=Paper=
{{Paper
|id=Vol-2753/paper14
|storemode=property
|title=Modeling of Infectious Disease Dynamics under the Conditions of Spatial Perturbations and Taking into account Impulse Effects
|pdfUrl=https://ceur-ws.org/Vol-2753/paper10.pdf
|volume=Vol-2753
|authors=Andrii Bomba,Sergii Baranovsky,Mykola Pasychnyk,Kateryna Malash
|dblpUrl=https://dblp.org/rec/conf/iddm/BombaBPM20
}}
==Modeling of Infectious Disease Dynamics under the Conditions of Spatial Perturbations and Taking into account Impulse Effects==
https://ceur-ws.org/Vol-2753/paper10.pdf
Modeling of Infectious Disease Dynamics under the Conditions
of Spatial Perturbations and Taking into account Impulse Effects
Andrii Bombaa, Serhii Baranovskya, Mykola Pasychnykb, Kateryna Malashc
a
National University of Water and Environmental Engineering, 11 Soborna Str., Rivne, 33028, Ukraine
b
Victor Polishchuk Regional Clinical Medical and Diagnostic Center by Rivne, 36 16th Lypnya Str., Rivne,
33028, Ukraine
c
Rivne State University of Humanities, 31 Plastova Str., Rivne, 33028, Ukraine
Abstract
The infectious disease mathematical model by Marchuk for conditions of diffusion
perturbations and taking into account impulse influences is generalized. The corresponding
singularly perturbed model problem with delays is reduced to a sequence of problems without
delay, for which the corresponding asymptotic developments of solutions are obtained. The
results of numerical experiments characterizing the impulse effects of infectious disease factors
on the immune response development in the conditions of spatially distributed diffusion
perturbations are presented. The model decrease of the antigens maximum level in the infection
epicenter due to their diffusion "erosion" in the viral disease process development is illustrated.
Keywords 1
Infectious disease model, dynamic systems, asymptotic methods, singularly perturbed
problems.
1. Introduction
The simplest (basic) infectious disease model which describes the most general laws of organism
humoral immune response to the viral antigens found is resulted in [1,2,3]. The infectious disease
process development in the model is determined by the nonlinear differential equations system with
delay, describing the rate of change in the viral antigens number, plasma cells, antibodies and the extent
of damage to the target organ. The relatively small number of active factors of the infectious disease
simplest model allows to establish strictly justified properties of its solutions, in particular, the stability
of inpatient solutions. In [1] it was shown that the stationary solution, which describes the healthy
organism state under certain conditions is asymptotically stable and retains this kind of resistance when
infecting a healthy organism with a dose of antigen V 0 , that does not exceed a certain level V * of
immunological barrier. The infectious disease basic model and its modifications in identifying their
parameters according to clinical observations allow to predict the nature of the course and outcome of
infectious disease, to investigate the general patterns of external influence on the process dynamics,
analyze and evaluate various treatment procedures. The generalization of an infectious disease simplest
model is antiviral and antibacterial immune response mathematical models[2,3]. In contrast to the
simplest model, in addition to the humoral immune response with the antibodies production, the cellular
type of immunity with the cytotoxic T-lymphocyte effectors accumulation is taken into account. As
mentioned in [2,3], antibodies are able to neutralize viral antigens that circulate freely in the blood or
lymph, but can’t penetrate into infected cells and neutralize viruses that multiply in them. Detection and
destruction of infected cells is carried out by cytotoxic T-killer lymphocytes. The antiviral immune
response model, as well as the basic model, is represented by a nonlinear differential equations system
IDDM’2020: 3rd International Conference on Informatics & Data-Driven Medicine, November 19–21, 2020, Växjö, Sweden
EMAIL: abomba@ukr.net (A. Bomba); svbaranovsky@gmail.com (S. Baranovsky); pasichnykua13@gmail.com (M. Pasychnyk);
katemalash@gmail.com (K. Malash)
ORCID: 0000-0001-5528-4192 (A. Bomba ); 0000-0002-8056-2980 (S. Baranovsky); ORCID: 0000-0001-7491-0371 (M. Pasychnyk);
ORCID: 0000-0003-4771-9349 (K. Malash )
©️ 2020 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)
with delay, describing the rates of change of free circulating in the body viruses Vf , antibodies F,
infected with viruses of target cells CV, T-killer lymphocytes Е, T-helper lymphocytes of cellular
immunity НЕ, T-lymphocytes helpers of humoral immunity HB, B-lymphocytes B, plasma P cells,
stimulated MV macrophages and non-functioning part of the virus-affected organ m.
