=Paper=
{{Paper
|id=Vol-3302/paper12
|storemode=property
|title=Modelling the Biostimulation Effect on the Development of an Infectious Disease in View of Diffusion Perturbations and the Organism’s Temperature Response
|pdfUrl=https://ceur-ws.org/Vol-3302/short6.pdf
|volume=Vol-3302
|authors=Serhii Baranovsky,Andrii Bomba,Oksana Pryshchepa
|dblpUrl=https://dblp.org/rec/conf/iddm/BaranovskyBP22
}}
==Modelling the Biostimulation Effect on the Development of an Infectious Disease in View of Diffusion Perturbations and the Organism’s Temperature Response==
https://ceur-ws.org/Vol-3302/short6.pdf
Modelling the Biostimulation Effect on the Development of an
Infectious Disease in View of Diffusion Perturbations and the
Organism’s Temperature Response
Serhii Baranovskya, Andrii Bombaa, Oksana Pryshchepaa
a
National University of Water and Environmental Engineering, 11 Soborna Str., Rivne, 33028, Ukraine
Abstract
The general model of a bioinfection was modified to predict the effect of biostimulation on
the dynamics of a viral infection taking into account small diffusion scattering in the
conditions of the organism’s temperature response. The solution of a singularly perturbed
model problem with a time-delay is reduced to a sequence of problem solutions without a
time-delay, for which numerical asymptotic approximations of the sought functions are
defined as a perturbation of the solutions of the corresponding degenerate problems. The
computer modelling results illustrate the ability of biostimulants to significantly influence the
dynamics of a chronic viral infection in the conditions of their diffusion dispersion and the
organism’s temperature response. It is emphasized that an excessive dose of bio-stimulating
agents can cause an increase in the antigen number to a level that significantly exceeds the
natural maximum and can pose a threat to the normal functioning of the target organ.
Keywords 1
The general model of a bioinfection, dynamical systems with a time-delay, asymptotic
methods, singularly perturbed problems.
1. Introduction
The formation of effective programs for infectious disease treatment requires the availability of a
reliable toolkit for predicting the dynamics of the disease taking into account the influence of various
types of therapeutic agents when they are used in various situational conditions. It is a common fact
that within the framework of the simplest infectious disease model, depending on the efficiency of the
immune system's response, four characteristic forms of the course of the disease are distinguished in
[1]: subclinical, acute, chronic, and lethal. Therefore, the presence of various forms of the disease
course, the establishment of which depends on the individual ability of the organism to produce an
immune response of the required strength, necessitates various therapeutic treatment strategies use.
It should be considered that the general methodology for constructing mathematical models of
both viral and bacterial infections presented in [1], along with the immune and other mechanisms of
the organism’s defence, is effectively used to predict the course of various diseases [2]. For instance,
in [3], this approach was used to build a model of antitumour immunity, and in [4] a modification of
the infectious disease model taking into account immunotherapy was proposed, which involves the
introduction of donor antibodies into the organism. In the general model of local tissue inflammation
presented in [5], Marchuk’s approach was used to describe the systemic reaction. An interesting
example of this methodology application is the mathematical model of the immune response to the
COVID-19 infection dynamics presented in [6], taking into account the influence of immunotherapy.
1
IDDM-2022: 5th International Conference on Informatics & Data-Driven Medicine, November 18-20, 2022, Lyon, France
EMAIL: svbaranovsky@gmail.com (S. Baranovsky); abomba@ukr.net (A. Bomba); o.v.pryshchepa@nuwm.edu.ua (O. Pryshchepa)
ORCID: 0000-0002-8056-2980 (S. Baranovsky); 0000-0001-5528-4192 (A. Bomba); 0000-0001-8032-1223 (O. Pryshchepa)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org)
We should also note that within the framework of the basic model, the presence of a constant
antigen concentration in a stationary state of the organism is a characteristic feature of the disease
course in a chronic form. The establishment of this disease form is caused by a sluggish immune
response. On the other hand, according to the model, a significant increase in the efficiency of the
immune response can be achieved due to a significant increase in the dose of an infection. In [1] an
assumption is made about the possibility of chronic infection treatment with an exacerbation, which
can be achieved by introducing a suitable biostimulant.
It is worth pointing out that due to the uneven distribution of antigens, antibodies, and other active
factors in the organism, the effect of their spatial dispersion can significantly affect the disease
dynamics. In [7,8], the authors presented the approach to consider diffusion disturbances of the
process in the conditions of concentrated administration of immunotherapy medicinal products into
the organism. A model reduction in the concentration of antigens in the infection zone because of
diffusion scattering is also demonstrated here. A similar effect occurs regarding medical drugs around
the areas they were used for. It is essential to consider this statement when predicting the dynamics of
a chronic disease in the conditions of using biostimulants.
