Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 64, pp. 1–17.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ASSESSING THE EFFICIENCY OF DIFFERENT CONTROL

STRATEGIES FOR THE COVID-19 EPIDEMIC

CÉSAR CASTILHO, JOÃO A. M. GONDIM,

MARCELO MARCHESIN, MEHRAN SABETI

Abstract. The goal of this work is to analyze the effects of control policies for

the coronavirus (COVID-19) epidemic in Brazil. This is done by considering an

age-structured SEIR model with a quarantine class and two types of controls.
The first one studies the sensitivity with regard to the parameters of the basic

reproductive number R0 which is calculated by a next generation method.

The second one evaluates different quarantine strategies by comparing their
relative total number of deaths.

1. Introduction

At the end of 2019 several cases of pneumonia of unknown etiology were detected
in Wuhan City in the Chinese province of Hubei. The Chinese Country Office
of the World Health Organization (Website https://www.who.int) was informed
and reported that a novel coronavirus (officially named COVID-19) was identified
on January 7, 2020, as the cause of such infection. The imminent potential for
worldwide spread was soon recognized and an international alert was issued.

COVID-19 was shown to be very lethal and easily spreading. Before the end of
January [19, 13] as many as 75, 000 infected cases were estimated in Wuhan City.
On March 11, because of the seriousness of the situation, the WHO declared it a
Pandemic.

The goal of this study is to assess through the analysis of a differential equa-
tions model the importance of different control policies for the Brazilian COVID-19
epidemic. Even though Brazil is considered for the scope of this paper, the tech-
niques and tools used in this study can be easily adapted for any other country.
The impact of different control strategies are qualitatively evaluated and mathe-
matically based guidelines concerning different protective measures and quarantine
strategies are formulated. The article is organized as follows: In Section 2, the
age-structured SEIR model with quarantine is formulated. Demographic data from
Brazil is introduced and discussed. In Section 3, the classical SEIR model with-
out vital dynamics and with a quarantine compartment is studied. The goals here
are, firstly, to adjust parameters and to fit the real data, and secondly, to study
the necessary quarantine efforts and times so to be able to influence the epidemic.
In Section 4, the parameters for the age-structured model are adjusted (using the

2010 Mathematics Subject Classification. 92D30, 93C15, 34H05.

Key words and phrases. Coronavirus; quarantine; epidemic; SEIR models.
c©2020 Texas State University.

Submitted April 6, 2020. Published June 24, 2020.

1



2 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

ones calculated on the previous section). The next generation approach is used to
calculate the basic reproduction number and a deterministic sensibility analysis of
R0 [5] is carried on. In Section 5, different quarantine strategies for different age
classes are considered and compared. We draw our conclusions in Section 6.

2. Age-structured SEIR model

The outbreak of COVID-19 has shown a markedly low proportion of children
among reported cases [17, 16]. More generally, it has been observed that the number
of cases and the risk of severe disease rise as age increases [4, 15, 20]. Understanding
the role of age in transmission and disease severity is critical for determining the
likely impact of control measures for decreasing transmission [6]. A classical SEIR
model is used with the addition of a quarantine class as proposed in [11]. Since age
is such an important factor on the COVID-19 epidemic, it will be assumed that
the population is age-structured (see [2, 9, 18] for continuous models and [21, 22]
for discrete models). Three age classes are used; infants with ages in the interval
[0, 19] (i = 1), adults with ages in the interval [20, 59] (i = 2), and elderly with ages
in the interval [60, 100] (i = 3). The proportion of each age class in the Brazilian
population is shown in Table 1 (see [10]).