The infectious disease basic model, the antiviral immune response model and other immunology
models [7,8,9] are built under the assumption that the "organism" is a homogeneous environment in
which all the process components are instantly mixed and, as a result, evenly distributed. On the other
hand, the antigens detection and the launch of the immune system of appropriate mechanisms of
response to them does not occur immediately after the antigens penetration into the body. That is, some
of the antigens that were not immediately neutralized by the immune system will penetrate into the
cells, multiply in them and spread further in the body. As a result, infection foci with higher antigens
concentrations forms around the affected cells. The antigens generated in the body will eventually be
redistributed from the initial infection foci to the surrounding uninfected areas, increasing the affected
area and decreasing the antigen concentrations values in the respective infection epicenters. An
approach for taking into account small spatially distributed diffusion effects on population dynamics is
presented in [4,5,6]. In particular, in [6] when modifying the simplest infectious disease model to take
into account the impact on the dynamics of the disease of certain drugs introduced into the body, articles
describing diffusion perturbations of the process active factors were added. The model decrease of the
maximum antigens concentration in the infection epicenter due to their diffusion "erosion" in the
infectious disease process development is illustrated. It is emphasized that even if the initial antigens
concentration in some infection area exceeds a certain critical value (immunological barrier), diffusion
"redistribution" for a certain period of time will reduce above critical values of antigen concentrations
to below critical level, and their subsequent disposal can be provided with the level of organism immune
protection available before infection.
But as in traditional infectious disease models of antiviral (antibacterial) immune response, and in
their modifications [5,6], which take into account spatially distributed diffusion perturbations, impulse
effects are not considered. The purpose of this work is to "fill" this gap.
2. Problem Statement
To take into account spatially distributed diffusion perturbations of infectious disease development,
it is proposed to modify the basic models by introducing components that describe small diffusion
spatially distributed effects ("redistributions"). Let us generalize this kind of modification of basic
models by G. Marchuk by introducing additional terms describing the influences that are close to the
impulse character. The spatio-temporal dynamics of the infectious disease process model components,
taking into account close to impulse influences in the domain GZ x,t : x ; 0 t will
be described by such a singularly perturbed dimensionless nonlinear differential equations system:
vt x,t h1 h2 f x,t v x,t h9 vxx x,t uV x,t ,
st x,t m h3v x,t f x,t h5 s x,t 1 2 h10 sxx x,t ,
(1)
ft x,t h4 s x,t f x,t h8v x,t f x,t h11 f xx x,t u F x,t ,
mt x,t h6 v x,t h7 m x,t 2 h12 mxx x,t ,
in conditions
s x,t0 s0 x , m x,t0 m0 x , v x,t v0 x,t ,
(2)
f x,t f 0 x,t , t0 t t0 ,
where v x,t V x,t Vm , s x,t С x,t C* , f x,t F x,t F ; V x,t , С x,t , F x,t , m x,t –
respectively, the concentration of antigens, plasma cells, antibodies and the value of the damage degree
to the target organ at point x at time t; Vm is some scale factor for the antigens concentration, for
