=Paper=
{{Paper
|id=Vol-3609/paper12
|storemode=property
|title=Generalizing the Antiviral Immune Response Model to Account for Adsorption Therapy in Conditions of Diffusion Perturbations and the Organism’s Temperature Response
|pdfUrl=https://ceur-ws.org/Vol-3609/short1.pdf
|volume=Vol-3609
|authors=Serhii Baranovsky,Andrii Bomba,Oksana Pryshchepa
|dblpUrl=https://dblp.org/rec/conf/iddm/BaranovskyBP23
}}
==Generalizing the Antiviral Immune Response Model to Account for Adsorption Therapy in Conditions of Diffusion Perturbations and the Organism’s Temperature Response ==
https://ceur-ws.org/Vol-3609/short1.pdf
Generalizing the Antiviral Immune Response Model to Account
for Adsorption, Diffusion Perturbations, and Temperature
Serhii Baranovskya, Andrii Bomba a, Oksana Pryshchepaa
a
National University of Water and Environmental Engineering, 11 Soborna Str., Rivne, 33028, Ukraine
Abstract
The antiviral immune response model has been generalized to take into account the effect of
adsorption therapy on the development of disease process under conditions of small diffuse
scattering of acting factors based on the synthesis of perturbation theory and modeling ideas
of the adsorption mass transfer process. A computational technology of step-by-step
asymptotic approximation of the solution to the corresponding model problem with time delay
as a perturbation of solutions to degenerate problems without delay has been developed. The
numerical experiment results demonstrate a predictive decrease in the concentration of viral
elements in the target organ through their neutralization into the organism by adsorbents
introduced. It is noted that the efficiency of adsorption drugs depends, in particular, on the
moment of time for their introduction, which must be considered when making decisions on
the formation of complex treatment programs using appropriate therapy.
Keywords 1
Antiviral immune response model, adsorption, dynamic systems with delay, asymptotic
methods, singularly perturbed problems.
1. Introduction
The practice of using adsorption agents for detoxification of the organism has a very long history
[1]. Charcoal, clay, ground tuffs and burnt horn were used to treat poisoning, dysentery, jaundice and
other diseases even in the time of Ancient Egypt, India, and Greece. The ability of adsorbents to bind,
retain and naturally remove from the organism not only various types of toxins, metabolic products and
heavy metals, but also pathogenic microorganisms and products of their vital activity ensures an
increase in the therapeutic effect with the complex application of traditional therapeutic procedures and
adsorption drugs. As is known, adsorption drugs, unlike pharmacological ones, have a different
mechanism of action [1, 2]. It is associated with the absorption of the substance as a result of its diffusion
into the pores of these adsorbents. At the same time, most of these drugs are not specific to certain types
of toxins or microorganisms. Adsorbents can connect only those substances whose molecules can
penetrate their internal pores. Considering the specifics of adsorption therapy application is important
for forecasting the infectious disease dynamics in decision-making systems in the development of
effective individual treatment programs.
The basic model of infectious disease and models of antiviral and antibacterial immune responses
[3] are proven and already classical tools for predicting the general patterns of infectious disease
courses. These models consider the antiviral humoral and cellular immune response and are the basis
of their new modifications and generalizations that make it possible to consider other factors and
mechanisms of the organism’s defence. Examples of effective application of viral and bacterial
infection mathematical models that are built according to the methodology described in [3] to consider
IDDM-2023: 6th International Conference on Informatics & Data-Driven Medicine, November 17-19, 2023, Bratislava, Slovakia
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)
© 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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Workshop ISSN 1613-0073
Proceedings
various protection mechanisms and features of various disease courses are given in [4]. Adaptation of
the simplest model of a viral disease to tumor-immune system interactions considering the role of
interleukins in immune processes was proposed in [5]. The basic model of the immune response was
modified to consider the features of the COVID-19 course and the conditions of the use of
immunotherapy [6]. The authors in [7] proposed to take into account the antigens' spatial dispersion
and other factors of the model as a diffuse perturbation of the process. This approach was used for
modifying the infectious disease model to consider the organism’s temperature response in [8] and for
generalizing the antiviral immune response model to comprehensively consider diffusion perturbations,
the temperature and the logistic dynamics of antigens in [9].