Let Si(t), Ei(t), Ii(t), Ri(t) and Qi(t) represent the number of susceptibles,
exposed, infected, removed and quarantined at age class i respectively at time
t ≥ 0. The equations are as follows

Q′i(t) = piSi(t)− λiQi(t), i = 1, 2, 3,

S′1(t) = Λ− (µ1 + ρ1)S1(t)− S1(t)
( 3∑
j=1

β1jIj(t)
)

− p1S1(t) + λ1Q1(t) ,

S′2(t) = ρ1S1(t)− (µ2 + ρ2)S2(t)− S2(t)
( 3∑
j=1

β2jIj(t)
)

− p2S2(t) + λ2Q2(t) ,

S′3(t) = ρ2S2(t)− µ3S3(t)− S3(t)
( 3∑
j=1

β3jIj(t)
)

− p3S3(t) + λ3Q3(t) ,

E′i(t) = Si(t)
( 3∑
j=i

βijIj(t)
)
− (σi + µi)Ei(t) ,

I ′i(t) = σiEi(t)− (γi + µi +mi)Ii(t) ,

R′i(t) = γIi(t)− µiRi(t) ,

(2.1)

The parameters are all non-negative (or positive) and are described in Table 2. pi
and λi are the quarantine entrance and exit rates for class i, respectively. Λ, µi and
ρi are the vital parameters. In the disease free situation the population is assumed
to be at demographic equilibrium. γi is the recovery rate, mi the disease induced
death rate and βij is the infection rate between class i and class j. Typically, it
will be assumed that βij = βji.



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 3

The class Qi has the effect of removing susceptible individuals from the infection
dynamics. If pi = λi = 0 there is no quarantine and the system reduces to an
age-structured SEIR model.

Table 1. Age Classes.

Class Age (years) % Population % Mortality (year) µi (year)

1 [0,19] 40.2 % 12.6 % 1.959/1000

2 [20,59] 50.5 % 33.2 % 4.109/1000

3 [60,100] 9.3 % 54.2 % 36.425/1000

According to [10] Brazil has 18.67 births and 6.26 deaths by 1000 inhabitants
per year, giving an annual population growth of 1.24 %. Let N denote the total
population and D the total deaths per year, thus

D

N
= µ =

6.25

1000
.

Similarly, let Di and Ni be the number of deaths per year and Ni be the population
of age class i respectively. Thus

µi =
Di

Ni
.

With this notation the data on Table 1 is denoted by

% Population =
Ni
N

and % Mortality =
Di

D
.

µi is calculated by

µi =
Di

Ni
=
Di

D

D

N

N

Ni
= µ

Di

D

N

Ni
= µ

(Di/D)

(Ni/N)
.

The disease free steady state is denoted by

S∗1 , S
∗
2 , S

∗
3 , Ei = Ii = Ri = 0 i = 1, 2, 3, (2.2)

where by S∗i we denote the number of individuals of age class i (all individuals are
susceptible, see Table 1). For the model without quarantine, adding the equations
for the disease free state gives

(S1(t) + S2(t) + S3(t))′(t) = (Λ− µ1S1(t)− µ2S2(t)− µ3S3(t)) .

Assuming that the total population is constant and on demographic equilibrium,
using the values for the population distribution as the equilibrium values, one must
have

Λ = µ1S
∗
1 + µ2S

∗
2 + µ3S

∗
3 =

6.25

1000
deaths/year.

The actual annual growth rate will be ignored. Since the time frame of interest
is small compared to the demographic time scale, this has no consequences on the
main conclusions of this work. The demographic equilibrium implies that ρ1 and
ρ2 satisfy

ρ2 =
µ3S

∗
3

S∗2
= 6.707× 10−3 and ρ1 =

(µ2 + ρ2)S∗2
S∗1

= 11.033× 10−3 .

If it is assumed that the demographic, disease and quarantine parameters are
equal for all age classes, the above system reduces to the classical SEIR system



4 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

Table 2. Parameters of the basic SEIR model with vital dynamics.

Parameter Description

pi quarantine entrance rate for class i.

λi quarantine exit rate for class i.

Λ recruitment rate.

µi natural death rate for class i.

ρi survival rate for class i to class i+ 1 i ≤ 2.

βij pathogen’s transmission rate between classes i and j.

σi rate at which exposed of class i convert into the infected class.

γi class i host’s recovery rate.

mi host’s pathogen-induced death rate at class i.

with the quarantine term as suggested by [11]. This will be important for what
follows. The parameters for the classical SEIR model will be estimated so that the
number of cases predicted by the model compares well with the data. This set of
parameters will be used later to adjust the age-structured model 2.1.

