=Paper=
{{Paper
|id=Vol-2516/paper12
|storemode=property
|title=Numerical Analysis of Results Simulation of Cyber-Physical Biosensor Systems
|pdfUrl=https://ceur-ws.org/Vol-2516/paper12.pdf
|volume=Vol-2516
|authors=Vasyl Martsenyuk,Andriy Sverstiuk,Aleksandra Klos-Witkowska,Kozodii Nataliia,Oksana Bagriy-Zayats,Ihor Zubenko
|dblpUrl=https://dblp.org/rec/conf/ictes/MartsenyukSKNBZ19
}}
==Numerical Analysis of Results Simulation of Cyber-Physical Biosensor Systems==
https://ceur-ws.org/Vol-2516/paper12.pdf
Numerical Analysis of Results Simulation of Cyber-
Physical Biosensor Systems
Vasyl Martsenyuk1[0000-0001-5622-1038], Andriy Sverstiuk2[0000-0001-8644-0776],
Aleksandra Kłos-Witkowska1[0000-0003-2319-5974], Kozodii Nataliia3[0000-0002-9096-4393],
Oksana Bagriy-Zayats2[0000-0002-5533-3561], and Ihor Zubenko4[0000-0001-5622-1038]
1 University of Bielsko-Biala, Department of Informatics and Automatics, Poland
2 I. Gorbachevsky Ternopil National Medical University,
Department of Medical Informatics, Ternopil, Ukraine
3 Ternopil Ivan Puluj National Technical University, Ukraine
4 National University of Ostroh Academy, Ukraine
vmartsenyuk@ath.bielsko.pl, sverstyuk@tdmu.edu.ua,
awitkowska@ath.bielsko.pl, bagrijzayats@tdmu.edu.ua
Abstract. The article deals with the comparative analysis of the results of nu-
merical modeling of mathematical models of cyber-physical biosensor systems
on hexagonal and rectangular lattices using lattice difference equations with de-
lay. The main attention is given to the mathematical description of the discrete
population dynamics in combination with the dynamic logic of the studied
models. The lattice difference equations with delay are proposed to simulate an-
tigen-antibody interaction within hexagonal and rectangular biopixels. Appro-
priate spatial operators have been used to model the interaction between biopix-
els similar to the phenomenon of diffusion. The paper presents the results of
numerical simulations in the form of phase plane images and lattice images of
the probability of antigen to antibody binding in the biopixels of cyber-physical
biosensor systems for antibody populations relative to antigen populations. The
obtained experimental results make it possible to carry out a comparative analy-
sis of the stability of mathematical models of cyber-physical biosensor systems
on hexagonal and rectangular lattices using lattice difference equations
Keywords: cyber-physical model, biosensor systems, stability of the model,
difference equations, hexagonal lattice, rectangular lattice
1 Introduction
Today, the concept of creating cyber-physical systems (CPS) for various fields of
human activity is actively developing. CPS is considered as an intelligent system that
integrates physical objects, external devices, processors, network equipment. The
main purpose of CPS is to monitor the behavior of physical objects as components of
such systems in real time. These are systems in which cybernetic tools (measuring,
Copyright © 2019 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
computing, communication, control, executive) interact with physical processes in
arbitrary objects [1].
Cyber-physical systems are identified with the manifestation of the fourth industri-
al revolution that takes place in the modern world [2]. Thus, there is also a physical
opportunity to use technologies of "Internet of Things" [3], where it is necessary to
use signals from sensors and measuring devices. Thus, more and more publications
[4] appear in the literature that draw attention to the modern concepts and offer the
innovative solutions. A. Platzer proposed an approach based on "dynamic logic",
which describes and analyzes cyber-physical systems [5-6]. In these works, the hybrid
programs (HPs) are used at the simple programming language with the simple seman-
tics. HPs allow the programmer to refer directly to the actual values of variables that
represent the real values and determine their dynamics.