Let us note that the wide use of various micro- and nanoporous catalytic environments in many
industries turned to the active study of adsorption processes and the development of corresponding
mathematical models. In particular, in [10, 11] the single-component adsorption process in a certain
catalytic medium of l length of microporous structure particles of the radius R is described by the
¢¢ -qintra (qr¢ ) r=R , qt¢=Dintra (qrr¢¢ + 2qr¢ r ) under conditions c(0,x) = 0 ,
following model problem: ct¢=Dinter cxx
c(t,l ) = c¥ , cx¢ (t,0) = 0 , q(0,x,r ) = 0 , q(t,x,R) =k ×c(t,x), qr¢ (t,x,0) = 0 . This approach is based on the idea
of taking into account the interaction of mass transfer in the interparticle space (first equation) and mass
transfer into the interior of particles with concentration q(t,x,r ) (second equation), which is related to
concentration c(t,x) . This approach (in [12]) has been used for modifying the basic model of viral
infection to consider adsorption therapy and diffuse scattering effects, which makes it possible to predict
an additional decrease in the antigens’ concentration due to their absorption by adsorbents.
This work aims to generalize the antiviral immune response model to consider the effect of adsorbing
substances on the disease dynamics in the conditions of diffusion perturbations and temperature of the
organism.
2. Generalizing the antiviral immune response model to consider adsorption,
diffusion perturbations and temperature
To predict the viral disease process development considering adsorption therapy we will generalize
the modification of the immune response model to viral pathogens (in [9]) by introducing additional
components that describe the diffusion mass transfer of antigens inside the adsorbent particles and its
relationship with the mass transfer of antigens in the environment of the target organ. We will describe
the corresponding immune response dynamics taking into account the effect of adsorbents under
conditions of diffusion scattering and temperature of the organism, in the set G ={(t,x): tÎR+ , xÎR} by
such a singularly perturbed system of nonlinear differential equations with small parameter µ :
¶V f C
=wV + h1 (Q) ×CV æç1- V** ö÷ +nh2 (Q)×CV E - h3 (Q)× FV f - h4 (Q)× MV f -
¶t è C V ø
¶2V f æ ¶Wf ö
-h5V f (C* - m - CV ) +µDV 2 -µDW* ç ÷ ,
¶x è ¶r ø r = R
¶W f 2 æ ¶2W f 2 ¶W f ö
=µ DW ç 2 + ÷,
¶t è ¶r r ¶r ø
¶MV ¶ 2 MV
= h4 (Q) M ×V f - h6 MV +µ2 DMV , (1)
¶t ¶x2
¶H E
= h7 (Q)x(m) H E (t - t1o , x) MV (t - t1o , x) - h8 (Q) MV H E -
¶t
¶2 H E