3. Unstructured SEIR model

The SEIR model without vital dynamics and with quarantine terms is given by

Q′(t) = pS(t)− λQ(t) ,

S′(t) = −βS(t)I(t)− pS(t) + λQ(t) ,

E′(t) = βS(t) I(t)− σ E(t) ,

I ′(t) = σ E(t)− γI(t) ,

R′(t) = γI(t) .

(3.1)

Ignoring the quarantine class (p = λ = 0) the parameters β, σ and γ can be adjusted
so that the SEIR curve fits the data. To achieve that, the difference between the
SEIR infected curve and the data curve for the number of infected is minimized
(see [12] for algorithm description). The parameters found were

β∗ = 0.8481, σ∗ = 0.2682, γ∗ = 0.0870 . (3.2)

The initial conditions used for the algorithm were β = 2.2/2.9, σ = 1/5.2 and
γ = 1/2.9. The figure 1 shows the data and the SEIR infected curve using the
parameters from (3.2). The considered time interval was 20 days.

Remark 3.1. The model must be considered with care. The curve I(t), as given
by the SEIR model, predicts the total number of infected individuals (symptomatic
and asymptomatic) at time t. However, to estimate the number of individuals that
will need medical care, one needs to know the proportion between the reported
and unreported cases. Estimates for the number of unreported cases can be found
at [14] and the severity of the reported cases can be found at [3]. Asymptomatic
cases can be as high as 75% [7] of all cases; also, ratio estimates of reported to
unreported cases goes from 1/1 to 1/20 [14]. These uncertainties must be taken into
consideration when using the model to make numerical previsions. The emphasis
of this paper is placed on understanding qualitatively efficient ways of controlling
the epidemic.



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 5

2 4 6 8 10 12 14 16 18 20

Time in days

0

50

100

150

200

250

N
um

be
r 

of
 c

as
es

Figure 1. The number of infected for the SEIR non-structured
model. The parameters values are as in 3.2.

Quarantines will be characterized by two values: the entrance rate p and the exit
rate λ. p is composed of two terms, γq and ξ. γq is the average time it takes for a
person to enter quarantine (see [11]) and ξ is a dimensionless multiplicative factor
representing the percentage of individuals that in fact voluntarily quarantine. With
this notation

p =
ξ

γq
.

As an example, suppose that 70% of the population quarantine in an interval of 2
days. Then p = 0.70/2 = 0.35. It will be assumed that p ∈ [0.0, 0.40]. p = 0 means
that there is no quarantine. As in [11] it will be assumed that the time to leave
quarantine will between 30 and 60 days, giving that λ ∈ [1/60, 1/30].

Remark 3.2. For future reference we observe that, from definition, p is smaller
than the percentage of quarantined population.

The effect of the quarantine on the prevalence curve is twofold: it decreases
the maximum I(t) value and postpones the date of its occurrence. To assess the
efficiency of the quarantine, the maximum of the prevalence curve and the time of
its occurrence were calculated and are shown in Figure 3 .

The important feature on Figure 3 is the existence of a threshold value for the
epidemic effort. For values greater than this critical value, the maximum number
of infected decreases extremely fast and the maximum time essentially stabilizes.
This is a common feature for all p and λ as shown in figure 5.

Critical values for quarantine efforts are clearly seen for the contour plots for the
maximum number of infected. The white region on Figure 6 divides the parameter
plane in two regions. The region above has a maximum number of infected smaller
than 1 × 106 infected (by the above rough estimates ≈ 5000 deaths). The region



6 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

0 1 2 3 4 5 6
Time (days) 105

0

1

2

3

4

5

6

7

8

9

10

N
um

be
r 

of
 in

fe
ct

ed

107 Quarantine - 30 days

p=0.00
p=0.05
p=0.08
p=0.10

0 1 2 3 4 5 6 7 8 9 10
Time (days) 105

0

1

2

3

4

5

6

7

8

9

10

N
um

be
r 

of
 in

fe
ct

ed

107 Quarantine - 60 days

p=0.00
p=0.05
p=0.08
p=0.10

Figure 2. Prevalence curve for different quarantine efforts. The
top figure assumes a 30 days quarantine and the bottom figure a
45 days quarantine.