With the growth of the pace of life and the need for more accurate methods for
monitoring various parameters, interest in cyber-physical systems and biosensors as
their components is growing in science and industry. Biosensors are an alternative to
well-known measurement methods that are characterized by poor selectivity, high
cost, poor stability, slow response, and can often be performed only by the highly
trained personnel. This is a new generation of sensors that use biological material in a
design that provides very high selectivity and allows you to quickly and simply
measure [7-8].
An important stage in the design of cyber-physical biosensory systems is the de-
velopment and research of their mathematical models that adequately reflect the im-
portant aspects of the spatial structure of biopixels important in terms of the research
tasks. After all, the quality of the biosensor model determines the effectiveness of its
processing methods in measuring systems. The design of cyber-physical biosensory
systems involves the selection of parameters that would ensure their operational sta-
bility. Such a task, in particular, arises in the development of a biosensor, which in-
cludes a two- and three-dimensional array of biopixels, and which consists in finding
appropriate parameters describing biological and diffusion processes. This problem
can be solved by developing and studying the stability of the corresponding cyber-
physical model of the biosensory system on hexagonal and rectangular lattices using
difference equations [9, 10].
2 Cyber-physical Biosensory System
2.1 Development of a functional scheme of discrete dynamics CPBSS on
hexagonal lattice using lattice difference equations with delay.
Cyber-physical Biosensory System (CPBSS). The definition of the term "Cyber-
physical sensory system (CPSS)" is given in [6]. This definition was introduced for
the industrial use of sensors. The general definition of the CPSS involves "a higher
degree of combination, system sharing, the ability to use embedded systems in the
field of automation and compliance with existing standards." The considered ap-
proach is used for the characterization of CPBSS, the functional scheme of which is
presented in Fig. 1 and allows to perform numerical simulation of the system under
study.
According to [6], the definitions and schemes for CPBS are used to define the CPS.
CPBSS converts physically measured immunological parameters into the digital in-
formation, which enables them to process signals in time using certain algorithms.
There is also an interaction with their own capabilities, requirements, internal data
and internal tasks in terms of distribution to the same or higher level of the hierarchy.
The concept of CPS at the basis of the CPBSS (the external rectangle in Figure 1),
with the account of the features of intellectual imaging sensors is used. With the addi-
tional skills (dotted line in Figure 1), the sensor extends to CPBS, which allows to
receive more diagnostic information about the object being studied.
Fig. 1. Functional scheme of CPBSS.
Four main types of detection are used in biosensory devices: electrochemical (poten-
tiometric, amperometric or conductivity (capacitive), optical and thermometric [10]).
All types of sensors can be used as direct (not marked) or as indirect (marked) biosen-
sors or immunosensors. Direct sensors are able to detect physical changes during the
formation of the immune complex, while indirect use different levels of the generated
signal that enable more sensible and universal detection in measuring systems.
CPBSS refers to the high-intelligence information systems. They use an affordable
set of interfaces that allow you to receive fast and accurate information of the status
and internal system data that should be available to other CPSs. According to [11]
CPBSS as the self-organizing system requires comprehensive knowledge of its own
dynamic structure and infrastructure of the general system. In order to make this, it is
necessary to determine the types of biosensory devices, taking into account their func-
tional application. For example, biosensors can be used to assess critical states in
cardiovascular diseases, insulin values when measuring glucose levels in blood and to
identify quantitative parameters in some pharmaceutical formulations.
In the article [11] the general structure of CPSS is proposed. While applying this
scheme, in the case of biosensors, three directions can be singled out: general infor-
mation about the biosensor; measurements of biological parameters and skills in rela-
tion to unit conversion and calibration; interaction with other biosensors. In this way,
the certain methods are described that allow the biosensor to be described. In the
study of CPBSS, the programming language R was used. Despite the great variety of
programming languages used in the development of CPS (Assembly, C, C++, D, Ja-
va, JavaScript, Python, Ada, etc. [12]), the language R is widely used in many indus-
tries involved in machine learning and visualization of data.
2.2 Discrete Dynamics CPBSS on hexagonal lattice using lattice difference
equations with delay.
For the CPBSS dynamics we use the mathematical description with the help of non-
linear difference equations with delay [10].
The model of the biosensor on the basis of a hexagonal lattice is considered. In this
case, for the numbering of immune pixels (i, j , k ) , i, j , k N , N , i j k 0 the
cubic coordinate system is used [12].
