 Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51
Inﬂuence of physical exercise on the
strengthening of immunity. Mathematical model.
Inﬂuencia del ejercicio f´ısico en el fortalecimiento de la inmunidad. Modelo
matem´atico.
Annia Ruiz anchez (anniaruiz@nauta.cu)
Daniela Sara Rodr´ıguez Salmon (daniela.rodriguezs@uo.edu.cu)
Sandy anchez Dom´ınguez (sandys@uo.edu.cu)
Mathematics Department, Faculty of Natural and Exact Sciences, University of Oriente
Cuba
Yuri Alc´antara Olivero (yalcantara@uo.edu.cu)
Department of Computer Science, Faculty of Natural and Exact Sciences, University of Oriente
Cuba
Physics department, Faculty of Natural and Exact Sciences, University of Oriente
Cuba
Isabel Mart´en Powell (isamp@infomed.sld.cu)
University of Medical Sciences of Santiago de Cuba
Cuba
Antonio Iv´an Ruiz Chaveco (iruiz2005@yahoo.es)
University of the State of Amazonas
Brazil
Abstract
In the present work we analyze how physical exercises can inﬂuence the increase of a
person’s immunity; a study of the diﬀerent types of pathogens is carried out, in particular
the characteristics of viruses, their manifestations and appearance are investigated; the char-
acteristics of the immune system as well as immunity, either innate or acquired, are studied.
The relationship between viruses and a person’s immune system is investigated, as well as
how the immune system can react to the presence of a virus.
The dynamics of the interaction of the virus vs the immune system is simulated by means
of a system of ordinary diﬀerential equations, the equilibrium points and the behavior of the
trajectories in a neighborhood of the equilibrium points are determined, additionally the
critical case of a zero and negative one eigenvalue, giving conclusions about the process in
the diﬀerent cases.
Key words and phrases: Mathematical model, epidemic, physical exercises, immunity.
MSC (2010): Primary 34Dxx; Secondary 34Cxx.
Autor de correspondencia: Sandy anchez Dom´ınguez Physical exercise and strengthening of immunity. Mathematical mode 41
Resumen
En el presente trabajo se analiza omo los ejercicios f´ısicos pueden inﬂuir en el aumento
de la inmunidad de una persona; se realiza un estudio de los diferentes tipos de pat´ogenos, en
particular se investigan las caracter´ısticas de los virus, sus manifestaciones y apariencia; se
estudian las caracter´ısticas del sistema inmunol´ogico as´ı como la inmunidad, ya sea innata o
adquirida. Se investiga la relaci´on entre los virus y el sistema inmunol´ogico de una persona,
as´ı como el sistema inmunol´ogico puede reaccionar ante la presencia de un virus.
La din´amica de la interacci´on del virus vs el sistema inmunol´ogico se simula mediante
un sistema de ecuaciones diferenciales ordinarias, se determinan los puntos de equilibrio y el
comportamiento de las trayectorias en una vecindad de las posiciones de equilibrio, adicio-
nalmente se estudia el caso cr´ıtico de un autovalor cero y uno negativo, dando conclusiones
sobre el proceso en los diferentes casos.
Palabras y frases clave: Modelo matem´atico, epidemia, ejercicios f´ısicos, immunidad.
1 Introduction
The immune system is a set of elements that exist in the human body. These elements interact
with each other and are intended to defend the body from diseases, viruses, bacteria, microbes,
among others. The human immune system serves as a protection, shield or barrier that protects
us from undesirable beings, antigens, that try to invade our body. Therefore, it represents the
defense of the human body.
There are confessions of patients who, due to the doctor’s suggestions, permanently started to
perform physical exercises to strengthen their immune system, in the face of infectious diseases,
perceiving a low immunity; gradually obtaining a change in your organism. The ﬁtness coach
stated that he was not the only case, as he had others with similar situations. This problem is
addressed further in specialized bibliographies (cf. [1, 2, 14]).
When the immune system does not function properly, it decreases its ability to defend our
body. Thus, we are more vulnerable to diseases such as tonsillitis or stomatitis, candidiasis, skin
infections, ear infections, herpes, colds and ﬂu. To strengthen the immune system and avoid
problems with low immunity, special attention is needed with food. Some fruits help increase
immunity, such as apples, oranges and kiwis, which are citrus fruits. The intake of omega 3 is
also an ally for the immune system.