A powerful tool to protect the organism from pathogens of infectious diseases is also a mechanism
of raising organism temperature. This mechanism provides both a decrease in the rate of pathogenic
microorganism reproduction and an increase in the activity of enzymes that stimulate immunological
reactivity. In [9], the appropriate generalization of the basic model of viral infection, taking into
account diffusion disturbances, concentrated effects, and the organism’s temperature reaction, is
proposed. The use of biostimulants for the treatment of chronic diseases can trigger the mechanism of
temperature increase, which will ultimately affect the development of the disease. Therefore,
considering this mechanism is also important when predicting the effect of biostimulants on the
disease course.
Thus, the purpose of this work is to modify the basic model of viral infection to predict the impact
of biostimulation on the disease dynamics taking into account diffusion disturbances and the
organism’s temperature response.
2. Modification of the Viral Infection Model to Consider the Influence of
Biostimulation and Diffusion Disturbances in the Conditions of the
Organism’s Temperature Response
The dynamics of a viral infection, taking into account the influence of biostimulation and diffusion
disturbances in the conditions of the body’s temperature response in the set
G (x,t ) : x , t , will be described by the following singularly perturbed system of
nonlinear differential equations (with delays 1 , 2 ):
V1 2V1
V1 +1 ( ) V1 1 FV
1 1 DV1 ,
t x 2
C1 V1 F1 2C
(m) 1 ( ) V1 (t 1 )F1 (t 1 ) С1 (C1 С1* ) 2 DC1 21 ,
dt V1F1 V2 F2 x
F1
dt C C
2 F
uF1 1C1 1 1 ** 2 ( f 1 1 1V1 )F1 DF1 21 ,
С x
(1)
V2 2V
uV2 2 F2V2 DV2 22 ,
t x
C2 V2 F2 2C
(m) 2 ( ) V2 (t 2 )F2 (t 2 ) С2 (C2 С2* ) 2 DC2 22 ,
dt V1 F1 V2 F2 x
F2
dt
C C
С 2 F
2C2 1 1 ** 2 ( f 2 2 2V2 )F2 DF2 22 ,
x
m 2m
V1 m ( ) m 2 Dm 2 ,
dt x
2
T V1 F1 T ( * ) u D 2
t x
for conditions
С1 (x,0) C10 (x) ,С2 (x,0) C20 (x) , m(x,0) m0 (x) , ( x,0 ) 0 (x) ,V1 (x,t ) V10 (x,t ) ,
(2)
F1 (x,t ) F10 (x,t ) ,V2 (x,t ) V20 (x,t ), F2 (x,t ) F20 (x,t ), t 0, max{1 , 2 }
where V1 (x,t ) , C1 (x,t ) , F1 (x,t ) are the number of viral infection antigens and their specific plasma
cells and antibodies, respectively; V2 (x,t ) , C2 (x,t ) , F2 (x,t ) are the numbers of elements of non-
pathogenic biostimulants, plasma cells and antibodies specific to them, respectively; m m(x,t ) is
relative characteristics of the lesion of the target organ; (x,t ) is the temperature at the point x at
the time t, 1 ( ) 1 (1 1 ( )) is the temperature-dependent rate of viral antigen reproduction,
0 1 *
11 const 0 ; 1 , 2 are coefficients related to the neutralization of viral antigens probability and
biostimulants, respectively, by their specific antibodies during their interaction;
k ( ) k0 (1 k1 ( * )) , k 1,2 are the temperature-dependent stimulation coefficients of
immunocompetent B-lymphocytes, k const 0 ; С1 , С2 are inverse values of the lifetime of
1
plasma cells specific for antigens and biostimulants, respectively; С1 , С2 are the number of plasma
* *
cells specific for antigens and biostimulants, respectively, in a healthy organism; 1 , 2 are rates of
one plasma cell specific to antigens and biostimulants antibody production, respectively; f 1 , f 2
are inverse values of the duration of the existence of antibodies specific to antigens and biostimulants,
respectively; 1 , 2 are the numbers of antibodies needed to neutralize one antigen and one
biostimulant element, respectively; is rate of damage to target the organ cells; DV1 , DV2 , 2 DC1 ,
2 DC2 , DF1 , DF2 , 2 Dm , D are the diffusion coefficients, respectively, of viral antigens and
biostimulants, plasma cells and antibodies specific to them, affected cells and thermal conductivity,
is a small parameter that characterizes the small influence of these components on the development
of the process; V1 (x,t ) , V2 (x,t ) , C1 (x) , C2 (x) , F1 (x,t ) , F2 (x,t ) , m (x) , (x ) are the bounded
0 0 0 0 0 0 0 0
sufficiently smooth functions. A monotonically decreasing smooth function provides consideration in
the model of the effect of reducing the productivity of plasma cell generation after reaching a certain
threshold value m* 0 of the level of damage to the target organ, (m )=1 , (1)=0 .