Let Vi , j , k (t ) is the concentration of antigens, Fi , j , k (t ) is the concentration of anti-
bodies in the biopixel (i, j , k ) ; i, j , k N , N , i j k 0 .
The model is based on such biological assumptions for an arbitrary biopixel
(i , j , k )
1. Antigens are detected, bind, and finally neutralized by antibodies with some prob-
ability velocity 0 .
2. It is assumed that when colonies of antibodies are absent, colonies of antigens are
regulated by a logistic equation with a delay:
Vi , j , k (n 1) (1 Vi , j , k (n r ))Vi , j , k (n) , (1)
where and – positive numbers, and r 0 mean latency of the negative re-
sponce of the antigens’ colonies.
3. The fertility rate 0 for the antigen population is introduced.
4. Antigens are neutralized by antibodies at a certain probability rate 0 .
5. The population of antigens tries to reach a certain limit of saturation with a speed
0 .
6. The diffusion of antigens from six adjacent pixels is considered (i 1, j , k 1) ,
(i 1, j 1, k ) , (i, j 1, k 1) , (i 1, j , k 1) , (i 1, j 1, k ) and (i, j 1, k 1) (Fig.
2) with diffusion speed D 2 , where D 0 – coefficient of diffusion; 0 –
distance between two adjacent pixels.
7. The constant mortality of antibodies f 0 is introduced.
8. As a result of the immune response the antibody density increases with a probabil-
istic velocity .
9. The antibody population is approaching a certain level of saturation with a speed
f 0.
10. The immune response occurs with some constant delay in a time r 0 .
11. Surface diffusion (motion of molecules on a solid surface for immobilized mole-
cules) is considered.
12. The definition of a conventional diffusion operator is used in the case of surface
diffusion with a diffusion imbalance coefficient n (0,1] . This means that only
n portion of the pixel antigens (i, j ) can be included in the diffusion process to
any adjacent pixel due to surface diffusion.
13. Antigen binding to antibodies results in fluorescence in the pixel. Fluorescence in-
tensity is assumed to be proportional to the number of contacts between antigens
and antibodies, i.e. k flVi , j ( n) Fi , j ( n) . It is also assumed that the pixel (i, j ) is
in fluorescence state if k flVi , j ( n) Fi , j (n) fl , where is some binding thresh-
old at which the fluorescence phenomenon occurs.
14. The output signal
s (n)
is proportional to the number of pixels in the fluores-
cence state.
15. Information on the number of biological measurements of values is calculated
based on the output signal.
Fig. 2. Hexagonal lattice, which binds six neighboring pixels in the model of the biopixel using
the cubic coordinates:
D D
1, 3, 5, 8, 9, 11 - 2 Vi , j , k (t ) ; 2 - 2 Vi 1, j , k 1 (t ) ;
D D D D
4 - 2 Vi 1, j 1, k (t ) ; 6 - 2 Vi , j 1, k 1 (t ) ; 7 - 2 Vi 1, j , k 1 (t ) ; 10 - 2 Vi 1, j 1, k (t ) ;
D
12 - 2 Vi , j 1, k 1 (t ) .
On the basis of the above information, we will write the mathematical model of late-
antigen-antibody interaction for a hexagonal array of biopixels based on the well-
known Marchuk model [13-15] and uses the spatial operator S proposed in [16] (ad-
ditional information is on page 10).
Vi , j , k (n 1) Vi , j , k (n) exp Fi , j , k n r Vi , j , k n r Sˆ Vi , j , k (n) ,
Fi , j , k (n 1) Fi , j , k (n) exp f Vi , j , k (n r ) f Fi , j , k (n)
(2)
where S Vi , j , k is a discrete diffusion for a spatial operator S .
D 2 Vi 1, j , k 1 Vi 1, j 1, k Vi , j 1, k 1 Vi 1, j , k 1 Vi 1, j 1, k Vi , j 1, k 1 6nVi , j , k
S Vi , j , k (3)
i, j , k N 1, N 1, i j k 0.