The immune system is made up of a complex of diﬀerent cells that receive and emit diﬀerent
signals directed at white blood cells, thus regulating the body’s defense mechanisms. The me-
diators of this interaction are proteins, peptides and other substances that for their activity are
called immunomodulators. Biological immunomodulators are made up of a group of molecules
with speciﬁc properties, many of them chemically and biologically very well characterized and
others to be discovered. (cf. [16,18]).
In the human organism there are own cells and inappropriate cells, among the inappropriate
are pathogens; These inappropriate cells can cause changes in the body, which can turn into
diseases and even cause the death of the person; pathogens can include viruses, bacteria, fungi,
and parasites; these can be intracellular or extracellular.
Viruses are simple structures, they are considered mandatory intracellular parasites, because
they depend on cells to multiply. Outside the intracellular environment, viruses are inert. How-
ever, once inside the cell, the replication capacity of viruses is surprising: a single virus is capable
of multiplying, in a few hours, thousands of new viruses. Viruses are capable of infecting living
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 42 A. Ruiz- D. Rodr´ıguez - S. anchez - Y. Alc´antara - A. Fern´andez - I. Mart´en - A. I. Ruiz
beings from all domains. In this way, represent the greatest biological diversity on the planet,
being more diverse than bacteria, plants, fungi and animals combined (cf. ).
When the human body is attacked by a virus, a reaction from the immune system to the person
quickly occurs to prevent this aggression; there are occasions when this reaction is suﬃcient to
free the organism from any infection, but in many cases this is not enough and it is necessary to
supply medication and other artiﬁcial substances capable of adding immunity such as interferons,
among others.
Immunity can be innate or acquired, acquired immunity is adaptive and is made up of lym-
phocytes; On the other hand, innate immunity is made up of cells and molecules with the great
function of defending the body from any aggressor, these have the ability to kill, this is an
instantaneous process, this being the ﬁrst defense of the body.
Interferons are glycoproteins that have several biological actions, including complex antiviral,
immunomodulatory and antiproliferative eﬀects. Its production and endogenous release occurs
in response to viruses and other inducers, with the exception of bacterial exotoxins, polyanions,
some low molecular weight compounds and microorganisms with intracellular growth (cf. ).
There are many diseases that are transmitted from person to person directly, and diﬀerent
forms of contagion are used for this purpose, often through speech or breathing or in some other
way; but in many other cases this transmission can be carried out by means of a vector being the
mosquito the most common. It is said that the cases of maximum risk are adults of the third age
and especially those who suﬀer from some chronic disease; but practice has shown that in the face
of this disease, there is no one safe, and it can have a slow evolution that acts in a fulminating
way.
Today the most worrying situation is COVID-19, caused by the SARS-CoV-2 coronavirus, a
respiratory disease that has claimed so many lives, there are many ideas on how to combat this
disease; but the method that most researchers agree with is the method of isolating those infected
to avoid possible transmission to other people [19, 20]. In  this process of contagion of the
coronavirus is simulated by means of a generalization of the logistic method to characterize the
process when it grows and when it decreases; indicating the moment of change of concavity of
the curve.
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. ).
In  diﬀerent real-life problems are dealt with using autonomous diﬀerential equations and
systems of equations, where examples are developed and other problems and exercises are pre-
sented for the reader to develop. The authors of  indicate a set of articles that form a collection
of several problems that model diﬀerent processes, using in their study the qualitative and ana-
lytical theory of diﬀerential equations for both autonomous and non-autonomous cases, in both
books the authors address the problem of epidemic development.
In , the probabilistic model is used to simulate population growth, which are applied to
the development of epidemics. There are multiple works devoted to the study of the causes and
the conditions under which an epidemic may develop, among which we can indicate (cf. ).
The problem of epidemic modeling has always been of great interest to researchers, such
as [6–11, 13] and . In this work we will apply the generalized logistic model, where the
case of the growth and decrease of the infected are applied in the same equation; to model the
development of epidemics, a model that allows forecasts of future behavior.
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 Physical exercise and strengthening of immunity. Mathematical mode 43
2 Model formulation
The human body works like a prefect machine, producing enzymes, hormones and substances
that will be used, according to the needs; but there are occasions that this is not enough due to
the causes of those needs, for example in the case of the appearance of a virus situation in which
in general artiﬁcial supplies are necessary to achieve an eﬀective coping with the situation.
In order to formulate the model using a system of diﬀerential equations, the following variables
will be introduced:
x1is the total concentration of the healthy cells at the moment t
x2is the total virus concentration at the time t.
In addition, ¯x1and ¯x2the values of the allowable concentrations of the healthy cells and the
virus respectively.
In this way the model will be given by the following system of diﬀerential equations.
