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Modeling Ebola Virus Dynamics by
Colored Petri Nets
Dmitry A. ZAITSEVa,1 and Peyman GHAFFARI
b and Virginia Sanz SANCHEZ
c
a
University of Information Technology and Management in Rzeszów, Poland
b
Department of Mathematics, University of Aveiro, Portugal
c
Faculty of Pharmaceutical Sciences, University of Iceland, Iceland
Abstract. We develop techniques to translate deterministic cellular automata
models into colored Petri nets based on an example of known cellular automata for
modeling and mimicking Ebola virus dynamics. Cellular automata for Ebola virus
dynamics use parametric specifications of rules that is peculiarity of the present
study. The simulation results completely coincide with known results obtained via
dedicated simulator. Having uniform language for models specification brings in
advantages for models mutual transformations and simulation.
Keywords. Ebola virus, immune response, cellular automata, colored Petri net,
simulation
1. Introduction
Ebola is one of the most lethal infectious diseases known to mankind. The Ebola virus
causes an acute, often fatal hemorrhagic illness. The incubation time is something
between two days and three weeks after becoming infected with the virus [1].
Cellular automata (CA) are widely applied for modeling processes of virus
spreading both on micro-level of cells of some organ [2] to macro-level of geo-spatial
representation [3].
A Petri net is a well-known formalism for specification of concurrent processes [4].
A family of Petri net classes offers a uniform language for modeling in various
domains with possibility of applying both model-checking and simulation techniques
of analysis.
In the present paper, we develop a technique for transformation of cellular
automaton models for virus spreading into colored Petri net (CPN) models using a
known cellular automaton that specifies Ebola virus dynamics [2].
2. Cellular automata and Petri nets
A cellular automaton [4] represents a homogenous structure of elements (cells) having
rather simple behavior such that their collective behavior represents a mass parallel
1 Corresponding Author, Dmitry A. Zaitsev, Information Technology Department, University of
Information Technology and Management in Rzeszów, ul. Sucharskiego 2, 35-225 Rzeszow, Poland; E-mail:
daze@acm.org.
Proceedings of CECNet 2021
A.J. Tallón-Ballesteros (Ed.)
© 2022 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms
of the Creative Commons Attribution Non-Commercial License 4.0 (CC BY-NC 4.0).
doi:10.3233/FAIA210464
707
process capable of modeling processes in many application domains. A state of cell in
the next tact of time depends on the cell current state and states of its neighbors.
Usually, an infinite square lattice of cells in multidimensional space is considered in
theoretical research though for practical applications, a finite part of the lattice is
simulated. Even one dimensional elementary (having binary state) cellular automata are
Turing-complete [5] that illustrates power of the concept. Traditionally, von Neumann
(sparse) and Moore (dense) neighborhoods of a cell are considered though recently a
generalized neighborhood, that fills the gap between two mentioned neighborhoods
within multidimensional lattices, has been introduced and studied [6], [7].
A Petri net [8] represents a bipartite directed graph with a dynamic process
introduced on it. The first part of vertices, depicted as circles or ovals and called places,
represents conditions. The second part of vertices, depicted as rectangles or bars and
called transitions, represents events. Dynamic elements, called tokens and depicted as
dots, are situated within places; they are consumed and produced as the results of firing
transitions. Within a conventional Petri net, tokens are elementary; the net state is
specified by the place marking represented by a vector which components contain the
number of tokens in the corresponding places. A conventional Petri net is applied for
modeling concurrent processes within a wide variety of application domains.
Comparable simplicity of Petri net concept allows us to apply analytical methods for
model checking [6] though its descriptive power is not too high for detailed
specification of systems and processes.
A colored Petri net [9], [10] represents a combination of a Petri net graph and a
functional programming language ML. Tokens are considered as elements of data types
traditionally called color sets. Arcs and transitions are inscribed with ML constructs.