-h9 (Q) MV H E E + h10 ( H E* - H E ) +µ2 DHE ,
¶x2
¶H B
= h11 (Q)x(m) H B (t - t2o , x) MV (t - t2o , x) - h12 (Q) MV H B -
¶t
¶2 H B
- h13 (Q) MV H B B + h14 ( H B* - H B ) +µ2 DH B ,
¶x2
¶E
= h15 (Q)x( m) E (t - t3o , x) H E (t - t3o , x) MV (t - t3o , x) - h16 (Q) MV H E E -
¶t
¶2 E
- h2 (Q)CV E + h17 ( E* - E ) +µ2 DE 2 ,
¶x
¶B
= h (Q)x( m) B (t - t4 , x) H B (t - t4 , x) MV (t - t4o , x) -
o o
¶t 18
¶2 B
- h19 (Q) MV H B B + h20 ( B* - B ) + µ2 DB 2 ,
¶x
¶P o o o ¶2 P
= h21 (Q)x(m) B (t - t5 , x) H B (t - t5 , x) MV (t - t5 , x) + h22 ( P* - P ) +µ2 DP 2 ,
¶t ¶x
¶F
¶t ( ) P
=wF + h23 P 1- ** - h3 (Q) ×V f F - h24 F +µDF 2 ,
P
¶2 F
¶x
¶СV ¶2CV
= h25V f (C - m - CV ) - h2 (Q) ECV - h26CV +µ DCV
* 2
,
¶t ¶x2
¶m ¶2 m
= h2 (Q)CV E + h26CV - h27 m +µ2 Dm 2 ,
¶t ¶x
¶Q o o o o ¶2 Q
= h28 F (t - t6 , x)V f (t - t6 , x)(1- h29 F (t - t6 , x )V f (t - t6 , x )) - h30 (Q-Q* ) +µDQ 2
¶t ¶x
under
V f (0, x) =V f0 ( x), MV (0, x) = MV0 ( x), H B (0, x ) = H B0 ( x), H E (0, x ) = H E0 ( x), E (0, x) = E 0 ( x),
P (0, x) = P0 ( x), B (0, x) = B0 ( x), F (0, x) = F 0 ( x), СV (0, x) = CV0 ( x), m(0, x) = m0 ( x), Q(0, x) =Q0 (x ),
H E (t, x) MV (t, x) =j1 (t, x), - t1o £ t < 0, H B (t, x ) MV (t, x) =j2 (t, x), - t2o £ t < 0, E (t, x ) H E (t, x )´ (2)
´MV (t, x) =j3 (t, x), - t3 £ t < 0, B (t, x ) H B (t, x) MV (t, x) =j4 (t, x ), - t £ t < 0, t = max{t4 ,t5 },
o o o o o
¶W f (t , x,0)
F (t, x)V f (t, x) =j5 (t, x), - t6o £ t < 0, W f (0, x,r )=W f0 ( x,r ), W f (t , x, R )=k×V f (t , x), =0
¶r
where Vf =Vf (t , x) , Wf =Wf (t , x) , MV = MV (t , x) , H E = H E (t , x) , H B = H B (t , x) , E = E(t , x) , B = B(t , x)
, P = P(t , x) , F = F (t , x) , СV = СV (t , x) , m = m(t , x) , Q=Q(t , x) are accordingly the number of antigens
that are present in the intercellular space of the target organ, the number of antigens in the middle of
particles of the adsorbent, the number of stimulated macrophages, the number of T-helper-lymphocytes
of cellular immunity, the number of T-helper-lymphocytes of humoral immunity, the number of T-
cell-effectors, the number of B-lymphocytes, the number of plasma cells, the number of antibodies, the
number of viruses infected cells, non-functional part of the damaged target organ and temperature at
the moment of time t at point x; M is the number of all macrophages in the organism, determined by
homeostasis; h1 (Q) = h10 (1+ h1* (Q-Q* )) is temperature dependent parameter ( h1* = const > 0 ), which is
related to antigen reproduction rate; hl (Q) = hl (1+ hl* (Q-Q* )) , l ={2,3,4,7,8,9,11,12,13,15,16,18,19,21}
