Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
https://produccioncientificaluz.org/index.php/divulgaciones/
DOI: https://doi.org/10.5281/zenodo.7487462
(CC BY-NC-SA 4.0)
c
Autor(s)
e-ISSN 2731-2437
p-ISSN 1315-2068
Qualitative study of a mathematical model for
the transmission of COVID-19
Estudio cualitativo de un modelo matem´atico para la transmisi´on del COVID-19
Yuri Alc´antara Olivero (yalcantara@uo.edu.cu)
ORCID: https://orcid.org/0000-0002-7208-4229
Department of Computer Science, Faculty of Natural and Exact Sciences
University of Oriente
Cuba.
Sandy anchez Dom´ınguez (sandys@uo.edu.cu)
ORCID: https://orcid.org/0000-0003-3788-8413
Mathematics Department, Faculty of Natural and Exact Sciences
University of Oriente
Cuba
Antonio Iv´an Ruiz Chaveco (iruiz2005@yahoo.es)
ORCID: https://orcid.org/0000-0002-3473-1704
University of the State of Amazonas
Brazil
Abstract
This paper presents an analysis of the characteristics of the model to simulate the process
of infection by COVID 19 in Wuhan China, a set of observations are indicated that represent
the bases for its modiﬁcation and a qualitative study is carried out.
Palabras y frases clave: Mathematical model, epidemic, qualitative analysis.
Resumen
En este trabajo se presenta un an´alisis del modelo para simular el proceso de infecci´on
por COVID-19 en Whuhan China, se indican un conjunto de observaciones que presentan
las bases para su modiﬁcaci´on y se realiza un estudio cualitativo.
Key words and phrases: Modelo matem´atico, epidemia, an´alisis cualitativo.
1 Introduction
The disease that has most aﬀected humanity in recent years has been COVID-19. In [12] the
authors make an exhaustive analysis of the situation presented in Wuhan, China, making a
Received 06/07/2021. Revised 09/09/2021. Accepted 31/08/2022.
MSC (2010): Primary 34C60; Secondary 34C20.
Corresponding author: Sandy anchez Dom´ınguez
24 Yuri Alc´antara - Sandy anchez - Antonio Ruiz
model that corresponded exactly to the presented situation, predicting the future of the disease
in correspondence with the cases presented.
In view of the situation presented in Santigo de Cuba, the authors of [13] adapted the model
presented in [12], managing to make sure that the prognoses made corresponded to the reality of
the epidemic in that region of the Cuban East. We propose to make a qualitative analysis of this
model in order to prove the adaptability of the model to other situations and to other countries
outside the country of origin.
Due to these great aﬀects produced by COVID-19 in the world, multiple results have been
published both from the point of view of biochemical characteristics, treatment, and from the
point of view of modeling to make predictions regarding the future of the pandemic, among others
can indicate the works [13, 10, 14, 17, 18], which represent models using ordinary diﬀerential
equations, which give conclusions regarding the future behavior of the infection process of the
population under consideration.
The qualitative study of these models is very important, as this allows us to draw conclusions
regarding the future situation of this process; allowing to determine necessary and suﬃcient
conditions under which a possible complication could or could not be prevented. (cf. [15, 16, 17]).
COVID-19 desease is caused by the SARS-CoV-2 coronavirus, a respiratory disease that so
many lives have claimed, there are many ideas on how to ﬁght this disease; but the method that
most researchers agree on, is given by the method of isolating the infected to prevent possible
transmission to other people [11]. One of the treatments that has already given results is interferon
alpha-2b, in addition to others already tested in the treatment of other diseases such as AIDS,
hepatitis, among others.
Interferon alpha-2b, was developed by the Cuban Genetic Engineering and Biotechnology
Center and has already been used in diﬀerent parts of the world with highly reliable results (cf.
[2]).
Today in the world vaccines are applied to raise the immune response of an individual and pro-
tect him from disease, among which have been certiﬁed are Pﬁzer-BioNTech, Moderna, Janssen
from Johnson & Johnson, Sputnik V and Sinovac-CoronaVac.
Currently in Cuba, they are working with ﬁve vaccine candidates against COVID-19, which
are passing through diﬀerent phases of the clinical trial, Soberana 02 and Abdala that are passing
through the third phase of the trial, Soberana 01, Soberana Plus and Mambisa that are passing
through the second phase of the trial [1, 9]. In the particular case of the candidate, Mambisa
explores the intranasal route, while the remaining candidates are intramuscularly [9], particularly
Soberana Plus is studied in convalescent patients.