Besides, time characteristics are introduced for the processes simulation. Supplied with
a range of random distribution functions, CPN Tools [10] allows us to provide
statistical analysis of timed and other characteristics. For composition of complex
models, hierarchical design is provided via the transition substitution when a transition
is substituted by a subnet. CPNs find wide application for the performance evaluation
of systems, especially networks, grids, and clouds [11]. For this purpose, classical
simulation technique is applied refined with special measuring components of models
to compute statistical characteristics on-the-fly directly in the process of simulation [9].
For comparably small models, application of state space technique for the model
properties analysis is possible.
3. Cellular automata models of Ebola virus dynamics
We use Ebola virus dynamics model [2]. A two dimensional CA with the cell states
within the set {0,1,2,3} is considered where 0 represents a healthy state, 1 and 2
represent an infected cell, and 3 represents a depleted (dead) cell. Here we present a
model for one of three rules studied in [2] that explains basic principles of the
ransformation.
Let us consider the mentioned rules description where asterisk symbol “*” denotes
any state and a matrix represents states of a cell (in the center) and its neighbors
according to Moore neighborhood [6]. Before a matrix, a rule symbol is written; the
rule application results in a new state of the central cell that is written after the equality
symbol.
CA Q with rule q specification follows:
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets708
ݍቆכככ
כͲכ
כככ
ቇൌ൜ͳǡכͳʹ
Ͳǡ
ݍቆכככ
כܽכ
כככ
ቇൌ͵ǡͳܽʹ
ݍቆכככ
כ͵כ
כככ
ቇൌͲǤ
The three above lines specify all possible four states of a cell within the set
{0,1,2,3}: first line – 0, second line – 1 or 2, and third line – 3. Note that, in second and
third line (for the current cell state 1, 2, or 3), the next state does not depend on
neighbors. The first line can be considered as a variant of totalistic rule with check that
the sum of all neighbor cell states is equal to or greater than 1.
CA Q specifies fast propagation of decease when, according to the first line of its
description, at least one deceased cell within Moore neighborhood causes the current
cell got infected. Together with other rules studied in [2], it serves for the subsequent
composition of stochastic automata that is beyond the scope of the present paper.
4. CPN model of a cell
We compose colored Petri net model of a cell functioning under rule q shown in Fig. 1.
In colored Petri net (CPN) of modeling system CPN Tools [10], [11] places are
depicted as ovals; each place has its name, written inside a place, data type of tokens
(color set), and initial marking. A token is not elementary in CPN; in the general case,
it is an element of abstract data types such as cortege, record, list, or union composed
of elementary types, such as integer or real numbers, strings and others. In the present
work, we use only color sets: an integer number (INT), and a timed integer number
(tint). Basic declarations look rather simple:
colset tint = INT timed;
var i,ii,j1,j2,j3,j4,j5,j6,j7,j8:INT;
var c:tint;
The reserved word “closet” serves for defining new sets of colors while the
reserved word “var” serves for declaration of variables. A color set is specified using
the equality sign “=”, a new color set tint is defined on basic integer color set INT
using modifier “timed”. Definition of variables consists of a list of names separated by
semicolon “:” symbol with the name of color set.
Within our model, we have only no more than one token inside each place though
in the general case the place marking is represented by a multiset composed using
double plus sign “++”. Recall that, each element belongs to a multiset in definite
number of copies (multiplicity), the multiset element is represented as k`e, where k
denotes multiplicity and e denotes an element. On the initial marking CPN Tools
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets 709
creates current marking of simulation system which is displayed in green colors and
changes as result of transitions firing.
Timed delays are associated either with transitions or with separate arcs. For timed
multisets, an element is represented as k`e@t, where t is the token activation time. A
transition fires instantaneously, increasing the step counter Step value, a delayed token
spends time before its activation in the corresponding output place of transition in
passive state not possible for enabling any transition. The timed delays are specified as
k`e@+d, where d species the delay. Here we use integer numbers to specify delay
though in the general case, a random distribution function can be applied, CPN Tools
providing a series of random distribution functions including Gauss, Poisson and others.
