parameters are used in the SEIR-Net Model:

Ntotal = Total number of inhabitants (N = 2500000 for Belo Horizonte)

ng = number of groups (4 in BH)
fracg = vector of the population fractions of each of the ng groups
((0.1170,0.2176,0.5233,0.1421) in BH)
N = vector of the total number of individuals (N = Ntotal * fracg)
S = vector of the total number of susceptible
E = vector of the total number of exposed
Ir = vector of reported number of infected
In = vector of the number of unreported infected
I = vector of the total number of infected (reported or not)
R = vector of the total number of removed
(Although S, E, I and R vary, the N = S + E + I + R ratio is always valid.)
mu = reducing factor for the transmission rate of the unreported infected.
(Here mu = 1 was used)
F = Contact Fraction Matrix, where the entry Fij indicates the contact intensity of virus transmission from
an individual in the group i to an individual in the group j
B = parameter of virus transmission to individuals in distancing
(B = 1,226, obtained by adjusting least squares for Belo Horizonte data)
alpha = Proportion of infected people who will be registered as reported cases
(alpha = 0.05: there are 20 times more current cases than reported cases)
Z = Average incubation period (Z = 3.69 days, according to [Li2020])
Dr = vector of average duration of the infectious period in reported cases
Dn = vector of Average duration of the infectious period in unreported cases
(Dr and Dn use values of 3.48 days, according to [Li2020])
nmax = duration in days of the simulation

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SEIR-Net System of Differential Equations

The SEIR-Net model is governed by a system of 4*ng differential equations. To exemplify, we

will explicitly show the equations for four groups in the population (ng = 4) with an interaction
network between the 4 age groups using the 4x4 matrix F = [F [i, j]], where F [i, j] is the social
contact factor that measures contact intensity between an individual in group i who transmits the
virus to an individual in group j. The symbol ()'indicates derivative in relation to time.

I[1] = Ir[1] + mu*In[1]

I[2] = Ir[2] + mu*In[2]
I[3] = Ir[3] + mu*In[3]
I[4] = Ir[4] + mu*In[4]
(S[1])'=-(F[1,1]*I[1] +F[2,1]*I[2] +F[3,1]*I[3] +F[4,1]*I[4])*(B/Ntotal) * S[1]
(S[2])'=-(F[1,2]*I[1] +F[2,2]*I[2] +F[3,2]*I[3] +F[4,2]*I[4])*(B/Ntotal) * S[2]
(S[3])'=-(F[1,3]*I[1] +F[2,3]*I[2] +F[3,3]*I[3] +F[4,3]*I[4])*(B/Ntotal) * S[3]
(S[4])'=-(F[1,4]*I[1] +F[2,4]*I[2] +F[3,4]*I[3] +F[4,4]*I[4])*(B/Ntotal) * S[4]
(E[1])'= (F[1,1]*I[1]+F[2,1]*I[2]+F[3,1]*I[3]+F[4,1]*I[4])*(B/Ntotal) * S[1] -E[1]/Z
(E[2])'= (F[1,2]*I[1]+F[2,2]*I[2]+F[3,2]*I[3]+F[4,2]*I[4])*(B/Ntotal) * S[2] -E[2]/Z
(E[3])'= (F[1,3]*I[1]+F[2,3]*I[2]+F[3,3]*I[3]+F[4,3]*I[4])*(B/Ntotal) * S[3] -E[3]/Z
(E[4])'= (F[1,4]*I[1]+F[2,4]*I[2]+F[3,4]*I[3]+F[4,4]*I[4])*(B/Ntotal) * S[4] -E[4]/Z
(Ir[1])'= alpha*E[1]/Z - Ir[1]/Dr[1]
(Ir[2])'= alpha*E[2]/Z - Ir[2]/Dr[2]
(Ir[3])'= alpha*E[3]/Z - Ir[3]/Dr[3]
(Ir[4])'= alpha*E[4]/Z - Ir[4]/Dr[4]
(In[1])'= (1-alpha)*E[1]/Z - In[1]/Dn[1]
(In[2])'= (1-alpha)*E[2]/Z - In[2]/Dn[2]
(In[3])'= (1-alpha)*E[3]/Z - In[3]/Dn[3]
(In[4])'= (1-alpha)*E[4]/Z - In[4]/Dn[4]

Initial conditions

The system described above is initialized with 1 exposed individual, divided proportionally
between the ng groups according to their proportions in the population:

E = fracg
S = N-fracg
I = [0, ..., 0]
R = [0, ..., 0]
h = 1.0 (one day step)
nmax = 120 days

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Numerical Solution

The SEIR-Net System is solved numerically with the Fourth-order Runge-Kutta method.

Program in R Language

The SEIR-Net method was implemented in the R language and can be downloaded from the
following link.

https://drive.google.com/open?id=19Jun9HP_7GBv-o_PcR1yUFjZAijAmN0O

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