*
It is natural to believe that an exacerbation of viral infection as the result of biostimulation use will
cause an increase in body temperature, which will affect further dynamics of the disease. To take this
effect into account in the system (1), unlike the well-known general model of bioinfection, we
introduced an equation that describes the temperature dynamics. As in [1] here:
0 , VF (VF )* , T* e (VF (VF ) )
*
T * ,
T , VF (VF ) 1 e (VF (VF ) )
* *
(VF )* is a certain threshold value of VF -complexes, exceeding which stimulates an increase in the
temperature, T const 0 ; (x) is the temperature distribution in an organism in a healthy state. We
* *
should also note that the rise in organism temperature is ambiguous. On the one hand, a higher
temperature has a depressing effect on the reproduction and penetration of viral elements into a cell
and, to some extent, stimulates immunological reactivity [1]. In addition to that, excessively high
temperatures cause a destructive effect on the cell functioning of the target organ. We propose to
consider such a diverse influence by introducing a temperature-dependent recovery rate of the
affected organ: m ( ) m (1 m ( )) . At the same time, we will assume that the destructive
0 1 *
effect of high temperature on the functioning of the target organ cells begins after exceeding a certain
threshold value ** :
0 , ** , m* e ( )
**
m1 * .
m ,
**
1 e ( )
**
The establishment of excessively high temperatures for a long period of time is quite dangerous for
the ability of the organism to ensure the vital activity of the target organ. In this case, it is important to
reduce the temperature to acceptable limits, which, as a rule, is carried out by various therapeutic
means. To consider such a controlling influence, a source function was introduced into the
temperature dynamics equation u (x,t ) . Other functions of the source V1 (x,t ) , uF1 (x,t ) , uV2 (x,t ) ,
similarly to [8,9], provide concentrated change consideration in the quantities of antigens and
different specificity antibodies.
3. Numerical Asymptotic Approximation of the Model Problem Solution
Let us note that 1 2 . Then, similarly to [8,9], we find the solution of the model problem (1)-
(2) with a time-delay , by the step method as a sequence of solutions on the intervals
r t (r 1) , r 0,1,... . Note that because of such a step-by-step procedure for each separate interval,
with the already found solution at the previous stage, we will get the problem without a time-delay.
We will ensure the required level of smoothness of the solutions at time points , 2 , ... by imposing
additional conditions for their coordination. We find the approximation of the solution of each
singularly perturbed problem from the obtained sequence by the numerical asymptotic method, which
is similar to the way it was done in [8,9]. For example, in this case (m)=1 we get the following
problems for finding unknown functions (the members of asymptotic) V1(r ,i ) , C1(r ,i ) , F1(r ,i ) , V2 (r ,i ) , C2(r ,i ) ,
F2 (r ,i ) , m(r ,i ) , (r ,i ) ( i 01
, ,...,n ):
V1 (r ,0) 10V1(r ,0)
V1(r ) + 1V1(r ,0) F1(r ,0) ,
t 1 11 ((r ,0) * )
C1 (r ,0)
p1(r ,0)10 (1 11 ((r ,0) * )) 1(r ) С1 (C1(r ,0) С1* ),
t
F1 (r ,0) C1(r ,0) C2 (r ,0)
t
uF1(r ) 1C1(r ,0) 1
С1 f 1 1 1V1(r ,0) F1(r ,0) ,
V2 (r ,0)
uV2 (r ) 2V2 (r ,0) F2 (r ,0) ,
t
(3)
C2 (r ,0)
p2 (r ,0) 20 (1 21 ((r ,0) * )) 2 ( r ) С2 (C2 ( r ,0 ) С2* ),
t
F2 (r ,0) C1(r ,0) C2 (r ,0)
t
2C2 (r ,0) 1
С2 f 2 2 2V2 (r ,0) F2 (r ,0) ,
m(r ,0) m0 m(r ,0)
V1(r ,0) ,
t 1 m1 ((r ,0) * )
(r ,0)
T V1(r ,0) F1(r ,0) T ((r ,0) * ) u (r ) ,
t
V1(r ,0) (x,r ) V1(r 1) (x,r ) ,V2(r ,0) (x,r ) V2(r 1) (x,r ) , С1(r ,0) (x,r ) C1(r 1) (x,r ) ,
С2(r ,0) (x,r ) C2(r 1) (x,r ) , F1(r ,0) (x,r ) F1(r 1) (x,r ), F2(r ,0) (x,r ) F2(r 1) (x,r ),
m(r ,0) (x,r ) m(r 1) (x,r ) , (r ,0) (x,r ) (r 1) (x,r ) , r t (r 1) ;
....................................................