In our models, we use a timed CPN to specify synchronous work of all the cells.
The marking change within a lattice is implemented in two tacts. At the first tact, all
the neighbors and the current cell itself can read the current “0,0” cell state using two
directed arcs without change of the marking. During this tact, a new state of the current
cell is computed and stored within additional place “new 0,0”. Then, at the second tact,
the current state of all cells is instantaneously replaced by their new states. For this
purpose, we have a clock token 1@t moving within a circle created by places clock1
and clock2 and transitions which connect the mentioned places. The clock extracted by
variable c inscribed on the arcs from place clock1 activates one of the four alternative
transitions having names “0”, “1”, “2”, and “3”, corresponding to the current “0,0” cell
state. The corresponding alternative values of the current state are inscribed on the
transitions incoming arcs. We use double directed arcs though one of the mentioned
transition fire only once because of the clock arc; after firing a transition, the clock
token is delayed by “1@+1” to give time for all cell to compute their new state.
The automaton Q specification in the Section 2, contains simple subrules for the
cell states equal to 1, 2, and 3: transform 1 or 2 into 3 and transform 3 into 0 without
taking into consideration state of the neighboring cells. Each of these transformations is
specified by a corresponding transition, putting the new current state value into place
“new 0,0”. For the current cell state equal to 0, the automaton Q, according to its first
subrule, checks states of its neighbors. There are two alternatives of this check
represented by transitions “1 or 2” and “all 0 or 3”. Now let us pay attention to the
naming of cells within the cell model. Because the model should suit to each cell of the
lattice, we do not use the cell index; instead of it we use neighbor offsets with regard to
the current state index. Thus, the current cell is named as “0,0”, the left upper cell is
named as “-1,-1” and so on because we use a matrix-like orientation of coordinate axes.
To extract values of the neighboring cells, we use a set of integer variables j1, j2, j3, j4,
j5, j6, j7, j8 corresponding to the clockwise enumeration of neighbors starting from the
upper neighbor. Note that, to avoid a tangled picture with manifold crossed lines, we
implement connections of either of transitions “1 or 2” and “all 0 or 3” with
neighboring cells as a bus merging all arcs in a single line; the bus encircles the cell
shape with endings of arcs coming to the neighbor cells states.
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets710
Figure 1. Model of cell with rule q.
To check the state of neighbors, we employ a guard function of a transition. The
guard function represents a logic expression that combines check of characteristics for
tokens extracted by various incoming arcs of a transition. The logic expression is
written in square brackets using the following notations: “orelse” denotes disjunction
and “andalso” denotes conjunction. Thus we interprete the first variant of the first
subrule for automaton Q, implemented by transition “1 or 2” as
ሺ݆ଵൌൌͳሻڀሺ݆ଵൌൌʹሻڀሺ݆ଶൌൌͳሻڀሺ݆ଶൌൌʹሻڀሺ݆ଷൌൌͳሻڀሺ݆ଷൌൌʹሻڀሺ݆ସൌൌ
ͳሻڀሺ݆ସൌൌʹሻڀሺ݆ହൌൌͳሻڀሺ݆ହൌൌʹሻڀሺ݆ൌൌͳሻڀሺ݆ൌൌʹሻڀሺ݆ൌൌͳሻڀሺ݆ൌൌ
ʹሻڀሺ଼݆ൌൌͳሻڀሺ଼݆ൌൌʹሻ.