There are multiple works dedicated to the study of the causes and the conditions under which
an epidemic may develop, among these we can indicate [7]. The problem of epidemic modeling
has always been of great interest to researchers such as the cases of (cf. [3, 4, 6, 5, 8]).
In [15] diﬀerent problems of real life are treated by means of equations and systems of diﬀer-
ential equations, all of them only in the autonomous case; where examples are developed, and
other problems and exercises are presented for them to be developed by the reader.
The objective of this work is to make a qualitative study of the SEIR model with three
additional classes: the total population size (N), the public perception of risk (D), and the
cumulative cases number (C) reported by Lin et al. [13] which simulates through a system
ordinary diﬀerential equations, the situation presented in Wuhan China, the process of infection
of COVID-19 in such a way that it can respond to the current situation in diﬀerent countries
and regions of the world; as it happened in Cuba and other countries, a possibility that had
already been indicated in [17], where in addition it was planted how to reverse this situation.This
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
Qualitative study for COVID-19 transmission 25
would give a justiﬁcation from the theoretical point of view of the conclusions presented in [13],
where the situation in Santiago de Cuba is precisely indicated, theoretically demonstrating the
2 Initial model presented
Starting from this original system, we will place some modiﬁcations depending on the character-
istics of the problem, in response to observations that will justify each of the adaptations made,
which has the following expression.
dS
dt =µS β(t)
NSI
dE
dt =(σ+µ)E+β(t)
NSI
dI
dt =(γ+µ)I+σE
dR
dt =µR +γI
dN
dt =µN
dD
dt =λD +I
dC
dt =σE
(2.1)
with
β(t) = β0(1 α)(1 D
N)k,(2.2)
where σ1,γ1,d,λ1,β(t), β0,αand kare the mean latent period, the mean infectious period,
the proportion of severe cases, the mean duration of public reaction, the dynamic transmission
rate, the initial transmission rate, the governmental measure strength and the intensity of indi-
Srepresents the susceptible population, Erepresents the exposed population, Irepresents the
infected population, Rrepresents the recovered population, Nrepresents total population, D
represents the public perception of risk and Crepresents the cumulative number of cases.
1. Under the same conditions, not all susceptible people are infected.
2. The recovery time is not the same for all patients.
3. The number of infected people is not exact, as there are asymptomatic patients on the
street without being detected.
4. The unknown functions in the system are being considered with some approximation.
5. Disturbances will be introduced to bring us closer to the real situation.
6. How β(t) is a limited function with values in the range [0, β0] a constant value within that
range will be considered here.
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
26 Yuri Alc´antara - Sandy anchez - Antonio Ruiz
7. To guarantee the validity of the qualitative study, we will make a distinction between the
total populations and the allowable concentrations.
3 Qualitative analysis of the modiﬁed model
For the qualitative study of the model, the system of diﬀerential equations will be modiﬁed for
which the following variables will be introduced:
˜
i1is the total infected population at the moment t.
˜s1is the total susceptible population at the moment t.
˜e1is the total exposed population at the moment t.
˜r1is the total recovered population at the moment t.
˜n1is the total population at the moment t.
˜
d1is the total of the population that have the risk at the moment t.
˜c1is the total of cases accumulated at the moment t.
In addition it will be denoted by ¯
i1, ¯r1¯s1, ¯e1, ¯n1,¯
d1and ¯c1the admissible values respectively, of
each of the populations. Here the variables will be introduced s1,e1,i1,r1,n1,d1and c1deﬁned
as follows: s1= ˜s1¯s1,e1= ˜e1¯e1,i1=˜
i1¯
i1,r1= ˜r1¯r1,n1= ˜n1¯n1,d1=˜
d1¯
d1and
c1= ˜c1¯c1, them as s10, e10, i10, r10, n10, d10 and c10 when t ,
the following conditions would be met: ˜s1¯s1, ˜e1¯e1,˜
i1¯
i1, ˜r1¯r1, ˜n1¯n1,˜
d1¯
d1
and ˜c1¯c1which would constitute the main objective of this work.