2.3 Discrete Dynamics CPBSS on rectangular lattice using lattice difference
equations.
Consider a simple competing antigen-antibody model for a two-dimensional biopixel
array that has been proposed and investigated in [17].
dVi , j (t )
( Fi , j (t ) Vi , j (t ))Vi , j (t ) S Vi , j
dt
(4)
dFi , j (t )
( f Vi , j (t ) f Fi , j (t ))Fi , j (t )
dt
The mathematical model (4) is given by the initial functions (5):
Vi , j (t ) Vi0,j (t ) 0, Fi , j (t ) Fi0,j (t ) 0, t [,0),
(5)
Vi , j (0), Fi , j (0) 0.
Discrete diffusion is used for the rectangular array N N for the spatial operator
used in the work [18]:
S Vi , j D 2 Vi 1, j Vi 1, j Vi , j 1 Vi , j 1 6Vi , j , i, j 1, N . (6)
Each colony is exposed to antigens produced in four adjacent pixels, which are sepa-
rated by equal distances .
We use boundary condition Vi , j 0 for array nodes i, j 0, N 1 .
The methods of sampling, permanence, and stability research used in the work are
based on the approach developed in [19] for predator-prey systems, extensible to fi-
nite lattice diffusion models.
System (4) without diffusion is approximated by the following differential equation
with piecewise constant argentations.
dVi , j
dt
Fi , j t / hh t / hh Vi , j t / hh t / h Vi , j (t ),
(7)
dFi , j (t )
dt
f Vi , j t / hh t / hh f Fi , j t / hh Fi , j (t )
for t nh, n 1h, n .
Let us denote that t / h n , t / h r .
Let's integrate the last system (7) by nh, t , where t (n 1)h, then (6) can be re-
written as:
dVi , j
dt
Fi , j nh rh Vi , j nh rh Vi , j (t ),
(8)
dFi , j (t )
dt
f Vi , j nh rh f Fi , j nh Fi , j (t )
Vi , j ( n) Vi , j (nh), Fi , j (n) Fi , j (nh),
The notation is entered which results in:
Vi , j (t ) Vi , j (n) exp Fi , j n r Vi , j n r ,
Fi , j (t ) Fi , j (n) exp f Vi , j (n r ) f Fi , j (n)
(9)
Considering t (n 1)h can simplify system (9) by adding diffusion to the first
equation. The result is a discrete analog continuous time system (4) in the form:
Vi , j (n 1) Vi , j (n) exp Fi , j , k n r Vi , j n r Sˆ Vi , j (n) ,
Fi , j (n 1) Fi , j (n) exp f Vi , j (n r ) f Fi , j (n)
(10)
Addition of diffusion is performed to obtain qualitative results in the study of the
persistence and stability of the model. Diffusion in a discrete space can be represented
as the product of matrices, according to [7].
It should be noted that the behaviour of system (10) may not coincide with the dif-
ferential equations (4). The equivalence of differential difference equations obtained
by direct Euler transform, Euler inverse transform or central difference schemes can
only be used for sufficiently small sampling intervals [20].
2.4 Dynamic logical simulation of CPBSS on hexagonal lattice using lattice
difference equations with delay.