0.1 0.15 0.2 0.25

 p - Quarantine effort 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

 M
ax

im
um

 n
um

be
r 

of
 in

fe
ct

ed
 

107

0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25

 p - Quarantine effort 

0

0.5

1

1.5

2

2.5

3

3.5

4

 M
ax

im
um

 n
um

be
r 

of
 in

fe
ct

ed
 

106

Figure 3. Maximum of infected as a function of the quarantine
effort p. The figure on the right-hand side details the fast decrease
after p = 0.21. Quarantine time is 30 days.

below has larger numbers of infected (and of deaths). The level sets accumulate
around a critical level set, showing again that, qualitatively, quarantines do work.



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

 p - Quarantine effort 

0

100

200

300

400

500

600

700

 T
im

e 
of

 m
ax

im
um

 (
da

ys
) 

Figure 4. The time for the maximum of the epidemic curve as a
function of the quarantine effort p. Quarantine time is 30 days.

0 0.05 0.1 0.15 0.2 0.25

p - Quarantine effort

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

m
ax

im
um

 n
um

be
r 

of
 in

fe
ct

ed

107

30 days
40 days
50 days
60 days

Figure 5. The maximum of the epidemic curve as a function of
the quarantine effort p for different quarantine values.

4. Control strategies for the age-structured model

The control measures for the age-structured model will be divided in two types.
The first type controls the epidemic parameters [2]. This will be made through an
R0 sensitivity analysis: the R0 for the age-structured model will be numerically
determined and its parameter dependence will be investigated. The second type
of control will be the age-oriented quarantines. The parameters pi determine the
quarantine effort for each class. Due to the different classes weights on the popula-
tion composition, and to the different epidemic parameters of each class, this study
allows us to assess the impact of each class quarantine on the epidemic dynamics.
Before we proceed, we need to adjust the parameters for the structured model.



8 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

Maximum number of infected - Level sets

0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 

p - Quarantine effort  

43

46

49

52

55

58

61

64

67

70

Q
ua

ra
nt

in
e 

tim
e

0.5

1

1.5

2

2.5

3

3.5

107

Figure 6. Level curves for the maximum number of infected as a
function of quarantine effort and quarantine time.

4.1. Data Fitting. There are 12 parameters to be determined for the structured
model: σi, γi, βij = βji for i, j = 1, 2, 3. The algorithm fits the parameters to the
available data of Brazil’s total number of reported cases for the first 19 days of
infection by a least squares method. The distance between the predicted curve

I(t) = I1(t) + I2(t) + I3(t)

and the data curve is minimized. The initial parameters for the minimization
search algorithm are based on the ones found for the unstructured SEIR model (3)
taking into consideration the population percentage of each age class. Let ci be the
population percentage of each class, that is (see Table 1), c1 = 0.402, c2 = 0.505
and c3 = 0.093. The initial values for the iteration are chosen as

σi = ci
σ∗i
3
, γi = ci

γ∗i
3

for i = 1, 2, 3.

βij = β∗ for i ≤ j = 1, 2, 3 ,

The resulting values are listed in Table 3 and a plot of the daily number of infections
and the number of reported cases is shown in Figure 7.

4.2. R0 Analysis. In this section R0 is calculated. To determine the relative
importance of model parameters to disease transmission and prevalence we study
the effects of these parameters on R0 through a deterministic sensitivity analysis
(refer to [5] for details). R0 is determined via the next generation approach [8]. It



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 9

Table 3. Fitted parameters for the age-structured SEIR model
without vital dynamics.

Parameter Value Parameter Value

β11 1.76168 σ1 0.27300

β12 0.36475 σ2 0.58232

β13 1.32468 σ3 0.69339

β22 0.63802 γ1 0.06862

β23 0.35958 γ2 0.03317

β33 0.57347 γ3 0.35577

2 4 6 8 10 12 14 16 18 20

Time in days

0

20

40

60

80

100

120

140

160

180

200

N
um

be
r 

of
 c

as
es

Figure 7. The number of infected for the SEIR structured model.
Parameter values as in Table 3