The system (2.1) can be generalized as follows
ds1
dt =µ1β1s1i1+S1(s1, e1, i1, r1, n1, d1, c1)
de1
dt =(σ+µ)e1+β1s1i1+E1(s1, e1, i1, r1, n1, d1, c1)
di1
dt =(γ+µ)i1+σe1+I1(s1, e1, i1, r1, n1, d1, c1)
dr1
dt =µr1+γi1+R1(s1, e1, i1, r1, n1, d1, c1)
dn1
dt =µn1+N1(s1, e1, i1, r1, n1, d1, c1)
dd1
dt =λd1+i1+D1(s1, e1, i1, r1, n1, d1, c1)
dc1
dt =σe1+C1(s1, e1, i1, r1, n1, d1, c1)
(3.1)
Where
S1(s1, e1, i1, r1, n1, d1, c1), E1(s1, e1, i1, r1, n1, d1, c1), I1(s1, e1, i1, r1, n1, d1, c1),
R1(s1, e1, i1, r1, n1, d1, c1), N1(s1, e1, i1, r1, n1, d1, c1), D1(s1, e1, i1, r1, n1, d1, c1)
C1(s1, e1, i1, r1, n1, d1, c1)
(3.2)
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
Qualitative study for COVID-19 transmission 27
are perturbations and from the mathematical point of view they are inﬁnitesimals of superior
order because they constitute series of power where the inferior degree of their powers is the
second; besides that β1[0,β0
N] like this,
S1(s1, e1, i1, r1, n1, d1, c1) = X
|p|≥2
s(p)
1sp1
1ep2
1ip3
1rp4
1np5
1dp6
1cp7
1,
with |p|=p1+p2+p3+p4+p5+p6+p7, the another series given on the equation (3.2) have a
similar development.
The characteristic equation corresponding to the matrix of the linear part of the system (3.1)
has the form,
k(k+λ)(k+µ)3(k+ (γ+µ))(k+ (µ+σ)) = 0
As it turns out, it has a zero eigenvalue and another six negatives, this is a critical case, it is
necessary to apply the analytical theory of diﬀerential equations to draw conclusions regarding
the future behavior of the infection process. For this, we will simplify the system, reducing it
to almost normal form. By means of a non-degenerate transformation X=SY , where X=
col(s1, e1, i1, r1, n1, d1, c1), Y= col(s2, e2, i2, r2, n2, d2, c2) and Sthe matrix of the eigenvalues of
the matrix of the linear part of the system
S=
0 0 0 0 0 1 0
0 0 0 0 0 0 µσ
σ
0 0 γ+λµ
γd 000 µσ
γσ
0 0 γλ+µ
γd 010 γ(µ+σ)
σ(γσ)
0 0 0 1 0 0 0
0 1 1 0 0 0 γd(µ+σ)
(γσ)(λ+µ+σ)
1 0 0 0 0 0 1
where additional satisfaction is required of additional algebraic conditions associated with the
proper subspaces to guarantee the reduction of the matrix to the diagonal form, so the system is
reduced to,
s0
2=k1s2+S2(s2, e2, i2, r2, n2, d2, c2)
e0
2=k2e2+E2(s2, e2, i2, r2, n2, d2, c2)
i0
2=k3i2+I2(s2, e2, i2, r2, n2, d2, c2)
r0
2=k4r2+R2(s2, e2, i2, r2, n2, d2, c2)
n0
2=k5n2+N2(s2, e2, i2, r2, n2, d2, c2)
d0
2=k6d2+D2(s2, e2, i2, r2, n2, d2, c2)
c0
2=C2(s2, e2, i2, r2, n2, d2, c2)
(3.3)
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
28 Yuri Alc´antara - Sandy anchez - Antonio Ruiz
Theorem 3.1. The exchange of variables
s2=s3+h1(c3)
e2=e3+h2(c3)
i2=i3+h3(c3)
r2=r3+h4(c3)
n2=n3+h5(c3)
d2=d3+h6(c3)
c2=c3+h7(c3) + ˜
h(s3, e3, i3, r3, n3, d3, c3)
(3.4)
transforms the system (3.3) into almost normal form,
s0
3=k1s3+S3(s3, e3, i3, r3, n3, d3, c3)
e0
3=k2e3+E3(s3, e3, i3, r3, n3, d3, c3)
i0
3=k3i3+I3(s3, e3, i3, r3, n3, d3, c3)
r0
3=k4r3+R3(s3, e3, i3, r3, n3, d3, c3)
n0
3=k5n3+N3(s3, e3, i3, r3, n3, d3, c3)
d0
3=k6d3+D3(s3, e3, i3, r3, n3, d3, c3)
c0
3=C3(c3)
(3.5)
where
h1(c3), h2(c3), h3(c3), h4(c3), h5(c3), h6(c3), h7(c3),˜
h(s3, e3, i3, r3, n3, d3, c3),
S3(s3, e3, i3, r3, n3, d3, c3), E3(s3, e3, i3, r3, n3, d3, c3), I3(s3, e3, i3, r3, n3, d3, c3),
R3(s3, e3, i3, r3, n3, d3, c3), N3(s3, e3, i3, r3, n3, d3, c3), D3(s3, e3, i3, r3, n3, d3, c3),
C3(s3, e3, i3, r3, n3, d3, c3)
S2(s2, e2, i2, r2, n2, d2, c2), E2(s2, e2, i2, r2, n2, d2, c2), I2(s2, e2, i2, r2, n2, d2, c2),
R2(s2, e2, i2, r2, n2, d2, c2), N2(s2, e2, i2, r2, n2, d2, c2), D2(s2, e2, i2, r2, n2, d2, c2),
C2(s2, e2, i2, r2, n2, d2, c2),
besides that
˜
h(s3, e3, i3, r3, n3, d3, c3), S3(s3, e3, i3, r3, n3, d3, c3), E3(s3, e3, i3, r3, n3, d3, c3),
I3(s3, e3, i3, r3, n3, d3, c3), R3(s3, e3, i3, r3, n3, d3, c3), N3(s3, e3, i3, r3, n3, d3, c3),
D3(s3, e3, i3, r3, n3, d3, c3)
are canceled when s3=e3=i3=r3=n3=d3= 0.