example, the biologically acceptable antigens concentration in the body; C* , F are the plasma cells
and antibodies concentration in a healthy organism; h1 , h2 F * , h3 Vm F * C* , h4 f , h5 C ,
h6 Vm , h7 m , h8 Vm ; is the antigens reproduction rate; is the coefficient that takes into
account the result of the antigens interaction with antibodies; is the period of time (delay) required to
form a plasma cells cascade; С is value inverse to the plasma cells lifespan; is immune system
stimulation factor; is the rate of antibodies production by one plasma cell; f is the value inverse of
the antibodies duration; is the cost of antibodies to neutralize one antigen; is the rate of the target
organ cells damage; m is the rate of the target organ recovery, h9 , h11 , 2 h10 , 2 h12 are diffusion
redistribution coefficients of antigens, antibodies, plasma and affected cells, respectively, is a small
parameter that characterizes the respective components small impact compared to the dominant
components of the process. The function (m) takes into account the effect of reducing the antibody
production productivity in significant damage to the target organ. If is the maximum value of the degree
of the target organ damage, at which the immune system normal functioning is still possible, then on
the segment 0 m m* value m is equal to one, regardless of the lesion, the immunological organs
function fully. If m* m 1 , the body efficiency is rapidly declining. Functions uV x,t , uF x,t ,
describing a close to pulse change, respectively, of the antigens and antibodies concentrations with
maximum values at points xV j , xF j at times tV j , tF j will be represented as
nV nF
uV x,t AV j e , uF x,t AF j e
V j x xV j V j t tV j F j x xF j F j t tF j
2 2 2 2
e e (3)
j 1 j 1
where AV j , AF j , V j , F j , V j , F j are parameters that determine the pulse intensity and "duration".
Note that in the future as initial, we will take, in particular, the conditions that characterize the
stationary solution, which corresponds to the healthy organism state, namely
s x,t0 1, m x,t0 0 , v x,t 0 , f x,t 1, t0 t t0 . (4)
At the beginning we consider the case when the level of damage to the target organ by antigens
remains such that it does not lead to a decrease in the productivity of antibody production, m 1 .
Then the solution of problem (1) - (2) with delay is reduced to a sequence of problems without delay
[10]:
v0t x,t h1 h2 f0 x,t v0 x,t h9 v0xx x,t uV x,t ,
s0t x,t h3v x,t f x,t h5 s0 x,t 1 h10 s0xx x,t ,
0 0 2
f x,t h s x,t f x,t h v x,t f x,t h f x,t u x,t ,
0t 4 0 0 8 0 0 11 0 xx F
(5)
m0t x,t h6 v0 x,t h7 m0 x,t h12 m0xx x,t ,
2
s x,t s 0 x , m x,t m0 x ,
0 0 0 0
v0 x,t0 v x,t0 , f0 x,t0 f 0 x,t0 , t0 t t0 ;
0
…………………………………………………………………………………
vkt x,t h1 h2 f k x,t vk x,t h9 vkxx x,t uV x,t ,
skt x,t h3vk 1 x,t f k 1 x,t h5 sk x,t 1 h10 skxx x,t ,
2
f x,t h s x,t f x,t h v x,t f x,t h f x,t u x,t ,
kt 4 k k 8 k k 11 k xx F
mkt x,t h6 vk x,t h7 mk x,t 2 h12 mkxx x,t , (6)
sk x,t0 k sk 1 x,t0 k , mk x,t0 k mk 1 x,t0 k ,
v x,t k v x,t k , F x,t k F x,t k ,
k 0 k 1 0 k 0 k 1 0
t0 k t t0 k 1 , k 1,2,... .
…………………………………………………………………………………
To ensure sufficient smoothness of the corresponding solutions at t t0 , t t0 2 , …, t t0 n
, in addition to the traditional smoothness conditions with respect to the initial conditions functions in
the infectious disease model, it is necessary to impose conditions of their consistency at t t0 , t t0 ,
… [11]. In particular, the condition must be met
s0t x,t0 h3v0 x,t0 f 0 x,t0 h5 s0 x,t0 1 2h10 s0xx x,t0 .