V1(r ,i )
d1(r ,0)(r ,i ) c1(r ,0)V1(r ,i ) 1 a1(r ,0) F1(r ,i ) b1(r ,0)V1(r ,i ) V1(r ,i ) ,
t
C1(r ,i )
p1(r ,i )1011(r ,i ) 1(r ) С1C1(r ,i ) C1(r ,i ) ,
t
F1(r ,i ) C1(r ,0) C2 (r ,0) C1(r ,0) (C1(r ,i ) C2 (r ,i )
1 C1(r ,i ) 1 C1(r ,i ) f 1 F1(r ,i )
t С** С**
1 1 a1(r ,0) F1(r ,i ) b1(r ,0)V1(r ,i ) F1(r ,i ) ,
V2 (r ,i )
2 a2 (r ,0) F2 (r ,i ) b2 (r ,0)V2 (r ,i ) V2 (r ,i ) ,
t
C2 (r ,i )
p2 ( r ,i ) 20 21(r ,i ) 2 ( r ) С2 C2 ( r ,i ) C2 (r ,i ) , (4)
t
F2 (r ,i ) C1(r ,0) C2 (r ,0) C2 (r ,0) C1(r ,i ) C2 (r ,i )
2 C2 (r ,i ) 1 C2 (r ,i ) f 2 F2 (r ,i )
t С** С**
2 2 a2 (r ,0) F2 (r ,i ) b2 (r ,0)V2 (r ,i ) F2 (r ,i ) ,
m(r ,i )
V1( r ,i ) dm (r ,0)(r ,i ) cm (r ,0) m(r ,i ) m (r ,i ) ,
t
(r ,i )
T a1(r ,0) F1(r ,i ) b1(r ,0)V1(r ,i ) T (r ,i ) (r ,i ) ,
t
V1(r ,i ) (x,r ) 0 ,V2 (r ,i ) (x,r ) 0 , С1(r ,i ) (x,r ) 0 , С2 (r ,i ) (x,r ) 0 , F1(r ,i ) (x,r ) 0 ,
F2 (r ,i ) (x,r ) 0 , m(r ,i ) (x,r ) 0 , (r ,i ) (x,r ) 0 , r t ( r 1 ) , i 1,2 ,...,n,
where C1(1) (x,0)=C10 (x) , C2(1) (x,0)=C20 (x) , m(1) (x,0)=m0 (x) , (1) (x,0)= 0 (x) , V1(1) (x,t )=V10 (x,t ) ,
V2(1) (x,t )=V20 (x,t ) , F1(1) (x,t )=F10 (x,t ) , F2(1) (x,t )=F20 (x,t ) , 1(r ) V1(r 1) (x,t ) ,
10
2 (r ) V2 (r 1) (x,t )F2 ( r 1 ) (x,t ) , a1(r ,0) V1(r ,0) , b1(r ,0) F1(r ,0) , c1(r ,0) ,
1 (0 ,r * )
1
1
10 11 V1(r ,0) m0 m0 m1 m(r ,0)
d1(r ,0) , cm (r ,0) , dm (r ,0) ,
(1 11 (0 ,r * ))2 1 m1 (0 ,r * ) (1 m1 (0 ,r * ))2
j 1 p1(r ,i j ) l 0 F1(r , j l )V1(r ,l ) F2 (r , j l )V2 (r ,l )
i i j
F1(r ,0)V1(r ,0) F V
j 0 1(r ,i j ) 1(r , j )
p1( r ,0 ) , p1( r ,i ) ,
l 1Fl (r ,0)Vl (r ,0) l 1Fl (r ,0)Vl (r ,0)
2 2
F2 (r ,i j )V2 (r , j ) j 1 p2 (r ,i j ) l 0 F1(r , j l )V1(r ,l ) F2 (r , j l )V2 (r ,l )
i i j
F2(r ,0)V2(r ,0) j 0
p2( r ,0 ) , p2 (r ,i ) .