We interpret the second variant of the first subrule for automaton Q as the above
expression negation with regard to the state value range, implemented by transition “all
0 or 3” as
ሺሺ݆ଵൌൌͲሻڀሺ݆ଵൌൌ͵ሻሻרሺሺ݆ଶൌൌͲሻڀሺ݆ଶൌൌ͵ሻሻרሺ݆ଷൌൌͲሻڀሺ݆ଷൌൌ͵ሻ
רሺ݆ସൌൌͲሻڀሺ݆ସൌൌ͵ሻרሺ݆ହൌൌͲሻڀሺ݆ହൌൌ͵ሻ
רሺ݆ൌൌͲሻڀሺ݆ൌൌ͵ሻרሺ݆ൌൌͲሻڀሺ݆ൌൌ ͵ሻ
רሺ଼݆ൌൌͲሻڀሺ଼݆ൌൌ͵ሻǤ
For the further composition of a lattice using the designed cell model, places
corresponding to the current and neighboring cell states are labeled by the input/output
0,0
In/Out
INT
In/Out
step1
INT
-1,-1
In/Out INT
In/Out
-1,0
In/Out INT
In/Out
-1,1
In/Out INT
In/Out
0,-1
In/Out
INT
In/Out
0,1
In/Out INT
In/Out
1,0
In/Out INT
0
In/Out
1,-1
In/Out INT
3
In/Out
1,1
In/Out INT
3
In/Out
new
0,0
INT
clock2
tint
clock1
tint
0
1
2
3
1 or 2
[j1=1 orelse j1=2 orelse
j2=1 orelse j2=2 orelse
j3=1 orelse j3=2 orelse
j4=1 orelse j4=2 orelse
j5=1 orelse j5=2 orelse
j6=1 orelse j6=2 orelse
j7=1 orelse j7=2 orelse
j8=1 orelse j8=2]
all 0 or 3
[(j1=0 orelse j1=3) andalso
(j2=0 orelse j2=3) andalso
(j3=0 orelse j3=3) andalso
(j4=0 orelse j4=3) andalso
(j5=0 orelse j5=3) andalso
(j6=0 orelse j6=3) andalso
(j7=0 orelse j7=3) andalso
(j8=0 orelse j8=0)]
replace
state
0
1
2
3
0
0
j1
j2
j8
j3
j4
j5
j6
j7
j1
j2
j3
j4
j5
j6
j7
j8
0
3
3
0
01
1@+1
1@+1
1@+1
1@+1
c
i
c
c
c
c
ii i
1@+1
11`011`3 11`3
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets 711
(In/Out) port tag drawn in blue color. They represent parameters of the model and will
be studied in detail when composing a lattice.
5. Composing CPN model of a lattice
To compose a lattice of CA Q, we create a separate place to store the corresponding
cell state, we use cell indices as names, for instance “0,0”, “0,1” etc. With each cell we
associate a cell model represented by a separate transition, to which name prefix “c” is
added, to obtain the main page of CA Q model, shown in Fig. 2. The arcs reflect
connections with neighboring cells; for Moore neighborhood applied in the present
model connections look rather dense. Note that, because the current cell state is not
actually localized within the cell model, we have actually 9 connections for each cell
represented by bidirected arcs. Neighbors only read the current cell state values while
the current cell model changes it.
Figure 2. Model of CA Q (with cell model in Fig. 1), finite 4x4 lattice with constant border conditions.