Proof. Deriving the transformation (3.3) along the trajectories of the systems (3.3) and (3.5) the
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
Qualitative study for COVID-19 transmission 29
system of equations is obtained,
S3=S2dh1
dc3
C3(c3)
E3=E2dh2
dc3
C3(c3)
I3=I2dh3
dc3
C3(c3)
R3=R2dh4
dc3
C3(c3)
N3=N2dh5
dc3
C3(c3)
D3=D2dh6
dc3
C3(c3)
C3=C2(c3)dh7
dc3
C3(c3)˜
h
s3
(k1s3+S3)˜
h
e3
(k2e3+E3)˜
h
i3
(k3i3+I3)
˜
h
r3
(k4r3+R3)˜
h
n3
(k5n3+N3)˜
h
d3
(k6d3+D3)˜
h
c3
C3(c3)
(3.6)
As the ˜
hseries has the form
˜
h=X
|p|≥2
˜
h(p)sp1
3ep2
3ip3
3rp4
3np5
3dp6
3cp6
3,then ˜
h
s3
s3=p1s3X
|p|≥2
˜
h(p)sp11
3ep2
3ip3
3rp4
3np5
3dp6
3cp6
3=p1˜
h.
Similarly, the expressions ˜
h
e3
e3=p2˜
h,˜
h
i3
i3=p3˜
h,˜
h
r3
r3=p4˜
h,˜
h
n3
n3=p5˜
h,˜
h
d3
d3=p6˜
h
and ˜
h
c3
c3=p7˜
h. Substituting these expressions in the equation (3.6), we obtain
S3=S2dh1
dc3
C3(c3)
E3=E2dh2
dc3
C3(c3)
I3=I2dh3
dc3
C3(c3)
R3=R2dh4
dc3
C3(c3)
N3=N2dh5
dc3
C3(c3)
D3=D2dh6
dc3
C3(c3)
C3+
6
X
i=1
piki˜
h=C2(c3)dh7
dc3
C3(c3)˜
h
s3
S3˜
h
e3
E3˜
h
i3
I3
˜
h
r3
R3˜
h
n3
N3˜
h
d3
D3˜
h
c3
C3(c3)
(3.7)
To determine the series that intervene in the systems and the transformation, we will separate
the coeﬃcients from the powers of degree p= (p1, p2, p3, p4, p5, p6, p7) in the following two cases.
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
30 Yuri Alc´antara - Sandy anchez - Antonio Ruiz
Case I: Doing s3=e3=i3=r3=n3=d3= 0 in the system (3.7) results the system,
S2(h1, h2, h3, h4, h5, h6, c3+h7) = dh1
dc3
C3(c3)
E2(h1, h2, h3, h4, h5, h6, c3+h7) = dh2
dc3
C3(c3)
I2(h1, h2, h3, h4, h5, h6, c3+h7) = dh3
dc3
C3(c3)
R2(h1, h2, h3, h4, h5, h6, c3+h7) = dh4
dc3
C3(c3)
N2(h1, h2, h3, h4, h5, h6, c3+h7) = dh5
dc3
C3(c3)
D2(h1, h2, h3, h4, h5, h6, c3+h7) = dh6
dc3
C3(c3)
C2(h1, h2, h3, h4, h5, h6, c3+h7)C3(c3) = dh7
dc3
C3(c3)
(3.8)
The system (3.8) allows the determination of the series coeﬃcients h1(c3), h2(c3), h3(c3), h4(c3),
h5(c3), h6(c3) and C3(c3) where for being the resonant case h7= 0, and the remaining series are
determined in a unique way.
