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)

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 S´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 S´anchez Dom´ınguez

24 Yuri Alc´antara - Sandy S´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

predictions made from the adaptation of the model.

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.











dt =−µS −β(t)

NSI

dt =−(σ+µ)E+β(t)

NSI

dt =−(γ+µ)I+σE

dt =−µR +γI

dt =−µN

dt =−λD +dγI

dt =σE

(2.1)

with

β(t) = β0(1 −α)(1 −D

N)k,(2.2)

where σ−1,γ−1,d,λ−1,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-

vidual response, respectively, in addition:

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 S´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 s1→0, e1→0, i1→0, r1→0, n1→0, d1→0 and c1→0 when t→ ∞,

the following conditions would be met: ˜s1→¯s1, ˜e1→¯e1,˜

i1→¯

i1, ˜r1→¯r1, ˜n1→¯n1,˜

d1→¯

and ˜c1→¯c1which would constitute the main objective of this work.

The system (2.1) can be generalized as follows











ds1

dt =−µ1−β1s1i1+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+dγi1+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

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







0 0 0 0 0 1 0

0 0 0 0 0 0 −µ−σ

0 0 −γ+λ−µ

γd 000 −µ−σ

γ−σ

0 0 γ−λ+µ

γd 010 γ(µ+σ)

σ(γ−σ)

0 0 0 1 0 0 0

0 1 1 0 0 0 γd(µ+σ)

(γ−σ)(−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,











2=k1s2+S2(s2, e2, i2, r2, n2, d2, c2)

2=k2e2+E2(s2, e2, i2, r2, n2, d2, c2)

2=k3i2+I2(s2, e2, i2, r2, n2, d2, c2)

2=k4r2+R2(s2, e2, i2, r2, n2, d2, c2)

2=k5n2+N2(s2, e2, i2, r2, n2, d2, c2)

2=k6d2+D2(s2, e2, i2, r2, n2, d2, c2)

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 S´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,











3=k1s3+S3(s3, e3, i3, r3, n3, d3, c3)

3=k2e3+E3(s3, e3, i3, r3, n3, d3, c3)

3=k3i3+I3(s3, e3, i3, r3, n3, d3, c3)

3=k4r3+R3(s3, e3, i3, r3, n3, d3, c3)

3=k5n3+N3(s3, e3, i3, r3, n3, d3, c3)

3=k6d3+D3(s3, e3, i3, r3, n3, d3, c3)

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)

admit a similar development

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=S2−dh1

dc3

C3(c3)

E3=E2−dh2

dc3

C3(c3)

I3=I2−dh3

dc3

C3(c3)

R3=R2−dh4

dc3

C3(c3)

N3=N2−dh5

dc3

C3(c3)

D3=D2−dh6

dc3

C3(c3)

C3=C2(c3)−dh7

dc3

C3(c3)−∂˜

∂s3

(k1s3+S3)−∂˜

∂e3

(k2e3+E3)−∂˜

∂i3

(k3i3+I3)

−∂˜

∂r3

(k4r3+R3)−∂˜

∂n3

(k5n3+N3)−∂˜

∂d3

(k6d3+D3)−∂˜

∂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 ∂˜

∂s3

s3=p1s3X

|p|≥2

h(p)sp1−1

3ep2

3ip3

3rp4

3np5

3dp6

3cp6

3=p1˜

Similarly, the expressions ∂˜

∂e3

e3=p2˜

h,∂˜

∂i3

i3=p3˜

h,∂˜

∂r3

r3=p4˜

h,∂˜

∂n3

n3=p5˜

h,∂˜

∂d3

d3=p6˜

and ∂˜

∂c3

c3=p7˜

h. Substituting these expressions in the equation (3.6), we obtain











S3=S2−dh1

dc3

C3(c3)

E3=E2−dh2

dc3

C3(c3)

I3=I2−dh3

dc3

C3(c3)

R3=R2−dh4

dc3

C3(c3)

N3=N2−dh5

dc3

C3(c3)

D3=D2−dh6

dc3

C3(c3)

C3+

i=1

piki˜

h=C2(c3)−dh7

dc3

C3(c3)−∂˜

∂s3

S3−∂˜

∂e3

E3−∂˜

∂i3

I3−

−∂˜

∂r3

R3−∂˜

∂n3

N3−∂˜

∂d3

D3−∂˜

∂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 S´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+˜

E3=E2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜

I3=I2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜

R3=R2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜

N3=N2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜

D3=D2(s3+h1, e3+h2, i3+h3, r3+h4, n3+h5, d3+h6, c3+h7+˜

i=1

pihi!˜

h=C2(s3, h1, e3, h2, i3, h3, r3, h4, n3, h5, d3, h6, c3+h7+˜

h)−∂˜

∂s3

S3−

−∂˜

∂e3

E3−∂˜

∂i3

I3−∂˜

∂r3

R3−∂˜

∂n3

N3−∂˜

∂d3

D3−∂˜

∂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

whose derivative along the trajectories of the system (3.5) has the following expression,

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

Vald´es Garc´ıa, Adriana Rodr´ıguez Vald´es, Manuel de Jes´us Salvador ´

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 P´ublica and managers of the provincial government of Santiago de

Cuba.

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preprint, nombre del repositorio or intituci´on, (a˜no)

[24] Nombre del sitio web, T´opico consultado, a˜no-elaboraci´on-sitio-web, feche-´ultima-

modiﬁcaci´on-sitio-web, direccin-electrnica-sitio-web.

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