Given that the diffusion redistributions of active factors are small compared to other components of
the infectious disease process, we use the asymptotic method to solve the corresponding singularly
perturbed model problems (4) - (5) [11,12]. In particular, the solutions of problems (4) - (5) are formally
n n
represented as asymptotic series v j (x,t ) i vij (x,t ) Rnjv x,t , , s j (x,t ) i sij (x,t ) Rnjs x,t , ,
i o i o
n n
f j (x,t ) i fij (x,t ) Rnjf x,t , , m j (x,t ) i mij (x,t ) Rnjm x,t , as perturbations of the
i o i o
corresponding degenerate problems solutions [4], where j 0,1,...,k,... , vij (x,t ), sij (x,t ) , fij (x,t ) ,
mij (x,t ) are the required functions (members of the asymptotics) Rnjv x,t, , Rnjs x,t, , Rnjf x,t, ,
Rnjm x,t, are relevant residual members. After substituting the asymptotic series and performing the
standard procedure of "equalization", we obtain the following problems to determine the functions
vij (x,t ) , sij (x,t ) , fij (x,t ) , mij (x,t ) i 0,1,...,n , j 0,1,...,k ,... :
v0 ,0 t x,t h1 h2 f0 ,0 x,t v0 ,0 x,t uV x,t ,
s x,t h v 0 x,t f 0 x,t h s x,t 1 ,
0 ,0 t 3 5 0 ,0
f0,0 t x,t h4 s0 ,0 x,t f0 ,0 x,t h8v0 ,0 x,t f 0 ,0 x,t u F x,t ,
(7)
m0 ,0 t x,t h6 v0 ,0 x,t h7 m0 ,0 x,t ,
s0 ,0 x,t0 s 0 x , m0 ,0 x,t0 m0 x ,
v0 ,0 x,t0 v x,t0 , f 0 ,0 x,t0 f x,t0 , t0 t t0 ,
0 0
v1,0 t x,t h1v1,0 x,t h2 a0 ,0 x,t f1,0 x,t b0 ,0 x,t v1,0 x,t v1,0 x,t ,
s x,t h a x,t f x,t b x,t f x,t h s x,t ,
1,0 t 3 0 ,0 1,0 0 ,0 1,0 5 1,0
f1,0 t x,t h4 s1,0 x,t f1,0 x,t h8 a0 ,0 x,t f1,0 x,t b0 ,0 x,t v1,0 x,t f 1,0 x,t ,
(8)
m1,0 t x,t h6 v1,0 x,t h7 m1,0 x,t ,
s1,0 x,t0 0 , m1,0 x,t0 0 ,
v1,0 x,t0 0 , f1,0 x,t0 0, t0 t t0 ,
......................................................
vi,0 t x,t h1vi ,0 x,t h2 a0 ,0 x,t fi ,0 x,t b0 ,0 x,t vi ,0 x,t V i ,0 x,t ,
s x,t h a x,t f x,t b x,t v x,t h s x,t x,t ,
i ,0 t 3 0 ,0 i ,0 0 ,0 i ,0 5 i ,0 si ,0
fi ,0 t x,t h4 si ,0 x,t fi ,0 x,t h8 a0 ,0 x,t fi ,0 x,t b0 ,0 x,t vi ,0 x,t f i ,0 x,t ,
(9)
mi,0 t x,t h6 vi ,0 x,t h7 mi ,0 x,t m i ,0 x,t ,
si ,0 x,t0 0 , mi ,0 x,t0 0 ,
vi ,0 x,t0 0 , fi ,0 x,t0 0 , t0 t t0 ,
v0 ,k t x,t h1 h2 f0 ,k x,t v0 ,k x,t ,
s x,t h v
0 ,k t 3 0 ,k 1 x,t f0 ,k 1 x,t h5 s0 ,k x,t 1 ,
f0,k t x,t h4 s0 ,k x,t f0 ,k x,t h8v0 ,k x,t f 0 ,k x,t ,
m0 ,k t x,t h6 v0 ,k x,t h7 m0 ,k x,t , (10)
s x,t k s x,t k , m x,t k m x,t k ,
0 ,k 0 0 ,k 1 0 0 ,k 0 0 ,k 1 0
v0 ,k x,t0 k v0 ,k 1 x,t0 k , f0 ,k x,t0 k f0 ,k 1 x,t0 k ,
t0 k t t0 k 1 ,
v1,k t x,t h1v1,k x,t h2 a0 ,k x,t f1,k x,t b0 ,k x,t v1,k x,t v1,k x,t ,
s x,t h a x,t f
1,k t 3 0 ,k 1,k 1 x,t b0 ,k x,t v1,k 1 x,t h5 s1,k x,t ,
f1,k t x,t h4 s1,k x,t f1,k x,t h8 a0 ,k x,t f1,k x,t b0 ,k x,t v1,k x,t f 1,k x,t ,
m1,k t x,t h6V1,k x,t h7 m1,k x,t , (11)
s x,t k s x,t k , m x,t k m x,t k ,
1,k 0 1,k 1 0 1,k 0 1,k 1 0
1,k
v x,t 0 k v 1,k 1 x,t0 k , f1,k x,t0 k f1,k 1 x,t0 k ,
t0 k t t0 k 1 ,
......................................................