F F
2 2
V
l 1 l (r ,0) l (r ,0)
V
l 1 l (r ,0) l (r ,0)
Functions V2 (r ,i ) , C2 (r ,i ) , F2 (r ,i ) , m (r ,i ) , (r ,i ) , similarly to [8,9], are expressed in equation terms
of asymptotic already found in the previous steps.
The solution to problems (3)-(4) at each stage will be found by numerical methods (for example,
Runge-Kutta methods), using the values of the functions V1(r 1) , F1(r 1) , V2(r 1) , F2 (r 1) already found at
the previous stages. Establishing space-time intervals of convergence when predicting real diseases
course is carried out similarly to [7-9].
4. Numerical Experiments Results
As mentioned above, the introduction of biostimulants into the organism will cause an
exacerbation of a chronic viral infection course, which is often accompanied by an increase in
temperature. Thus, an important problem in the formation of an effective treatment program is the
qualitative prediction of the exacerbation level and the disease course when introducing a certain dose
of biostimulants. Computer experiments were focused on the study of these aspects.
Fig. 1 illustrates a chronic viral infection model dynamics in the conditions of the organism’s
temperature reaction with the concentrated introduction of various doses of biostimulants into the
locus of the infection. Here, the effect of introducing biostimulants was modelled by the source
(V2 (t tV2 )2 V2 (x xV2 )2 )
function uV2 AV2 e ( tV2 100 ). The presented results of computer modelling prove the
ability of biostimulants introduced into the organism to significantly influence the chronic infection
dynamics. As should be expected, increasing the dose of biostimulants leads to a significant
exacerbation of the disease, which can exceed the natural level of its course and become quite
dangerous if the dosage of biostimulants is incorrect.
The effect of the protective mechanism of temperature increase on the model course of viral
infection with the concentrated use of biostimulants is shown in fig. 2, a), b). Here, the introduction of
biostimulants was modelled in the same way as in the previous case. Computer experiments were
conducted at different values Т of the rate of temperature increase. As expected, at higher rates of
organism temperature increase, the severity of viral infection decreases. At the same time, taking into
account such a temperature protection mechanism ensures the predictable stabilization of a chronic
disease at lower values of the number of viral elements. We should also note that according to the
assumptions of the model (1)-(2) biostimulants are not capable of multiplying independently in the
organism, and the increase in their number is determined by the procedure of introducing the
appropriate injection solution. Therefore, the mechanism of temperature increase practically does not
affect the dynamics of biostimulants.
5. Conclusions
The paper presents a general bioinfection model modification to predict the impact of
biostimulation on the dynamics of a viral infection, considering small diffusion scattering in the
conditions of the organism’s temperature reaction. It is proposed to apply an effective step-by-step
procedure to find out the solution of the model singular of the perturbed problem. According to the
procedure, firstly, we reduce the original problem with a time-delay to a sequence of problems
without a time-delay. And further on, step by step, considering the conditions of the required level of
smoothness, we find the numerical asymptotic approximation of the obtained problem solutions at
each time interval as a perturbation of the solutions corresponding degenerate problems.
Fig. 1. Dynamics of Chronic Infection Antigens in the Locus of Infection
with the Introduction of Different Doses of Biostimulants
Fig. 2. Dynamics of a) Chronic Infection Antigens;
b) Biostimulants in the Locus of Infection at Different Rates of Organism Temperature Increase
The given results of computer modelling illustrate the ability of the introduced in the organism
biostimulants to significantly affect the dynamics of the initial chronic viral infection even in the
conditions of their diffusion scattering and temperature reaction of the organism. Increasing the dose
of biostimulants leads to a significant exacerbation of the viral infection. It is emphasized that an
excessive dose of biostimulating agents can cause an increase in the number of antigens to a level that
significantly exceeds the natural maximum and may pose a threat to the normal functioning of the
target organ. It has also been demonstrated that high temps of increasing the temperature within
certain limits can effectively reduce the severity of the disease flow and provide predictable
stabilization of chronic disease in smaller values of the viral element number.
Let us note that the results presented in the paper were obtained in the cases of one-dimensional
homogeneous model environments, which is a certain limitation. In our opinion, it is promising to
generalize the proposed approach for predicting the dynamics of mixed infections with a
comprehensive account of diffusion perturbations in environments with essentially spatial effects, the
temperature response of the organism, and various kinds of concentrated effects [10–12] in
immunotherapy and pharmacotherapy, including those based on antiviral and antiviral immune
response models.
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