In order to simulate the dynamic logic of CPBSS, we use the syntax proposed by
A. Platser for the general CPS [5]. The CPS uses the HP, which has more features
than difference equations. The first level of HP is a dynamic program that is defined
by the following grammar
a :: Vi , j , k (n 1) Vi , j , k (n) exp Fi , j , k (n r ) Vi , j , k (n r ) Sˆ Vi , j , k (n) ,
(11)
Fi , j , k (n 1) Fi , j , k (n) exp f Vi , j , k (n r ) f Fi , j , k (n) & t
where t is an evolutionary domain constraint in the form of a formula for the logic
of the first order of real arithmetic
def
t V min Vi , j , k (n) V max
(12)
F min Fi , j , k (n) F max i, j , k N , N n 0, i j k 0
The functioning of the biopixel (i, j , k ) is determined by two states, with respect to
fluorescence. Namely, s f l is a state of fluorescence and snon f l is one of the non-
fluorescence states. The use of the first order of semantics of logic and the satisfaction
ratio s L for the first-order formula L of real arithmetic and state s can be deter-
mined for some pixels (i, j , k ) ; i, j , k N , N , i j k 0 states s f l and snon f l as
s ft k flVi , j , k (n) Fi , j , k (n) fl ,
(13)
snonfl k flVi , j , k (n) Fi , j , k (n) fl
Discrete changes occur in computer programs when they accept new values for varia-
bles. This situation occurs when a fluorescence phenomenon occurs in a pixel
(i, j , k ) ; i, j , k N , N , i j k 0 . The state s f l , i , j , k : 1 is assigned a value of 1 to
the variable s f l ,i , j , k . This leads to a discrete, jump-like change, as the value s f l ,i , j , k
does not change smoothly, but rapidly when it suddenly changes from 1 to s f l ,i , j , k ,
causing a discrete jump of values s f l ,i , j , k . In this way, we obtain a discrete model of
change s f l , i , j , k : 1 , except for the model of change (13).
2.5 Investigation of stability model of CPBSS on hexagonal lattice. Constant
states.
In general, the state of equilibrium
i , j , k Vi , j , k , Fi , j , k , i, j , k N , N , i j k 0 for the system (2) can be found as
a solution of an algebraic system:
Vi , j , k Vi , j , k exp Fi , j , k Vi , j , k Sˆ Vi , j , k
Fi , j , k Fi , j , k exp f Vi , j , k f Fi , j , k .
(14)
Considering Vi , j , k , Fi , j , k , i, j, k N , N , i j k 0 , we have the following cases.
Stable state without antigens and antibodies i0,0
, j ,k
0,0
(0,0) ,
i, j , k N , N , i jk 0.
Stable state without antibodies i*,0
, j ,k
*,0
, 0 , i, j , k N , N , i j k 0
Identical endemic steady state. In the case if Vi , j , k V ident 0 , i, j , k N , N ,
i j k 0 , Sˆ Vi , j , k 0 , we receive the stable state i , j , k ident V ident , F ident ,
where
f f f
V ident , F ident .
f
2
2 f
So, if f 0 , then ident is an endemic state.
Non-identical endemic steady state. In the general case, we need to solve the alge-
braic system (14) and find an endemic stable state, which will be called non-identical
stationary state non ident (Vi ,non
j ,k
ident
, Fi ,non
j ,k
ident
) , i, j , k N , N , i j k 0 . In case all
(Vi ,non
j ,k
ident
, Fi ,non
j ,k
ident
) 0 , then non ident is an endemic state. Values V ident and F ident can
be used as the initial approximations for numerical methods for solving a nonlinear
algebraic system (14).
3 Numerical Simulation Cyber-physical Biosensory System
3.1 Results of numerical simulation of mathematical model of CPBSS on
hexagonal lattice using lattice difference equations with delay.
Model (2) is considered at h 0.012 ; 2h ; 2h ; f h ; 0.01184 / ;
0.5h ; f 0.5h ; D / 2 2.22 h ; N 4 .
Similar to the model based on the differential equations [17], in a system with the
discrete time when the delay time value is changed r we observe the qualitative
changes in the behavior of biopixels and the model under study as a whole. Numerical
modeling is performed at the values of the parameters given above. In this case, the
long-term behavior of the system (2), which describes a hexagonal array of biopixels
at N 4 for r 5 ; r 17 ; r 22 . Phase diagrams of antibody and antigen popula-
tions for pixel and adjacent pixels at different values are shown in fig. 35.
a) b) c)
Fig. 3. Results of numerical modeling of the system (2) at r 5 (a), r 17 (b), r 22 (c).
The image of the phase planes in coordinates Vi , j , k , Fi , j , k for the pixel (0, 0, 0) . Designation:
○ identical stable state, ● non-identical steady state
Thus at r 16 there are trajectories that correspond to a stable focus for all pixels
(Fig. 3(a)). At a value r 17 Hopf bifurcation occurs the following trajectories
correspond to stable ellipsoidal boundary cycles for all pixels (Fig. 3(b)). The results
of numerical modeling are consistent with the theoretical results on the basis of the
theorem on the Hopf bifurcation [21], which confirms the appearance of small invari-
ant cycles of the radius Ο ( h ) . Fig. 3(c) for r 22 shows the phase diagrams,
which are the limit cycles with two extremums (one local maximum and one local
minimum).