vi,k t x,t h1vi ,k x,t h2 a0 ,k x,t fi ,k x,t b0 ,k x,t vi ,k x,t vi ,k x,t ,
s x,t h a x,t f
i ,k t 3 0 ,k i ,k 1 x,t b0 ,k x,t vi ,k 1 x,t h5 si ,k x,t si ,k x,t ,
fi ,k t x,t h4 si ,k x,t fi ,k x,t h8 a0 ,k x,t fi ,k x,t b0 ,k x,t vi ,k x,t f i ,k x ,t ,
mi,k t x,t h6 vi ,k x,t h7 mi ,k x,t m i ,k x,t , (12)
s x,t k s x,t k , m x,t k m x,t k ,
i ,k 0 i ,n1 0 i ,k 0 i ,k 1 0
vi ,k x,t0 k vi ,k 1 x,t0 k , fi ,k x,t0 k fi ,k 1 x,t0 k ,
t0 k t t0 k 1 ,
where a0 ,j x,t v0 ,j x,t , b0 ,j x,t f0 ,j x,t ;
2 v0 , j x,t 2 f0 , j x,t
v1, j x,t h9 , f 1, j x,t h11 ;
x2 x2
i 1 2 vi1, j x,t
vi , j x,t h2 vk , j x,t fik , j x,t h9 ,
l 1 x2
i 1 2 si2 , j x,t
s i , j x,t h3vk , j x,t fik , j x,t h10 ,
l 1 x 2
i 1 2 fi1, j x,t
f i , j x,t h8 vk , j x,t fik , j x,t h11 ,
l 1 x 2
2 mi2 , j x,t
m i , j x,t h12 , i 2,3,...,n , j 0,1,...,k ,... .
x2
Note that the proposed approach is easy to transfer to other, in particular, finite domains GZ . In this
case, of course, instead of those described above, more complex schedules should be used (see, for
example, [9,10,11]). Estimation of residual terms Rnjv x,t, , Rnjs x,t, , Rnjf x,t, , Rnjm x,t, and
establishment of spatio-temporal intervals of convergence for forecasting of concrete processes are
carried out on the basis of the principle of type of a maximum similarly to [4,11,12].
3. Numerical Experiments Results
Numerical experiments within this model investigated the features of the body's humoral immune
response to viral antigens and the corresponding spatiotemporal dynamics of infectious disease for
different situational conditions under diffusion disturbances and taking into account close to pulse
changes in antigen and antibody concentrations in certain areas of the body.