(x0
1=x1f(x1, x2)
x0
2=x2g(x1, x2)(1)
The main objective of this work is to determine the equilibrium positions and to study the trajec-
tories of the system in the vicinity of the equilibrium positions. Suppose that the and functions
can be expressed by the following development, which is in correspondence with the relationship
between the virus and the home immune system, because between them there is a coping rela-
tionship, ﬁghting for survival.
f(x1, x2) = a1a2x2a3x1+f1(x1, x2)
g(x1, x2) = b1+b2x1+b3x2+g1(x1, x2)
Remark 1. Here are considering that the initial encounter is favorable to viruses, otherwise there
would be no viral process. The signs of the coeﬃcients of the previous development correspond to
the characteristics of the problem addressed.
The functions f1(x1, x2)and g1(x1, x2)from a physiological point of view represent external
inﬂuences, in particular the eﬀect of physical exercise, these disturbances from a mathematical
point of view are inﬁnitesimal of order superior in a neighborhood of the origin (0,0), that is to
say in its development only terms of superior degree appear.
Therefore, the system (1) takes the form:
(x0
1=a1x1a2x1x2a3x2
1+x1f1(x1, x2)
x0
2=b1x2+b2x1x2+b3x2
2+x2g2(x1, x2)(2)
If the functions f1(x1, x2) and g1(x1, x2) are identically null, then system (2) takes the form,
(x0
1=a1x1a2x1x2a3x2
1
x0
2=b1x2+b2x1x2+b3x2
2
(3)
The equilibrium positions of the system (3) are the points, P1(0,0), P20,b1
b3,P3a1
a3
,0and
P4a2b1a1b3
a2b2a3b3
,a1b2a3b1
a2b2a3b3.
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 44 A. Ruiz- D. Rodr´ıguez - S. anchez - Y. Alc´antara - A. Fern´andez - I. Mart´en - A. I. Ruiz
Analysis at point P1
To the point P1, if you have that the characteristic equation of the matrix of the linear part of
the system has the form, λ2+ (b1a1)λa1b1= 0. Here there is a positive eigenvalue, and
therefore the equilibrium position P1is unstable.
Example 1. Be the system
(x0
1= 2x1x1x23x2
1
x0
2=x2+ 2x1x2+ 2x2
2
5 10 15 20
t
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x1
Figure 1: Graph of y1(t) in the Example 1
5 10 15 20
t
0.01
0.02
0.03
0.04
x2
Figure 2: Graph of y2(t) in the Example 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
x1
0.02
0.04
0.06
0.08
0.10
x2
Figure 3: Graph of y1(t) vs y2(t) in the Example 1
Here we see that in spite of the origin of coordinates both the solution in x1(t) how in x2(t)
remain in tune, but this position of equilibrium is unstable as it changes in the graph of x1(t)
against x2(t). This is in correspondence with the results obtained from the theoretical point of
view.
It is not important to study the point P2because it is made explicit here that the concentra-
tions of healthy cells would disappear, and therefore the person will die.
Analysis at point P3
To the point P3it is necessary to make a change of variables to analyze the behavior of trajectories
in a neighborhood at that point. The transformation of coordinates,
x1=y1+a1
a3
x2=y2
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 Physical exercise and strengthening of immunity. Mathematical mode 45
Reduces the system (3) in the next system,
y0
1=a1y1a1a2
a3
y2a2y1y2a3y2
1
y0
2=a1b2a3b1
a3
y2+b2y1y2+b3y2
2
(4)
The characteristic equation of the matrix of the linear part of the system (4) has the form,
λ2+a3(a1+b1)a1b2
a3
λ+a1a3b1a2
1b2
a3
= 0
Theorem 1. The equilibrium position P3it is asymptotically stable if and only if a1b2< a3b1is
fulﬁlled.
Proof. Like a3(a1+b1)a1b2
a3
=a1b2a3b1
a3
a1and a1a3b1a2
1b2
a3
=a1(a1b2a3b1
a3
) then,
the eigenvalues associated with the system (4) are λ1=a1b2a3b1
a3
and λ2=a1, therefore if
the condition a1b2< a3b1therefore if the condition, λ1and λ2are negative and the system (4)
is asymptotically stable.
Remark 2. It follows that if the condition of Theorem 1 are satisﬁed, the virus will disappear,
with no consequences for the patient whenever the point coordinates correspond to the optimal
concentration values, otherwise it will be necessary to take the necessary prophylactic measures
to prevent the patient from falling into a coma.