Lattice graphs were used for numerical modeling of the cyber-physical model of
the biosensor. Firstly, the corresponding graphs were constructed, where the probabil-
ity of antigen-antibody contact was given for each pixel, and as Vi , j , k Fi , j , k at r 5 ,
r 17 , r 22 , are shown at Fig. 4 (а-c).
a) b) c)
Fig. 4. Lattice images of the probability of antibody bonds with antibodies in pixels of the
system (2) at r 5 (а), r 17 (b), r 22 (c).
As it was shown by the numerical analysis fluorescing states in biopixels are changed
according to the laws of discrete dynamics. Analyzing the obtained results, it was
concluded that when changing the values of r , the behavior of pixels and CPBSS
changes qualitatively.
3.2 Results of numerical simulation of mathematical model of CFBSS on
rectangular lattice using lattice difference equations with delay.
Consider model (10) for: N 16 , 2 min 1, 2 minmLg , f 1 min 1 ,
2
0.8 / , 0.5 minmLg , f 0.5 minmLg D 0.2 nm
min
, 0.3 nm .
The results of numerical simulations were implemented for different values r of
time delay (Fig. 5(a-c)).
a) b) c)
Fig. 5. Image of phase planes of system (10) for antibody Fi , j , populations relative to antigen
populations Vi , j , as a result of numerical simulation at r 8 (a), r 12 (b), r 16 (c).
Designation: - initial state, ○ - identical steady state, ● - non-identical steady state.
The results of the numerical simulations presented in Figures 7(a-c) were constructed
for n [0, 5000 ] . As Figure 5(a) shows, the solution converges to a non-identical
steady state, which is a stable focus.
In Figure 5(b), the solution converges to a stable boundary cycle with two local ex-
trema in the cycle.
Figure 5 (a) for r [0, 12 ) shows trajectories corresponding to a steady focus for
all pixels. Hopf bifurcation [21] occurs for values r 12 and the following trajecto-
ries correspond to stable boundary cycles of the ellipsoidal shape for all pixels.
Phase diagrams for r 12 show that the solution is a boundary cycle with two lo-
cal extrema (one local maximum and one local minimum per cycle). Chaotic behav-
iour is observed for r 16 (Figure 5 (c)), i.e. no periodic behaviour over a large time
interval. Initial conditions were disturbed to test the sensitivity of the system to verify
that the solution is chaotic for r 16 . Comparisons of solutions for the population of
antigens V1,3 with the initial conditions V1,3 (t ) 1 , V1,3 (t ) 1.001, n [r ,0] and all
other identical initial conditions, show chaotic behaviour. Namely, at the initial time,
the two solutions appear to be the same, but with the increase of time there is a differ-
ence between the solutions, which confirms the conclusion that the behaviour of the
system is chaotic at r 16 .
The model of the biosensor was analyzed using a lattice graph representing the
probability of binding of antigens to antibodies in the pixels of system (10) (Figure 6).
It was accepted fl 1,5 .
Fig. 6. Lattice images of the probability of binding of antigens to antibodies in pixels of system
(10) at r 8 .
The study of phase diagrams and lattice images of the binding of antigens to antibod-
ies in the pixels of system (10) is completely consistent with previous studies [9-10]
regarding the stability of the array of biopixels in CРBSS.
3.3 Comparative analysis of results of numerical modeling of mathematical
models of cyber-physical biosensor systems on hexagonal and rectangular
lattices using lattice difference equations
The results of comparative analysis of numerical modeling of the studied mathemati-
cal models of CPBSS in the form of phase diagrams of populations of antigens, anti-
bodies (Fig. 3a, 5a) and lattice images of the binding of antigens to antibodies from
biopixels of the studied systems (Figs. 4a, 6a) that for r 5 (hexagonal lattice) and
r 8 (rectangular lattice) the solutions of the respective systems (2) and (10) tend to
non-identical endemic states, which in this case are stable focuses. A similar depend-
ence was observed for all biopixels of the CPBSS model on the hexagonal lattice for
r 0, 17 (Fig. 3a, 4a), and in the case of using a rectangular lattice a non-identical
endemic state was observed for r 0, 12 (Figs. 5a, 6).