equals the spectral radius of FV −1, where

F =


0 0 0 β11S

∗
1 β12S

∗
1 β31S

∗
1

0 0 0 β21S
∗
2 β22S

∗
2 β23S

∗
2

0 0 0 β31S
∗
3 β32S

∗
3 β33S

∗
3

0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0


and

V =



D1 0 0 0 0 0
0 D2 0 0 0 0
0 0 D3 0 0 0

−σ1 0 0 D̃1 0 0

0 −σ2 0 0 D̃2 0

0 0 −σ3 0 0 D̃3

 ,



10 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

where Di = σi + µi and D̃i = γi + µi +mi for i ∈ {1, 2, 3}. Thus

FV −1 =

(
K11 K12

K21 K31

)
,

where the block K11 is
β11σ1S

∗
1

γ1(σ1+µ1)(γ1+µ1+m1)
β12σ2S

∗
1

γ2(σ2+µ2)(γ2+µ2+m2)
β13σ3S

∗
1

γ3(σ3+µ3)(γ3+µ3+m3)

β21σ1S
∗
2

γ1(σ1+µ1)(γ1+µ1+m1)
β22σ2S

∗
2

γ2(σ2+µ2)(γ2+µ2+m2)
β23σ3S

∗
2

γ3(σ3+µ3)(γ3+µ3+m3)

β31σ1S
∗
3

γ1(σ1+µ1)(γ1+µ1+m1)
β32σ2S

∗
3

γ2(σ2+µ2)(γ2+µ2+m2)
β33σ3S

∗
3

γ3(σ3+µ3)(γ3+µ3+m3)

 ,

the block K12 is 
β11S

∗
1

γ1+µ1+m1

β12S
∗
1

γ2+µ2+m2

β13S
∗
1

γ3+µ3+m3
β21S

∗
2

γ1+µ1+m1

β22S
∗
2

γ2+µ2+m2

β23S
∗
2

γ3+µ3+m3
β31S

∗
3

γ1+µ1+m1

β32S
∗
3

γ2+µ2+m2

β33S
∗
3

γ3+µ3+m3


and K21 and K22 are the 3×3 zero matrix. Due to the block structure of FV −1, its
eigenvalues are easily calculated. However, due to the high number of parameters,
their expression is too cumbersome to be of any analytical use. The sensitivity
analysis is therefore computed numerically. Figures 8, 9, 10 and 11 show the R0

parameter dependence.

0.05 0.1 0.15 0.2 0.25 0.3
9

10

11

12

13

14

15

16

R
0

1

2

3

Figure 8. R0 as a γi function. Different curves show which γi is
varying while the others are kept constant.

The use of the parameters of the classical SIR model as control variables was
studied in [1]. We will follow its interpretation. Measures as keeping social distance,
wearing protective masks, washing hands, etc have the effect of reducing the contact
rates βij . Identifying infected through tests, body temperature checks, etc and
putting them into quarantine has the effect of increasing the removal rates γi. σi
is a parameter that can not be controlled.

The results can be summarized as follows:



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
12

12.2

12.4

12.6

12.8

13

13.2

R
0

1

2

3

Figure 9. R0 as a σi function. Different curves show which σi is
varying while the others are kept constant.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10

12

14

16

18

20

22

24

26

28

30

R
0

11

22

33

Figure 10. R0 as a βii function. Different curves show which βii
is varying while the others are kept constant.

(i) Class 1 is the most sensitive to screening measures (see Figure 8). Young-
sters should be preferentially screened.

(ii) Considering the direct contacts within the same class, class 2 is the more
sensitive (see Figure 10). Social distance between adults has the biggest
impact on R0.

(iii) For the direct contact between different class, β12 has the greatest impact
on R0.



12 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10

12

14

16

18

20

22

24

26

28

R
0

12

13

23

Figure 11. R0 as a βij , i < j function. Different curves show
which βij is varying while the others are kept constant.

5. Effects of different quarantine policies

In this section we study the impacts of different quarantine strategies. The
disease induced mortality rate was taken into account by considering the number
of deaths as a fraction of the recovered class. Death rates for all age groups are
estimated using the data from [3] (Table 4). As mentioned in Section 3, I1, I2 and I3
include symptomatic and asymptomatic infected individuals as well as unreported
cases, so death rates will be multiplied by a factor of 0.25 (since only 25% of the
infected are symptomatic [7]) and by 1/20 (due to unreported cases [14]). This
leaves us with a multiplicative factor of φ = 0.25 ∗ (1/20) = 0.0125 to estimate the
number of deaths. Since we will be working with relative proportions, the actual
value of φ will be of no importance.