are temperature dependent parameters ( hl* = const > 0 ), which are related to the immune system. The
parameter h28 characterizes the mechanism for increasing the organism’s temperature, which is related
to the number of Vf F -complexes: if their number does not exceed the certain threshold value (Vf F )* ,
then the temperature does not increase and h28 = 0 . If the value of Vf F exceeds this threshold, then the
temperature increases h28 = h28
*
=const > 0 . The other parameters hi of the model are determined
according to [3]. C* , H E* , H B* , E* , B* , P* , Q* are the number of the target organ cells, immunological
cells of the corresponding type and the temperature value maintained in the organism by homeostasis
respectively; CV** , P** are the maximum numbers of damaged and plasma cells respectively that
produce antibodies; Vf0 ( x) , MV0 ( x) , HE0 ( x) , HB0 ( x) , E0 ( x) , B0 ( x) , P0 ( x) , F 0 ( x) , CV0 ( x) , m0 ( x) ,
Q0 (x) , j1 (t, x) , j2 (t, x) , j3 (t , x) , j4 (t, x) , j5 (t , x) , Wf0 ( x,r ) is smooth limited functions of necessary
order. The function x(m) considers the effect of productivity decrease on the immune system due to
damage to the target organ, 0 £x(m) £1 . And eDV , e2 DW , e2 DMV , e2 DHE , e2 DHB , e2 DE , e2 DB , e2 DP ,
eDF , e2 DCV , e2 Dm are diffusion coefficients of the corresponding components of the process; µDQ is
the thermal conductivity coefficient of the target organ medium; eDW* is the coefficient that
characterizes the interaction of the diffusion redistribution of antigens in micropores of the adsorbent
particles on their diffusional scattering in the intercellular space; µ is a parameter intended to describe
the small influence of diffusion components compared to the influence of other process components.
Functions wV (t , x) , wF (t , x) are intended to describe changes concentrated in space and time according
to the number of antigens and antibodies [9].
3. Computing technology of numerical-asymptotic approximation of the
problem solution
Let us come to building a procedure for a step-by-step numerical asymptotic approximation of the
solution to the problem (1)-(2). We represent the delay values specified in the model as tio = si ×t , where
si Î N , i=1,6 , t> 0 . It should be noted that according to the results of the parameter identification of
the immune response model to viral infection in [3] using the data of clinical observations, the duration
of formation periods of new helper cells H E , H B and T-effectors and B-lymphocytes are practically
