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 S´anchez (anniaruiz@nauta.cu)

Daniela Sara Rodr´ıguez Salmon (daniela.rodriguezs@uo.edu.cu)

Sandy S´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

Adolfo Arsenio Fern´andez Garc´ıa (adolfof@uo.edu.cu)

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.

Recibido 08/04/2021. Revisado 30/04/2021. Aceptado 12/07/2021.

MSC (2010): Primary 34Dxx; Secondary 34Cxx.

Autor de correspondencia: Sandy S´anchez Dom´ınguez

Physical exercise and strengthening of immunity. Mathematical mode 41

Resumen

En el presente trabajo se analiza c´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. 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. [17]).

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. [22]).

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 [21] 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. [5]).

In [3] 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 [4] 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 [17], 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. [12]).

The problem of epidemic modeling has always been of great interest to researchers, such

as [6–11, 13] and [15]. 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) = a1−a2x2−a3x1+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=a1x1−a2x1x2−a3x2

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=a1x1−a2x1x2−a3x2

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

P4a2b1−a1b3

a2b2−a3b3

,a1b2−a3b1

a2b2−a3b3.

Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51

44 A. Ruiz- D. Rodr´ıguez - S. 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+ (b1−a1)λ−a1b1= 0. Here there is a positive eigenvalue, and

therefore the equilibrium position P1is unstable.

Example 1. Be the system

(x0

1= 2x1−x1x2−3x2

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=−a1y1−a1a2

a3

y2−a2y1y2−a3y2

1

y0

2=a1b2−a3b1

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+a1a3b1−a2

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

=a1b2−a3b1

a3

−a1and a1a3b1−a2

1b2

a3

=−a1(a1b2−a3b1

a3

) then,

the eigenvalues associated with the system (4) are λ1=a1b2−a3b1

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=−0.3y1−0.12 y2−0.2y1y2−0.5y2

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) = a3b3−a2b2

a3

y2

1+b2y1y2and

Y2(y1, y2) = a2(a3b3−a2b2)

a2

3

y2

1+a2(a3+b2)

a3

y1y2−a3y2

2

Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51

46 A. Ruiz- D. Rodr´ıguez - S. 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) = Y1−dh1

dz1

Z1−∂h0

∂z1

Z1−∂h0

∂z2

Z2

λ2h2+Z2=Y2−dh2

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) = a3b3−a2b2

a3

z2

1−a2b2(a3b3−a2b2)

a1a2

3

z3

1+. . .

h2(z1) = −a2(a3b3−a2b2)

a1a2

3

z2

1+2a2

2(a3+b2)(a3b3−a2b2)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

Z1−∂h0

∂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(a3b3−a2b2)

a2

3

z2

1+a2(a3+b2)

a3

z1z2−a3z2

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) = a3b3−a2b2

a3

z3

1−a1z2

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. 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=−z1−z2−z1z2−2z2

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+a2b1−a1b3

a2b2−a3b3

x2=u2+a1b2−a3b1

a2b2−a3b3

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(a2b1−a1b3)

a2b2−a3b3

u1−a2(a2b1−a1b3)

a2b2−a3b3

u2−a2u1u2−a3u2

1

u0

2=b2(a1b2−a3b1)

a2b2−a3b3

u1−b3(a3b1−a1b2)

a2b2−a3b3

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)

a2b2−a3b3

λ+(a1b2−a3b1)(a2b1−a1b3)

a2b2−a3b3

= 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 (a1b2−a3b1)>0, (a2b1−a1b3)>0 and (a2b2−a3b3)>0, therefore

(a1b2−a3b1)(a2b1−a1b3)

a2b2−a3b3

>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=−0.0149754u1−2.99507u2−2u1u2−0.01u2

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. 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

Jes´us Salvador ´

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 P´ublica and managers of

the provincial government of Santiago de Cuba.

References

[1] Alak, K., Pilat, Kruger K. Current Vol. Kanowledge and new chailengesin exercise immunol-

ogy, Deutsche Zeitschriftfur Sportmedizin, 70(10) (2019), 250–260.

[2] Boch. W, Immunsistem und Sport - E ine wechselhafte Bziehung, Deutsche Zeitschriftfur

Sportmedizin, 70(10) (2019), 217–218.

[3] Ruiz Chaveco, A. I. et al., Modelagem matem´atica de problemas diversos, Curitiba: Appris,

Brazil, 2018.

[4] Ruiz Chaveco, A. I. et al., Applications of Diﬀerential Equations in Mathematical Modeling,

Curitiba: CRV, Brazil, 2016.

[5] Del Sol G. Y, The interferon that treats covid-19,https://www.granma.cu, 2020.

[6] Earn D. J., Rohani P., Bolker B. M., and Grenfell B. T., A simple model for complex

dynamical transitions in epidemics, Science, 287 (2000), 667–670.

[7] Esteva L. and Vargas C., Analysis of a dengue disease transmission model, Math. Biosci.,

150 (1998), 131–151.

[8] Greenhalgh. D and Das. R, Some threshold and stability results for epidemic models with a

density dependent death rate, Theoret. Population Biol., 42 (1992), 130–151.

[9] Gripenberg. G, On a nonlinear integral equation modelling an epidemic in an age-structured

population, J. Reine Angew. Math., 341 (1983), 147–158.

[10] Halloran. M. E, Watelet. L and Struchiner. C. J, Epidemiological e[U+FB00]ects of vaccines

with complex direct eﬀects in an age-structured population, Math. Biosci., 121 (1994), 193–

225.

[11] Halloran. M. E, Cochi. S. L, Lieu. T. A, Wharton. M, and Fehrs. L, Theoretical epidemiologic

and morbidity eﬀects of routine varicella immunization of preschool children in the United

States, Am. J. Epidemiol., 140 (1994), 81–104.

[12] Hamer. W. H, Epidemic disease in England, Lancet, 1(1906), 733–739.

[13] Hethcote. H. W, A thousand and one epidemic models, in Frontiers in Theoretical Biology,

Lecture Notes in Biomath. 100, Springer - Verlag, Berlin, 1994.

[14] Inkabi. S.E, Richter. P and Attakora. K, Exercise immunology: involved componentes and

varieties in diferente types of physical exercise, Scientect Journal of Life Sciennces, 1(1)

(2017), 31–35.

Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51

Physical exercise and strengthening of immunity. Mathematical mode 51

[15] Hethcote. H. W, Qualitative analysis of communicable disease models, Math. Biosci., 28

(1976), 335–356.

[16] Janeway. C.A, Jr., et al., Immunobiology, Garland Science, 2005.

[17] Maitland. H. B and Maitland. M. C, Cultivation of vaccinia virus without tissue culture,

Lancet., 212 (1928), 596–597. Doi: 10.1016/S0140-6736(00)84169-0

[18] Mayer. G, Immunology Chapter Two: Complement. Microbiology and Immunology, On Line

Textbook. USC School of Medicine, 2006.

[19] Montero. C. A, Covid 19 with science in China, On http://www.cubadebate.cu, 2020.

[20] Rodney. C. B. Mathematical Modeling, S˜ao Paulo, Brazil, 2004.

[21] Ruiz. A, et al., Coronavirus, A Challenge For Sciences, Mathematical Modeling, IOSR Jour-

nal of Mathematics (IOSR-JM), 16(3) (2020), 28–34. Doi:10.9790/5728-1603012834

[22] Sen. G, Viruses and interferons, Annu. Rev. Microbiol. 55 (2001), 294–300.

Divulgaciones Matem´aticas Vol. 22, No. 1 (2021), pp. 40–51