As a mathematical model, CAs are thought of as infinite lattices and this definition
is applied for deriving analytical results with regard to CA properties. For computer
simulations, we always use a finite part of lattice obtained according to given size of
CA, usually square for a two dimensional automaton. Thus, a question regarding the
borders of finite lattice arise because if we simply cut a square finite lattice, cells on the
borders will use indices of nonexistent cells to implement their rules. The simplest
0,0
INT
1
0,1
INT
1
0,2
INT
0
0,3
INT
0
1,0
INT
0
1,1
INT
0
1,2
INT
1
1,3
INT
1
-1,0
INT
0
-1,1
INT
0
-1,2
INT
0
-1,3
INT
0
2,0
INT
0
2,1
INT
0
2,2
INT
0
2,3
INT
1
0,-1
INT
0
1,-1
INT
0
2,-1
INT
0
3,-1
INT
0
0,4
INT
0
1,4
INT
0
2,4
INT
0
3,4
INT
0
3,0
INT
1
3,1
INT
0
3,2
INT
0
3,3
INT
1
-1,-1
INT
0
-1,4
INT
0
4,-1
INT
0
4,0
INT
0
4,1
INT
0
4,2
INT
0
4,3
INT
0
4,4
INT
0
c0,0
cellcell
c0,1
cellcell
c0,2
cellcell
c0,3
cellcell
c1,0
cellcell
c1,1
cellcell
c1,2
cellcell
c1,3
cellcell
c2,0
cellcell
c2,1
cellcell
c2,2
cellcell
c2,3
cellcell
c3,0
cellcell
c3,1
cellcell
c3,2
cellcell
c3,3
cellcell
`
`
`
`
`
`
`
`
11`1 11`1 11`0 11`0
11`0 11`0 11`1 11`1
11`0 11`0 11`0 11`0
11`0 11`0 11`0 11`1
11`0
11`0
11`0
11`0
11`0
11`0
11`0
11`011`1 11`0 11`0 11`1
11`0 11`0
11`0 11`0 11`0 11`0 11`0 11`0
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets712
solution, implemented in our model of lattice (Fig. 2), consists in adding places storing
states of neighbors for one more layer of cells on each border; it corresponds to
constant border conditions, though individual for each border cell. Thus in Fig. 2,
actually cell state of 5x5 lattice is stored while cell state of 4x4 lattice is modified
according to the rule q. Note that we can implement other border conditions, for
instance, connect opposite edges to obtain a torus [12] or use additional subnets which
implement varying border conditions.
Now let us consider in detail how hierarchical models are composed within CPN
Tools. The basic operation is substitution of a transition by a subnet. In Fig. 2, each
transition, which implements modification of the corresponding cell state, is tagged by
the transition substitution tag “cell”. It means that the subnet shown in Fig. 1 works
inside each transition with specific data according to the cell neighborhood state. To
use parameters while substituting a transition, socket-port mapping is provided by CPN
Tools [11]. Places of a submodel, which map on the corresponding places of a model,
are called ports and indicated by port tags, for instance In/Out tags in Fig. 1. Ports are
mapped to places of the model, which are called sockets, using special tools of
Hierarchy palette of CPN Tools. It is supposed that transition substitution and port-
socket mapping are implemented manually.
When the CA Q model is composed, we can simulate CA dynamics for various
input state of lattice and border conditions. For this purpose CPN Tools offers step-by-
step mode of simulation for model debugging. Then simulation of specified number of
steps can be launched. It is convenient to use time break-points to check the lattice state
when all the cells implemented change of state at the current timed tact.
6. Analysis of simulation results
Specifying here Ebola virus dynamics model in CPN language, we apply mainly
simulation to study the model behavior and properties. Besides, though it is feasible for
rather small models without using high performance resources, we can build complete
state space of model, obtain automatic report on it containing conclusions of such basic
properties as boundedness, liveness, and fairness [7], [11]. There is an additional
possibility to compose requests to state space for finding specific properties which can
be interesting, for instance, when modeling viruses. For 1x1 CA Q shown in Fig. 3,
CPN Tools built state space drawn in Fig. 4.
In Fig. 3, we renamed vertices because state space palette of tools has more strict
requirements to syntax: we replaced zero by “O” and one by “I” and sign minis by “m”
to have names starting from a letter and containing no special characters. Thus, we
have a cell with state 1 that has 7 neighbors with state 0 and one neighbor with state 1.
Built state space of CA Q 1x1 (Fig. 3) represented in Fig. 4 contains 9 states; after an
initial part of first two transitions, the models falls in a loop of states from 3 to 9.
Initially drawn state space does not reflect detailed descriptions of states and
transitions; firing a transition is represented by an arc connecting state, a state is
specified by its number and the number of incoming and outgoing arcs. Fig. 4 contains
attributes of all transitions expanded, as well as expanded state description for state 7.