Table 4. Death rates for the age-structured model (data taken
from [3]).

Age group Number of cases Deaths % Death

1 350 1 0.29%

2 9541 36 0.38%

3 9068 768 8.47%

With these values on hand, we can study the impact of a quarantine with param-
eters λ and pi, for i ∈ {1, 2, 3}. Calling p the quarantine effort for the unstructured
model, it is assumed that the total quarantine effort equals the effort for the un-
structured model that is

p1 + p2 + p3 = p .

Remark 5.1. As mentioned in 3.2, calling qi, i = 1, 2, 3, the percentage of quar-
antined on each age class, it follows that

p < q1 + q2 + q3 ≤ 1 .



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 13

Four different choices for the pi’s will be used, as detailed in Table 5. These
are choices for the quarantine effort of each age group. Strategy S1 splits the ef-
fort equally among the three groups. Strategy S2 emphasizes a stronger isolation
of the elderly (twice as much as the other groups). Strategy S3 enforces isolation
of the youngsters and adults twice as much as it does for the elderly. Finally,
strategy S4 doubles the quarantine effort on the adults in comparison to the oth-
ers. To assess the efficiency of these different control strategies, for a fixed con-
trol effort p, each control strategy will be calculated for different quarantine times
λ ∈ {1/30, 1/45, 1/60}.

Table 5. Quarantine strategies.

Strategy Choices for the pi

S1 p1 = p/3, p2 = p/3, p3 = p/3

S2 p1 = p/6, p2 = p/6, p3 = 2p/3

S3 p1 = 2p/5, p2 = 2p/5, p3 = p/5

S4 p1 = p/6, p2 = 2p/3, p3 = p/6

The estimation of the number of deaths can be made by multiplying the number
of recovered at the end of the epidemic in each of the three age groups by the
death rates from Table 4 and by the multiplicative factor φ. However, due to
parameters uncertainties and lack of estimations for the parameter p, a different
approach is taken. We arbitrarily chose one of the values as unit and calculated all
the other results proportionally. The results for p = 0.2 are available in Table 6.
(For reference only, the number of deaths chosen as unit was 2869).

Table 6. Proportion of deaths for each age group for different
quarantine strategies and durations.

λ Age group S1 S2 S3 S4

1 1 1.02 0.99 1.03

1/30 2 1.61 1.67 1.59 1.47

3 7.20 6.43 7.46 7.51

Total 9.81 9.12 10.04 10.01

1 0.95 0.99 0.93 1.01

1/45 2 1.51 1.60 1.47 1.29

3 6.77 5.75 7.18 7.26

Total 9.23 8.34 9.58 9.56

1 0.90 0.96 0.88 0.98

1/60 2 1.41 1.54 1.36 1.14

3 6.38 5.21 6.90 7.01

Total 8.69 7.71 9.14 9.13

One could argue that the optimal control would occur if we put all the quarantine
effort in the isolation of the elderly and no isolation at all for youngsters and adults.
With our terminology, this means considering a strategy S5 defined by p1 = p2 = 0
and p3 = p. However, this leads to two main problems: first, due to the small



14 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

percentage of the elder class the quarantine effort would be too small (in fact
smaller than 0.1) to be of any significance. Second, it would allow for a much
higher number of infected individuals (see Figure 12), hence a great increase in
the total of hospitalizations, which would collapse the Health System. Therefore,
in order to achieve better quarantine results, the total effort needs to include all
age-groups, with more emphasis on the elderly since they have a higher fatality rate
due to the disease.

0 50 100 150 200 250 300 350 400 450 500

Time in days

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N
um

be
r 

of
 c

as
es

 (
as

 p
er

ce
nt

ag
e 

of
 to

ta
l p

op
ul

at
io

n)

S2
S5

Figure 12. Plots of the total number of infected, as percentages
of the total population, for strategies S2 and S5. The quarantine
parameters were p = 0.2 and λ = 1/60.