According to the results of the phase diagrams of antigen populations, antibodies
and lattice images, the probability of antigen to antibody binding in CPBSS biopixels,
we can conclude that for r 17 (in the case of hexagonal lattice (Figs. 3b, 4b)) and
r 12 (in the case of a rectangular lattice (5b)) Hopf bifurcation occurs and all sub-
sequent trajectories correspond to stable boundary cycles for all pixels (Figs. 3c, 4c,
5c).
The results of numerical analysis, the probability of binding of antigens to antibod-
ies in the biopixels of the studied models, change according to the laws of discrete
dynamics. Analyzing the results, it is concluded that for r the behavior of the biopix-
els and CPBSS changes qualitatively.
4 Conclusions
In the work a comparative analysis of CPBSS models was performed on hexagonal
and rectangular lattices using differenсе equations. The general scheme of the cyber-
physical sensor system proposed in [11] was used. The basic model has been modi-
fied to take into account the features of biosensors. Lattice images in biopixels are
modified according to the laws of discrete dynamics. The developed models take into
account the interaction of biopixels with each other by antigen diffusion.
The mathematical description of the CPBSS contains discrete population dynam-
ics, which is combined with the dynamic logic used for discrete events. The paper
uses a class of time-lattice difference equations that model the interaction of antigens
and antibodies in biopixels. Spatial operators model the interaction of diffusion type
between biopixels. Dynamic mathematical modeling is insufficient to simulate dis-
crete dynamics in the systems under study. To address this drawback, we used the
dynamic logic syntax proposed for Platzer cyber-physical systems to describe the
discrete states of a biopixel as a result of fluorescence.
In the paper represents the results of numerical simulations in the form of phase
plane images and lattice images of the probability of antigen to antibody binding in
the biopixels of cyber-physical biosensor systems for antibody populations relative to
antigen populations. The obtained experimental results make it possible to carry out a
comparative analysis of the stability of mathematical models of cyber-physical bio-
sensor systems on hexagonal and rectangular lattices. We can conclude that for
r 17 (in the case of hexagonal lattice) and r 12 (in the case of a rectangular
lattice) Hopf bifurcation occurs and all subsequent trajectories correspond to stable
boundary cycles for all pixels.
The numerical simulation results obtained in the paper make it possible to carry out
stability analysis and comparisons of the studied models, taking into account the time
delay. Future research plans to study cyber-physical biosensor systems using fast
dynamic wireless networks [22]. Also, as records are accumulated in the systems
under study, it is planned to analyze them in order to optimize the distributed database
structure, according to [23].
References
1. Lee, E.: Cyber physical systems: Design challenges. In: International Symposium on Ob-
ject/Component/Service-Oriented Real-Time Distributed Computing, pp. 10 (2008).
2. Lee, J., Bagheri, B., Kao, H.: A cyber-physical systems architecture for industry 4.0-based
manufacturing systems. Manufacturing Letters, vol. 3, 18-23 (2015).
3. Krainyk, Y., Davydenko, Y., Tomas, V.: Configurable Control Node for Wireless Sensor
Network. In: 3rd International Conference on Advanced Information and Communications
Technologies (AICT), pp. 258-262. IEEE Press, Lviv, Ukraine (2019).
https://doi.org/10.1109/AIACT.2019.8847732
4. Kim, K.-D., Kumar, P.: Cyber–physical systems: A perspective at the centennial. In: Pro-
ceedings of the IEEE, vol. 100, pp. 1287-1308, (2012).
5. Platzer, A.: Differential dynamic logic for hybrid systems. Journal of Automated Reason-
ing, 2(41), 143-189 (2008).
6. Platzer, A.: Logical Foundations of Cyber-Physical Systems. Springer International Pub-
lishing, pp. 1-24, (2018).