Case II For the case when s36= 0, e36= 0, i36= 0, r36= 0, n36= 0 and d36= 0 of the system
(3.7) it follows that,
S3=S2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜
h)
E3=E2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜
h)
I3=I2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜
h)
R3=R2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜
h)
N3=N2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜
h)
D3=D2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜
h)
6
X
i=1
pihi!˜
h=C2(s3, h1, e3, h2, i3, h3, r3, h4, n3, h5, d3, h6, c3+h7+˜
h)˜
h
s3
S3
˜
h
e3
E3˜
h
i3
I3˜
h
r3
R3˜
h
n3
N3˜
h
d3
D3˜
h
c3
C3
(3.9)
Because the series of the system (3.5) are known expressions, the system (3.9) allows calculating
the series ˜
h(s3, e3, i3, r3, n3, d3, c3), S3(s3, e3, i3, r3, n3, d3, c3), E3(s3, e3, i3, r3, n3, d3, c3),
I3(s3, e3, i3, r3, n3, d3, c3), R3(s3, e3, i3, r3, n3, d3, c3), N3(s3, e3, i3, r3, n3, d3, c3) and
D3(s3, e3, i3, r3, n3, d3, c3). This proves the existence of variable exchange.
In the system (3.5) the function C3(c3) admits the following development in power series:
C3(c3) = αcn
3+. . .
Where αis the ﬁrst non-zero coeﬃcient and nis the corresponding power.
Theorem 3.2. If α < 0and nis odd, so the trajectories of the system (3.5) are asymptotically
stable, otherwise they are unstable.
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33
Qualitative study for COVID-19 transmission 31
Proof. Consider the Lyapunov funtion deﬁned positive,
V(s3, e3, i3, r3, n3, d3, c3) = 1
2(s2
3+e2
3+i2
3+r2
3+n2
3+d2
3+c2
3)
whose derivative along the trajectories of the system (3.5) has the following expression,
dV
dt =k1s2
3+k2e2
3+k3i2
3+k4r2
3+k5n2
3+k6d2
3+αcn+1
3+R(s3, e3, i3, r3, n3, d3, c3)
As in Rappear the powers of degrees greater than the second with respect to s3,e3,i3,r3,n3and
d3and higher degree n+ 1 with respect to c3, the expression of the derivative of Vis negative
deﬁnite, this allows us to aﬃrm that the equilibrium position is asymptotically stable. This
result suggests that the limited progression of the COVID-19 epidemic may be due to opportune
epidemiological investigations and eﬀective control measures in each source of infection.
4 Acknowledgments
The authors appreciate the technical support and invaluable feedback provided by Luis Eugenio
Alvarez and Hilda Moran-
deira Padr´on. We also thank to Universidad de Oriente, Direcci´on de DATYS-Santiago de Cuba,
Direcci´on Provincial de Salud ublica and managers of the provincial government of Santiago de
Cuba.
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Qualitative study for COVID-19 transmission 33
[19] Primer-Apellido Inicial-Segundo-Apellido.(opcional), Primer-Nombre; T´ıtulo del art´ıculo,
Revista, Volumen(N´umero) (A˜no), agina inicial - agina ﬁnal.
[20] Primer-Apellido Inicial-Segundo-Apellido.(opcional), Primer Nombre; T´ıtulo de disertaci´on,
Tipo de sisertaci´on, Instituci´on, A˜no.
[21] Primer-Apellido, Primer-Nombre; T´ıtulo del libro, Editorial, Numero-de-Edici´on,vLugar-de-
edici´on, a˜no.
[22] Primer-Apellido Inicial-Segundo-Apellido.(opcional), Primer Nombre; T´ıtulo del art´ıculo.
En: T´ıtulo del libro o proceeding (nombre de los editores), Editorial, (a˜no), agina-inicial
agina-ﬁnal.
[23] Primer-Apellido Inicial-Segundo-Apellido.(opional), Primer Nombre; T´ıtulo del art´ıculo,
preprint, nombre del repositorio or intituci´on, (a˜no)
[24] Nombre del sitio web, opico consultado, no-elaboraci´on-sitio-web, feche-´ultima-
modiﬁcaci´on-sitio-web, direccin-electrnica-sitio-web.
Divulgaciones Matem´aticas Vol. 22, No. 2 (2021), pp. 23–33