Fig. 1 presents the spatial and temporal dynamics of antigen concentrations with the development
of infectious disease in the chronic form according to model (1) - (2) in cases without taking into account
(Fig. 1, a)) and taking into account (Fig. 1, b)) small spatially distributed diffusion influences under
conditions that the initial distribution of antigens concentration is uneven in space
v x,t0 v0 x e (x ) (there is a separate center of infection of an organism with a maximum
2
antigens concentration in a point x0 ). These results show that in the case without diffusion
"redistribution", the development of the process according to the "scenario" of chronic disease is in
some way "localized" in some area, which corresponds to the area with higher relative to some
immunological barrier values of antigen concentration at the initial time. The influence of diffusion
"redistribution" of the antigens initial concentration smooths out such "localization" of the model
process. The corresponding model change with time of antigen concentration in the conditions of the
chronic form of the disease at different intensities of diffusion "redistribution" in the infection epicenter
is shown in Fig. 2. Under conditions without diffusion "redistribution" (ε = 0) the antigens
v v
t t
x x
a) b)
Figure 1: Spatial-temporal dynamics of antigen concentration under conditions of non-uniform in
space distribution of antigen concentration at the initial time t0 at a) 0,000 ; b) 0,025
Figure 2: Dynamics of antigen concentration of model (1) - (2) in the infectious epicenter disease in
chronic form at different levels of diffusion intensity
v v
t t
x x
a) b)
Figure 3: Spatial-temporal dynamics of antigen concentration under conditions of pulsed exposure at
a) 0,000 ; b) 0,025
concentration in the body increases to some maximum level, then decreases and is established over time
at some stationary level. As the intensity of diffusion "redistribution" increases, the maximum value of
antigen concentration in the epicenter decreases, and starting from some value of intensity the maximum
antigens concentration will not increase, ie the level of immune protection adopted in the model in the
presence of diffusion redistribution moment of time to prevent a model increase in the maximum
antigens concentration in the infection epicenter and over time without "exacerbations" to reduce their
concentration to some stationary level.
Spatial and temporal dynamics of antigen concentration with the development of infectious disease
in a situation where at the initial time the values of the active factors of the process correspond to the
values of the stationary solution, which characterizes the state of a healthy organism is shown in Fig. 3.
The change in the concentration of antigens in the body is close to the pulse nature with the maximum
value at some point xV at time tV . As already mentioned, the humoral type of immune response
provides antibody neutralization of viral antigens that circulate freely in the blood or lymph. Depending
on the immune system state, individual antigens can enter the cells of the target organ, where they can
multiply and cause its destruction. As a result, a cell with a high antigens concentration appears at the
site of the destroyed cell, which causes a close to impulse effect. The obtained results, as in the case of
the initial condition with uneven distribution of antigen concentration, illustrate a certain "localization"
in some area of the "scenario" of the disease in the chronic form (Fig. 3, a)) in the case without diffusion
"redistribution". As in the previous case, the diffusion "redistribution" of the antigens concentration
in the region of the impulse effect smooths the "localization" of the model process. Note that the
presented generalization of the mathematical model of infectious disease taking into account the
impulse effects under diffusion perturbations allows us to investigate the effects caused by several close
to pulse sources of antigens with maximum values at different points xV j and in different time tV j .
Quite an effective procedure for the treatment of infectious diseases is the use of immunotherapy
[6]. Donor antibodies can be administered by injection, which in this model will be presented as close
to pulsed sources of donor antibodies with maximum values at points xF j in time tF j . Figure 4 presents
the spatiotemporal dynamics of antigen concentrations with the development of infectious disease in
the chronic form in the presence of close to pulsed sources of antigens and donor antibodies with
maximum values at one point ( xV xF ), but different time ( tV tF ) under conditions excluding
diffusion "redistribution" (Fig. 4, a)) and taking into account the diffusion "redistribution" (Fig. 4, b).
The results illustrate, as expected, the decrease in antigen concentration due to the introduction of donor
antibodies in the appropriate area. In conditions without diffusion "redistribution", the effect of donor
antibodies in the cell of their introduction is longer. In terms of diffusion "redistribution" introduced
donor antibodies over time "blur" to a larger "territory" of the body, resulting in faster consumption of
antigen neutralization and their impact on the disease process is less long.