the same. Therefore, we will assume that s1 = s2 , s3 = s4 ( t1o =t2o and t3o =t4o ) and s6 < s1 < s3 < s5 .
We will also assume that equations in the system (1) is dimensionless [7-9, 12]. Then, similarly to
[7-9,12], we will find a solution to the model problem (1)-(2) with delay as a sequence of solutions to
problems without delay on the intervals (s -1)t£t £ st , s =1,2,... . To ensure the necessary order of
smoothness of partial solutions for t =t , t = 2t , … we add conditions for their consistency in the same
way as was done in [7-9,12]. According to a [7-9,12], we find the approximation of solutions obtained
as a result of singularly perturbed problems on each of the intervals (s -1)t£t £ st (s =1,2,...) , by the
asymptotic method, formally presenting them as asymptotic series: Vf ( s ) = å i=0µiVf ( s ,i ) + Rn (fs ) ,
n V
Wf ( s ) = å i=0µiWf ( s ,i ) + Rn ( fs ) , MV ( s ) = å i=0µi MV ( s ,i ) + RnM( Vs ) , H E( s ) = å i=0µi H E ( s ,i ) + RnH(Es ) ,
n W n n
H B( s ) = å i=0µi H B ( s ,i ) + RnH(Bs ) , E( s ) = å i=0µi E( s ,i ) + RnE( s ) , B( s ) = å i=0µi B( s ,i ) + RnB( s ) , P( s ) = å i=0µi P( s ,i ) + RnP( s ) ,
n n n n
F( s ) = å i=0µi F( s ,i ) + RnF( s ) , CV ( s ) = å i=0µi CV ( s ,i ) + RnC(Vs ) , m( s ) = å i=0µi m( s ,i ) + Rnm( s ) , Q( s ) = å i=0µi Q( s ,i ) + RnQ( s ) ,
n n n n
where Vf ( s ,i ) =Vf ( s ,i ) (t , x) , Wf ( s,i ) =Wf ( s ,i ) (t , x) , MV ( s ,i ) = MV ( s ,i ) (t , x) , HE ( s ,i ) = HE ( s ,i ) (t , x) ,
HB( s ,i ) = HB( s ,i ) (t , x) , E( s ,i ) = E( s ,i ) (t , x) , B( s ,i ) = B( s ,i ) (t , x) , P( s ,i ) = P( s ,i ) (t , x) , F( s ,i ) = F( s ,i ) (t , x) ,
CV ( s ,i ) =CV ( s ,i ) (t , x) , m( s ,i ) = m( s ,i ) (t , x) , Q( s ,i ) =Q( s ,i ) (t, x) ( i = 0,1,...,n ) required functions,
V V W W
Rn (fs ) = Rn (fs ) (t , x,µ) , Rn ( fs ) = Rn ( fs ) (t , x,µ) , RnM( Vs ) = RnM( Vs ) (t , x,µ) , RnH(Es ) = RnH(Es ) (t , x,µ) , RnH(Bs ) = RnH(Bs ) (t , x,µ) ,
RnE( s ) = RnE( s ) (t , x,µ) , RnB( s ) = RnB( s ) (t , x,µ) , RnP( s ) = RnP( s ) (t , x,µ) , RnF( s ) = RnF( s ) (t , x,µ) , RnC(Vs ) = RnC(Vs ) (t , x,µ) ,
Rnm( s ) = Rnm( s ) (t , x,µ) , RnQ( s ) = RnQ( s ) (t , x,µ) are relevant residual members. We will perform a regularizing
transformation r = r µ (0£ r £ R = R µ) [12] since the sizes of adsorption particles are small. We will
obtain problems for finding unknown functions (asymptotics) as a result of the "procedure of
equalization" [7-9, 12]. For example, for the period 0 £ t £ t6o , at x(m) =1 problems for finding