Example 2. Given the following system that satisﬁes the conditions of Theorem 1
(y0
1=0.3y10.12 y20.2y1y20.5y2
1
y0
2=0.64 y2+ 0.1y1y2+ 0.1y2
2
Where a1b2=a3b1,λ1= 0 so the system (4) constitutes a critical case, then the matrix of
the linear part of the system (4) to have a zero eigenvalue and a negative one, in this case the
second method of Liapunov will be applied once this system is reduced to the quasi-normal form.
By means of a non-degenerate transformation z=Sy, the system (4) can be transformed into
the system,
(y0
1=Y1(y1, y2)
y0
2=λ2y2+Y2(y1, y2)(5)
when Sis the matrix
a2
a3
1
0 1
,
Y1(y1, y2) = a3b3a2b2
a3
y2
1+b2y1y2and
Y2(y1, y2) = a2(a3b3a2b2)
a2
3
y2
1+a2(a3+b2)
a3
y1y2a3y2
2
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 46 A. Ruiz- D. Rodr´ıguez - S. anchez - Y. Alc´antara - A. Fern´andez - I. Mart´en - A. I. Ruiz
2 4 6 8 10 t
0.1
0.2
0.3
0.4
0.5
y1
Figure 4: Graph of y1(t) in the Example 2
2 4 6 8 10 t
0.2
0.4
0.6
0.8
1.0
y2
Figure 5: Graph of y2(t) in the Example 2
0.02 0.04 0.06 0.08 0.10 y1
0.02
0.04
0.06
0.08
0.10
y2
Figure 6: Graph of y1(t) vs y2(t) in the Example 2
Theorem 2. The exchange of variables,
(y1=z1+h1(z1) + h0(z1, z2)
y2=z2+h2(z1)(6)
transforms the system (5) into almost normal form,
(z0
1=Z1(z1)
z0
2=λ2z2+Z2(z1, z2)(7)
Where h0and Z2cancel each other out z2= 0.
Proof. Deriving the transformation (6) along the trajectories of the systems (5) and (7) the
system of equations is obtained,
p2λ2h0+Z1(z1) = Y1dh1
dz1
Z1h0
z1
Z1h0
z2
Z2
λ2h2+Z2=Y2dh2
dz1
Z1
(8)
To determine the series that intervene in the systems and the transformation, we will separate
the coeﬃcients of the powers of degree p= (p1, p2) in the following two cases:
Case I): Doing in the system (9) z2= 0, is to say for the vector p= (p1,0) results the system,
Z1(z1) = Y1(z1+h1, h2)dh1
dz1
Z1
λ2h2=Y2(z1+h1, h2)dh2
dz1
Z1
(9)
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 Physical exercise and strengthening of immunity. Mathematical mode 47
The system (9) allows determining the series coeﬃcients, Z1,h1and h2, where for being the
resonant case h1= 0, and the remaining series are determined in a unique way.
Z1(z1) = a3b3a2b2
a3
z2
1a2b2(a3b3a2b2)
a1a2
3
z3
1+. . .
h2(z1) = a2(a3b3a2b2)
a1a2
3
z2
1+2a2
2(a3+b2)(a3b3a2b2)2
a1a2
3
z3
1+. . .
Case II): For the case when z26= 0 of the system (8) it follows that,
p2λ2h0=Y1(z1+h1, z2+h2)h0
z1
Z1h0
z2
Z2
Z2=Y2(z1+h1, h2+z2)
(10)
Because the series from system (7) are known expressions, the system (10) allows you to calculate
the series h0and Z2.
Z2(z1, z2) = a2(a3b3a2b2)
a2
3
z2
1+a2(a3+b2)
a3
z1z2a3z2
2+. . .
h0(z1, z2) = b2
a2
1
z1z2+. . .
This proves the existence of variable exchange.
Theorem 3. When a3b3< a2b2, the solution of the system (7) is stable, otherwise it is inestable.
Proof. Consider the Lyapunov function deﬁned positive,
V(z1, z2) = 1
2(z2
1+z2
2)
The derivative along the trajectories of the system (7) has the following expression,
dV
dt (z1, z2) = a3b3a2b2
a3
z3
1a1z2
2+R(z1, z2).
Therefore, taking into account that the point P3is in the ﬁrst quadrant, dv(z1, z2)
dt is negative
deﬁnite, because in function Rwe only have terms of a degree greater than 2 concerning z1and
higher than the second with respect to z2, therefore the system (3) is asymptotically stable.