Notice that strategy S2 is, by far, the best among these. All other strategies end
up with, at least, 7.5% more deaths. We can also analyze the strategies by plotting
them. Let µi, i ∈ {1, 2, 3}, be the death rates from Table 4, so

Dj(t) = φ

3∑
i=1

µiRi(t)

converges to the total amount of deaths that result from strategy Sj , j ∈ {1, 2, 3, 4}.
Figure 13 plots the graphs of Dj(t), normalized by

lim
t→∞

D2(t),

produced by the four strategies for different values of p. Notice yet that, in all four
cases, the strategy that produces the smallest limit value (hence the smallest death
toll) is S2.

6. Conclusions

In this paper we introduced an age-structured SEIR model with a quarantine
compartment. Three age classes were used: infants (0 to 19 years), adults (20 to



EJDE-2020/64 CONTROL STRATEGIES FOR THE COVID-19 EPIDEMIC 15

0 100 200 300 400 500

Time in days

0

0.5

1

1.5
p = 0.1

S1
S2
S3
S4

0 100 200 300 400 500

Time in days

0

0.5

1

1.5
p = 0.2

S1
S2
S3
S4

0 100 200 300 400 500

Time in days

0

0.5

1

1.5
p = 0.3

S1
S2
S3
S4

0 100 200 300 400 500

Time in days

0

0.5

1

1.5
p = 0.4

S1
S2
S3
S4

Figure 13. Plots of Dj(t) for λ = 1
30 , j ∈ {1, 2, 3, 4}.

59 years) and elderly (60 to 100 years). First we studied the associated classical
unstructured SEIR model without vital dynamics. The parameters were fitted by
a least squares algorithm and the impact of the quarantine parameters p and λ
was studied. Our main findings concern the existence of a numerical threshold
value for the quarantine parameters: above a certain curve on the (p, λ)-plane, the
maximum number of infected decreases in an accentuated way. This shows that an
abrupt decline on the number of cases should be observed if the quarantine is being
efficient. If this decline is not being observed, quarantine effort and time should be
increased.

The parameters obtained for the unstructured SEIR model were used to adjust
the parameters for the age-structured SEIR model. Using this data, the basic
reproduction number R0 was calculated and its dependence on the epidemic values
was studied. Our findings for the R0 analysis are as follows:

(i) Class 1 is the most sensitive to screening measures (see Figure 8). Young-
sters should be preferentially screened.

(ii) Considering the direct contacts within the same class, class 2 is the more
sensitive (see Figure 10). Social distance between adults has the biggest
impact on R0.



16 C. CASTILHO, J. A. M. GONDIM, M. MARCHESIN, M. SABETI EJDE-2020/64

(iii) For the direct contact between different class, β12 has the greatest impact
on R0.

Finally we studied the impact of age-oriented campaigns considering different strate-
gies and different values of p for the total campaign effort. Recalling that p is
bounded by the percentage of quarantined population (see remarks 3.2 and 5.1),
our findings show that the highest possible quarantine must be made, and then, this
effort must concentrate on putting into quarantine the total of elders and assuring
equal proportions of adults and youngsters.

Acknowledgements. The authors would like to thank Renato Mello (IFSP -
Campus Salto) for discussions during the preparation of this manuscript.

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César Castilho

Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE CEP
50740-540 Brazil

Email address: castilho@dmat.ufpe.br

João A. M. Gondim

Unidade Acadêmica do Cabo de Santo Agostinho, Universidade Federal Rural de Per-

nambuco, Cabo de Santo Agostinho, PE CEP 54518-430 Brazil
Email address: joao@dmat.ufpe.br

Marcelo Marchesin
Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte,

MG CEP 31270-000 Brazil

Email address: mdm@mat.ufmg.br

Mehran Sabeti

Instituto de Ciências Exatas e Tecnológicas, Universidade Federal de Viçosa Campus-
Florestal, Viçosa, MG CEP 35690-000 Brazil

Email address: mehran@ufv.br


	1. Introduction
	2. Age-structured SEIR model
	3. Unstructured SEIR model
	4. Control strategies for the age-structured model
	4.1. Data Fitting
	4.2. R0 Analysis

	5. Effects of different quarantine policies
	6. Conclusions
	Acknowledgements

	References