corrections to solutions of degenerate problems that take into account the effects of diffusion scattering
in the intercellular space and adsorption of antigens look as:
¶V f ( s ,i ) V V V V V V
=[aCVf ( s )CV ( s ,i ) + aQf( s ) Q( s ,i ) ] +nh2 [bE (f s ) E( s ,i ) + bCVf ( s )CV ( s ,i ) + bQf( s ) h2*Q( s ,i ) ] - h3 [dV ff ( s )V f ( s ,i ) +
¶t
V V V V V V V
+ dF f( s ) F( s ,i ) + dQf( s ) Q( s ,i ) ] - h4 M [qV ff ( s )V f ( s ,i ) - qQf( s ) Q( s ,i ) ] - h5 [ gV ff ( s )V f ( s ,i ) - gCVf ( s ) (CV ( s ,i ) + m( s ,i ) )] +Y( sf,i ) ,
¶Wf ( s ,i ) æ ¶2Wf ( s ,i ) 2 ¶Wf ( s ,i ) ö
=DW ç + ÷,
¶t è ¶r2 r ¶r ø
¶MV ( s ,i ) V V
= h4 M (qV ff ( s )V f ( s ,i ) + qQf( s ) Q( s ,i ) ) - h6 MV ( s ,i ) ,
¶t
¶H E ( s ,i )
= h7 F(HsE) Q( s ,i ) - h8 [aHHEE ( s ) H E ( s ,i ) + aMHVE ( s ) MV ( s ,i ) + aQH(Ek ) Q( s ,i ) ] - h9 [bEH(Es ) E( s ,i ) + bHHEE( s ) H E ( s ,i ) +
¶t
+bMHVE ( s ) MV ( s ,i ) + bQH(Es ) Q( s ,i ) ] - h10 H E ( s ,i ) + Y(HsE,i ) ,
¶H B ( s ,i )
= h11F(HsB) Q( s ,i ) - h12 [aHHBB ( s ) H B ( s ,i ) + aMHVB ( s ) MV ( s ,i ) + aQH(Bs ) Q( s ,i ) ] - h13 [bBH(Bs ) B( s ,i ) + bHHBB( s ) H B ( s ,i ) +
¶t
+bMHVB ( s ) MV ( s ,i ) + bQH(Bs ) Q( s ,i ) ] - h14 H B ( s ,i ) + Y(HsB,i ) ,
¶E( s ,i ) V
(3)
= h15F(Es ) Q( s ,i ) - h16 [aEE( s ) E( s ,i ) + aHE E ( s ) H E ( s ,i ) + aME V ( s ) MV ( s ,i ) + aQE( s ) Q( s ,i ) ] - h2 [bE (f s ) E( s ,i ) +
¶t
V V
+bCVf ( s )CV ( s ,i ) + bQf( s ) Q( s ,i ) ] - h17 E( s ,i ) + Y(Es ,i ) ,
¶B( s ,i )
= h18F(Bs ) Q( s ,i ) - h19 [aBB( s ) B( s ,i ) + aHB B ( s ) H B ( s ,i ) + aMB V ( s ) MV ( s ,i ) + aQB( s ) Q( s ,i ) ] - h20 B( s ,i ) + Y(Bs ,i ) ,
¶t
¶P( s ,i )
= h21F(Ps ) Q( s ,i ) - h22 P( s ,i ) + Y(Ps ,i ) ,
¶t
¶F( s ,i ) V V V
= h23 aP ( s ) P( s ,i ) - h3 [dV ff ( s )V f ( s ,i ) + dF f( s ) F( s ,i ) + dQf( s ) Q( s ,i ) ] - h24 F( s ,i ) + Y(Fs ,i ) ,
F
¶t
¶СV ( s ,i ) V V V V V
= h25 [ gV ff ( s )V f ( s ,i ) - gCVf ( s ) (CV ( s ,i ) + m( s ,i ) )] - h2 [bE (f s ) E( s ,i ) + bCVf ( s )CV ( s ,i ) + bQf( s ) Q( s ,i ) ]-
¶t
-h26CV ( s ,i ) +YC( sV,i ) ,
¶m( s ,i ) V V V
= h2 [bE (f s ) E( s ,i ) + bCVf ( s )CV ( s ,i ) + bQf( s ) Q( s ,i ) ]+ h26CV ( s ,i ) - h27 m( s ,i ) +Y(ms ,i ) ,
¶t
¶Q( s ,i )
=-h30Q( s ,i ) +Y(Qs ,i ) ,
¶t
V f ( s ,i ) (( s -1)t, x) = 0, MV ( s ,i ) (( s -1) t, x) = 0, H E( s ,i ) (( s -1) t, x) = 0, H B( s ,i ) (( s -1) t, x) = 0,
E( s ,i ) (( s -1)t, x) = 0, B( s ,i ) (( s -1) t, x) = 0, P( s ,i ) (( s -1) t, x) = 0, F( s ,i ) (( s -1) t, x) = 0,
CV ( s ,i ) (( s -1)t, x) = 0, m( s ,i ) (( s -1) t, x) = 0, Q( s ,i ) (( s -1) t, x) = 0, W f ( s ,i ) (( s -1) t, x, r )=0, (4)