7. Adley C.: Past, present and future of sensors in food production. Foods, 3(3), pp. 491-510,
(2014).
8. Bahadır E., Sezgintürk M.: Applications of commercial biosensors in clinical, food, envi-
ronmental, and biothreat/biowarfare analyses. Analytical Biochemistry, vol. 478, pp. 107-
120 (2015).
9. Martsenyuk V., Sverstiuk A., Gvozdetska I. Using Differential Equations with Time Delay
on a Hexagonal Lattice for Modeling Immunosensors. Cybernetics and Systems Analysis,
vol. 55 (4), p. 625-636 (2019).
10. Liu L., Liu Z.: Asymptotic behaviors of a delayed nonautonomous predator-prey system
governed by difference equations. Discrete Dynamics in Nature and Society, vol. 2011, pp.
1–15, (2011).
11. Berger C., Hees A., Braunreuther S., Reinhart G.: Characterization of Cyber-Physical Sen-
sor Systems. Procedia CIRP, vol. 41, pp. 638-643, (2016).
12. Internet-resurs: https://www.redblobgames.com/grids/hexagons/.
13. McCluskey, C.: Complete global stability for an SIR epidemic model with delay – distrib-
uted or discrete. Nonlinear Analysis: Real World Applications, 1 (11), 55-59, (2010).
14. Nakonechny, A., Marzeniuk, V.: Uncertainties in medical processes control. Lecture Notes
in Economics and Mathematical Systems, vol. 581, pp. 185-192 (2006).
15. Piotrowska, M.: An immune system–tumour interactions model with discrete time delay:
Model analysis and validation, Communications in Nonlinear Science and Numerical Sim-
ulation, vol. 34, pp. 185-198, (2016).
16. Prindle, A., Samayoa, P., Razinkov, I., Danino, T., Tsimring, L., Hasty, J.: A sensing array
of radically coupled genetic ‘biopixels’, Nature, 481(7379), 39-44 (2011).
17. Martsenyuk, V., Kłos-Witkowska, A., Sverstiuk, A.: Stability, bifurcation and transition to
chaos in a model of immunosensor based on lattice differential equations with delay. Elec-
tronic Journal of Qualitative Theory of Differential Equations, 27, pp. 1-31 (2018).
18. Prindle A., Samayoa P., Razinkov I., Danino T., Tsimring L., Hasty J.: A sensing array of
radically coupled genetic ‘biopixels’. Nature, vol. 481(7379), pp. 39-44 (2012).
19. Liu L., Liu Z.: Asymptotic behaviors of a delayed nonautonomous predator-prey system
governed by difference equations. Discrete Dynamics in Nature and Society, vol. 2011, pp.
1-15, 2011.
20. Letellier C., Elaydi S., Aguirre L., Alaoui A.: Difference equations versus differential
equations, a possible equivalence for the Rossler system. Physica D: Nonlinear Phenome-
na, vol. 1-2(195), pp. 29-49, (2004).
21. Hofbauer, J., Iooss, G.: A hopf bifurcation theorem for difference equations approximating
a differential equation, Monatshefte fur Mathematik, 2(98), pp. 99-113 (1984).
22. Burlachenko, I., Zhuravska, I., Davydenko, Y., Savinov, V.: Vulnerabilities Analysis and
Defense Based on MAS Method in Fast Dynamic Wireless Networks. In: 4th International
Symposium on Wireless Systems within the International Conferences on Intelligent Data
Acquisition and Advanced Computing Systems (IDAACS-SWS), pp. 98-102. IEEE Press,
Lviv, Ukraine (2018). https://doi.org/10.1109/IDAACS-SWS.2018.8525692
23. Fisun, M., Dvoretskyi, M., Shved, A., Davydenko, Y.: Query parsing in order to optimize
distributed DB structure. In: 9th IEEE International Conference on Intelligent Data Acqui-
si-tion and Advanced Computing Systems: Technology and Applications (IDAACS), pp.
172-178. IEEE Press, Bucharest, Romania (2017).
https://doi.org/10.1109/IDAACS.2017.8095071