Fig. 5 presents the spatial and temporal dynamics of viral antigens concentrations with the
development of infectious disease in the chronic form in the presence of one close to the pulse source
of antigens and two close to pulse sources of donor antibodies with maximum values at point ( xV xF )
and at different time ( tV tF1 tF 2 ), without taking into account (Fig. 5, a)) and taking into account the
diffusion "redistribution" (Fig. 5, b)). As can be seen from the presented results, if the diffusion
"redistribution" is taken into account, the intensity and duration of action of donor antibodies introduced
into the body are smaller than in the model situation without such "redistribution" caused by diffusion
"erosion" of donor antibodies from their injection site.
v v
t t
x x
a) b)
Figure 4: Spatial-temporal dynamics of antigen concentration in the presence of close to pulsed
sources of antigens and donor antibodies
v v
t t
x x
a) b)
Figure 5: Spatial-temporal dynamics of antigen concentration in the presence of several close to
pulsed sources of donor antibodies
Figure 6: Dynamics of the main active factors of model (1) - (2) of an infectious disease in a chronic
form in the presence of several pulse sources of donor antibodies at different levels of diffusion
influence intensity
Figure 6 illustrates the change in the model dynamics of active infectious disease factors in the
chronic form in the infection epicenter in a situation with one pulsed source of viral antigens and two
pulsed sources of donor antibodies depending on the intensity of diffusion "redistribution" (parameter
ε). The presented results show a decrease in the maximum value of the number of antigens, antibodies,
plasma and affected cells in the infection epicenter with increasing intensity of diffusion
"redistribution", which leads to a decrease in model "severity" of infectious disease. The introduction
of close to the pulse of several sources of donor antibodies allows in this model to further reduce the
antigens concentration in the infection epicenter. In particular, in a situation without taking into account
the diffusion "redistribution", the effect of donor antibodies causes a decrease in the concentration of
antigens to values close to zero. Given the diffusion effect, the action of donor antibodies due to their
"redistribution" is less effective and leads to a smaller decrease in the antigens concentration. Therefore,
to achieve the desired therapeutic effect, it is necessary to change the treatment procedure, for example,
to increase the frequency of administration of antibodies, or their number in one injection.
4. Conclusions
Based on the modification of the simplest infectious disease model, an approach is presented to take
into account close to impulse influences on the development of an infectious disease in the conditions
of small spatially distributed diffusion perturbations. The corresponding model problem with delay is
reduced to a sequence of problems without delay, for which representations of the required functions
in the form of asymptotic series as perturbation of solutions of the corresponding degenerate problems
are constructed.
The numerical experiments results illustrate the decrease in the maximum value of the antigens
concentration in the infection epicenter due to their diffusion "redistribution", including for different
situational conditions. It has been shown that even when the initial antigen concentration V 0 or the
intensity of the pulsed antigen source in some area of the infection zone exceeds a certain critical value
V * the diffusion "redistribution" over a period of time can reduce the critical antigen concentration to
a level below the critical the reduction can be provided by the available level of antibodies, as well as
a more economical mode of administration of donor antibodies by injection. That is, under this model,
the "severity" of the infectious disease in such cases will decrease, so to speak, at low cost.
The developed computational procedure can be an element of designing specialized expert systems
for making a wide range of decisions such as: can we in this case according to the values of relevant
input data, in particular, on the intensity of diffusion "redistribution", on the level of immune protection
available in the body, or, otherwise, to carry out external therapeutic effects. And in a situation where
it is decided that such an effect should be exercised, in particular by means of injections of donor
antibodies, to establish the most rational frequency of their introduction and an acceptable concentration
of antibodies for each injection.
It is also promising to take into account such impulse effects in spatially distributed diffusion
perturbations in the study of viral and bacterial diseases on the basis of more general models, in
particular, models of antiviral and antibacterial immune response by Marchuk and Petrov [5].
5. References
[1] Marchuk G. I. Mathematical models in immunology, Moskow: Nauka, 1980.
[2] Marchuk G. I. Mathematical models in immunology. Computational methods and experiments,
Moskow: Nauka, 1991.
[3] Marchuk G.L. Mathematical models of immune response in infectious diseases, Dordrecht:
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