¶W f ( s ) (t , x,0)
= 0, W f ( s ,i ) (t , x, R )=k×V f ( s ,i ) (t , x ), ( s -1) t£ t £ st, s =1,2,..., s6 ,
¶r
Vf h10 (CV** - 2CV ( s ,0) ) Vf h10 h1*CV ( s ,0) (CV** - CV ( s,0) ) V
where aCV ( s ) = ** ; aQ( s ) =- ** ; bCVf ( s ) = (h2* (Q( s ,0) -Q* ) +1)E( s ,0) ;
CV (1+ h1 (Q( s ,0) -Q ))
* *
CV (1+ h1 (Q( s ,0) -Q ))
* * 2
V V
V
bE f( s ) = (1+ h2* (Q( s ,0) -Q* ))CV ( s ,0) ; bQf( s ) = h2*CV ( s ,0) E( s ,0) ; dV ff ( s ) = (1+ h3* (Q( s ,0) -Q* ))F( s ,0) ;
V V V V
dF f( s ) = (1+ h3* (Q( s ,0) -Q* ))V f ( s ,0) ; dQf( s ) = h3*V f ( s ,0) F( s ,0) ; qV ff ( s ) = (h4* (Q( s ,0) -Q* ) +1); qQf( s ) = h4*V f ( s ,0) ;
V V
gV ff ( s ) = (C* - m( s ,0) - CV ( s ,0) ); gCVf ( s ) =V f ( s ,0) ;
i -1 éT
ö h1 CV ( s ,0) (CV - CV ( s ,0) )
i -r * **
( s ,r ) æ
Y( sf,i ) = åê ** çCV ( s ,i-r ) (1- CV ( s ,0) ) - åCV ( s ,k )CV ( s ,i -r -k ) ÷ - ** *
V
Q( s ,i -r ) ´
ø CV (h1 (Q( s ,0) -Q ) +1)
*
r =1 ë CV è k =1
T( s ,0) ù é i-1 æ r
ö
´T( s ,r ) - ** CV ( s ,r )CV ( s ,i-r ) ú +nh2 êå E( s ,i-r ) çCV ( s ,r ) (h2* (Q( s ,0) -Q* ) +1) + h2* åQ( s ,k )CV ( s ,r -k ) ÷ + h2* E( s ,0) ´
CV û ë r =1 è k =1 ø
i -1
ù é i-1 æ r
ö i -1
ù
´åQ( s ,r )CV ( s ,i -r ) ú - h3 êåV f ( s ,i -r ) ç F( s ,r ) (h3* (Q( s ,0) -Q* ) +1) + h3* åQ( s ,k ) F( s ,r -k ) ÷ + h3*V f ( s ,0) åQ( s ,r ) F( s ,i -r ) ú -
r =1 û ë r =1 è k =1 ø r =1 û
i -1 i -1 ¶ V f ( s ,i-1)
2
æ ¶W f ( s ,i -1) ö
- h4 Mh4* åQ( s ,r )V f ( s ,i-r ) + h5 åV f ( s ,i -r ) (m( s ,r ) + CV ( s ,r ) ) + DV - DW* ç ÷ , T( s ,i ) =
r =1 r =1 ¶x 2
è ¶r ø r = R
h1* å r =0Q( s ,i-r )T( s ,r )
i -1
h10 ¶2Vf ( s ,0) æ ¶Wf ( s ,0) ö
=- * , i = 2,3,...; T( s ,0) = * Vf
; Y( s ,1) = DV - DW* ç ÷ ; Vf ( s ,0) ,
h1 (Q( s ,0) -Q ) +1
*
h1 (Q( s ,0) -Q ) +1*
¶x 2
è ¶r ø r =R
Wf ( s,0) , MV ( s ,0) , HB ( s ,0) , HE ( s ,0) , B( s ,0) , E( s ,0) , F( s ,0) , P( s ,0) , m( s ,0) , CV ( s ,0) , Q( s ,0) are the known solutions
of the corresponding degenerate problems. Other coefficients and involved functions in (3) are also
known and expressed similarly through the previously found asymptotics. We apply numerical methods
(for example, Runge-Kut methods) to find the solution of the corresponding degenerate problem for
each interval (s -1)t£t £ st , s =1,2,... and problems for finding corrections using already found values
of the solutions to such problems at the previous stages. We establish the spatiotemporal intervals of
convergence in predicting the dynamics of real viral diseases in a similar way [7-9,12].