Remark 3. In this case, the total concentration of healthy cells converges to the optimal concen-
tration, however, the total concentration of the virus converges to an acceptable concentration;
otherwise, the patient would enter a state of crisis at any time, in which case measures would
have to be taken to avoid worse consequences. For the convergence of total concentrations to per-
missible values, the action of external agents is feasible, where the inﬂuence of physical exercise
is included.
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 48 A. Ruiz- D. Rodr´ıguez - S. anchez - Y. Alc´antara - A. Fern´andez - I. Mart´en - A. I. Ruiz
Example 3. Given the following system that veriﬁes the conditions of the theorem 3
(z0
1=z1z2z1z22z2
1
z0
2= 2z1z2+z2
2
is obtained:
20 40 60 80 100
t
-0.10
-0.08
-0.06
-0.04
-0.02
0.02
z1
Figure 7: Graph of z1(t) in the Example 3
20 40 60 80 100
t
0.02
0.04
0.06
0.08
z2
Figure 8: Graph of z2(t) in the Example 3
-0.10 -0.08 -0.06 -0.04 -0.02 0.02 0.04 y1
0.02
0.04
0.06
0.08
0.10
0.12
y2
Figure 9: Graph of z1(t) vs z2(t) in the Example 3
As it is perceived in this system, the conditions of the theorems are fulﬁlled, and the conver-
gence of the total concentrations to the optical concentrations is graphically demonstrated, this
allows to see the reality of the theory developed in this paper.
Analysis at point P4
For biological interest, let’s assume the next conditions of existence a2b1> a1b3,a1b2> a3b1and
a2b2> a3b3, which guarantees that P4is in the ﬁrst quadrant. The transformation of coordinates,
x1=u1+a2b1a1b3
a2b2a3b3
x2=u2+a1b2a3b1
a2b2a3b3
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 Physical exercise and strengthening of immunity. Mathematical mode 49
Reduces the system (3) in the next system,
u0
1=a3(a2b1a1b3)
a2b2a3b3
u1a2(a2b1a1b3)
a2b2a3b3
u2a2u1u2a3u2
1
u0
2=b2(a1b2a3b1)
a2b2a3b3
u1b3(a3b1a1b2)
a2b2a3b3
u2+b2u1u2+b3u2
2
(11)
Whose characteristic equation of the matrix of the linear part of the system (11) has the form,
λ2+a3b1(a2+b3)a1b3(a3+b2)
a2b2a3b3
λ+(a1b2a3b1)(a2b1a1b3)
a2b2a3b3
= 0 (12)
Theorem 4. The equilibrium position P4it is asymptotically stable if and only if the condition
a3b1(a2+b3)> a1b3(a3+b2)is fulﬁlled.
Proof. The proof of this theorem is obtained from the conditions of Hurwitz’s theorem, by the
conditions of existence (a1b2a3b1)>0, (a2b1a1b3)>0 and (a2b2a3b3)>0, therefore
(a1b2a3b1)(a2b1a1b3)
a2b2a3b3
>0, so if a3b1(a2+b3)> a1b3(a3+b2) is fulﬁlled the coeﬃcients of
the characteristic equation are positive and the eigenvalues have negative real part.
Example 4. Given the following system that veriﬁes the conditions of the theorem 4
u0
1=0.0149754u12.99507u22u1u20.01u2
1
u0
2= 0.985025u1+ 0.00492512u2+ 0u1u2+ 0.01u2
2
is obtained:
2 4 6 8 10
t
0.02
0.04
0.06
0.08
0.10
u1
Figure 10: Graph of u1(t) in the Example 3
2 4 6 8
t
0.15
0.20
0.25
0.30
0.35
0.40
u2
Figure 11: Graph of u2(t) in the Example 3
Remark 4. It follows that if condition of Theorem 4 are fulﬁlled, the concentration values
of the virus and the cells will remain in the vicinity of the point P4, and if the coordinates of
that point are close to the optimal values of the concentrations, there will be no consequences for
the patient, so the concentrations of viruses and cells would be close to the values allowed for
the human body, so no there will be consequences, otherwise the necessary prophylactic measures
must be taken to avoid a fatal outcome.
Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51 50 A. Ruiz- D. Rodr´ıguez - S. anchez - Y. Alc´antara - A. Fern´andez - I. Mart´en - A. I. Ruiz
Acknowledgments
The authors appreciate the technical support and invaluable feedback provided by Luis Eugenio
Vald´es Garc´ıa, Digna de la Caridad Bandera Jim´enez, Adriana Rodr´ıguez Vald´es, Manuel de
Alvarez and Hilda Morandeira 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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