4. Numerical experiment
As noted above, the introduction of adsorption agents can reduce the number of antigens in the
organism as a result the severity of the viral infection will also decrease. Therefore, the assessment of
the effect of certain drugs on the disease dynamics is an important task in developing a comprehensive
treatment program using additional adsorption therapy. In general, we have focused the computer
experiments on studying the features of the influence of adsorption agents on the predictive dynamics
of the antiviral immune response.
Figure 1 illustrates the model dynamics of viral infection antigens in the infection locus in various
situational conditions: without taking into account the diffusion scattering and the absence of adsorption
therapy (curve 1); taking into account the effect of diffusion scattering and the absence of adsorption
antigens (curve 2); taking into account diffusion scattering and the effect of adsorption therapy (curve
3). Here, the number of antigens at the initial moment was equal to zero ( Vf0 ( x) = 0 ), but at the moment
tV =1 at the point xV = 0 there is a concentrated increase in the model number of antigens:
2 2
wV = AV e-aV (t -tV ) e-bV (x-xV ) . As expected, the lowest level of predicted growth of antigens is observed in
a situation when we consider the effects of diffusion scattering and absorption of antigens by adsorbent.
Thus, the presented results confirm the practicability of the adsorption drugs for additional
neutralization of antigens, which makes it possible to reduce, in particular, the critical level of disease
exacerbation. In addition, it should be noted that the timely introduction of adsorption drugs into the
body can more effectively control the growth of the antigen population. Figure 2 shows the predicted
Fig. 1. Dynamics of antigens at: µDV = 0.00 and Fig. 2. Dynamics of antigens at: tV = 0.0
µDW* = 0.00 (curve 1); µDV = 0.05 and µDW* = 0.00 (curve 1); tV = 2.5 (curve 2); tV = 5.0 (curve 3).
(curve 2); µDV = 0.05 and µDW* =0.05 (curve 3).
dynamics of antigen concentration in cases if there is an increase in their number at different times:
tV = 0.0 (curve 1); tV = 2.5 (curve 2); tV = 5.0 (curve 3). According to the numerical experiments, the
largest increase in the predicted number of antigens was obtained in the case when the interval between
the introduction of the adsorption drug and the concentrated increase in the number of antigens was the
largest. It means a decrease in the effectiveness of the corresponding therapy. Let us also note that the
results presented concern cases of one-time introduction of adsorbents into the organism at the initial
point in time. However, similar results will be expected even with repeated introduction of adsorption
drugs.
5. Conclusions
Based on the synthesis of approaches to modeling adsorption in the medium of porous particles and
the ideas of perturbation theory, the basic model of the immune response to viral pathogens is
generalized, that considers the effect of adsorption therapy in conditions of diffusion perturbations. We
have formed a step-by-step computational technology for asymptotically approximating the solution as
a perturbation of corresponding solutions to the degenerate problems by reducing the original model
singularly perturbed problem with delay to a sequence of problems without delay.
The results of computer modeling illustrate the effect of an additional reduction in the predicted
number of antigens as a result of their removal and neutralization by adsorbents. It has been shown that
the use of adsorption drugs, in particular during periods of disease exacerbation, provides an additional
reduction in the supercritical amount of antigens and accordingly alleviates the disease course. At the
same time, it is emphasized that an important condition for increasing the effectiveness of adsorption
substances is the timeliness of their introduction into the organism. Taking into account features of
adsorption influence is important for decision-making systems for the formation of complex treatment
programs using adsorption drugs to increase the effectiveness of traditional therapeutic agents.
In our opinion, it is promising to adapt and expand the limits of application of the presented our
approach to consider the conditions of competitive adsorption and diffusion, convection mass transfer,
mixed infections and biostimulation. It is also promising to take into account random factors [13,14]
and more complex function dependencies, which take into account the decrease in the immune system
efficiency with a significant number of target organ cells affected by the virus, and conditions of quasi-
periodic introduction of adsorption drugs.
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