Transition specification indicates the transition name and values of tokens which were
taken at transition firing (in curly brackets). Compound names of vertices are written
including a prefix with the subnet name followed by, separated by quotation symbol,
the node name. State specification lists marking of all transitions. The obtained report
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets 713
concludes that the net is bounded, its initial marking is a home marking, dead markings
are absent though some instances of transitions are dead, for example “cell'all0or3”,
and classifies transitions into impartial, fair, just and unfair, no unfair transitions found.
Figure 3. CA Q 1x1.
Figure 4. State space of CA Q 1x1 (Fig. 3).
Thus, the CA Q 1x1 model with the specified initial state, coming from CA state 1,
falls into a loop of CA states 3->0->1->3 stored in place “cell’OO”. With our model,
we observed other variants of cyclic behavior of CA studied in [2] for 3x3 and 7x7
lattices that indirectly acknowledges correctness of our constructs.
For statistical analysis of composed models having big size of the cell matrix,
simulation on prolonged intervals of time is applied that also acknowledges
conclusions of [2] with regard to parameters choice to provide recovery after the
decease spreading. Combining state space and simulation techniques allows us to study
the constructed models comprehensively that is an advantage of applying a colored
Petri net and modeling system CPN Tools.
OO
INT
1
OI
INT
1
IO
INT
0
II
INT
0
mIO
INT
0
mII
INT
0
OmI
INT
0
ImI
INT
0
mImI
INT
0
c00
cellcell
11`1 11`1
11`0 11`0
11`0 11`0
11`0
11`0
11`0
1
0:1
2
1:1
1:1->2 @ 0 cell'v1 1: {c=1}
3
2:1
2:2->3 @ 1 cell'replace 1: {ii=1,i=3,c=1}
4
1:1
3:3->4 @ 1 cell'v3 1: {c=1}
5
1:1
4:4->5 @ 1 cell'replace 1: {ii=3,i=0,c=1}
6
1:1
5:5->6 @ 1 cell'v0 1: {c=1}
7
1:1
7 @ 1:
lattice'OO 1: 1`0
lattice'OI 1: 1`1
lattice'IO 1: 1`0
lattice'II 1: 1`0
lattice'mIO 1: 1`0
lattice'mII 1: 1`0
lattice'OmI 1: 1`0
lattice'ImI 1: 1`0
lattice'mImI 1: 1`0
cell'OO 1: 1`0
cell'step1 1: empty
cell'm1m1 1: 1`0
cell'm10 1: 1`0
cell'm11 1: 1`0
cell'O 1: 1`0
cell'O1 1: 1`1
cell'I0 1: 1`0
cell'Im1 1: 1`0
cell'I1 1: 1`0
cell'new00 1: 1`1
cell'clock2 1: 1`1@1
7 @ 1:
lattice'OO 1: 1`0
lattice'OI 1: 1`1
lattice'IO 1: 1`0
lattice'II 1: 1`0
lattice'mIO 1: 1`0
lattice'mII 1: 1`0
lattice'OmI 1: 1`0
lattice'ImI 1: 1`0
lattice'mImI 1: 1`0
cell'OO 1: 1`0
cell'step1 1: empty
cell'm1m1 1: 1`0
cell'm10 1: 1`0
cell'm11 1: 1`0
cell'O 1: 1`0
cell'O1 1: 1`1
cell'I0 1: 1`0
cell'Im1 1: 1`0
cell'I1 1: 1`0
cell'new00 1: 1`1
cell'clock2 1: 1`1@1
6:6->7 @ 1 cell'some1or2 1: {j1=0,j2=0,j3=1,j4=0,j5=0,j6=0,j7=0,j8=0}
9
1:1
9:9->3 @ 1 cell'replace 1: {ii=1,i=3,c=1}
8
1:1
7:7->8 @ 1 cell'replace 1: {ii=0,i=1,c=1}
8:8->9 @ 1 cell'v1 1: {c=1}
D.A. Zaitsev et al. / Modeling Ebola Virus Dynamics by Colored Petri Nets714
7. Conclusions
In this paper we presenter technique to transform so called SEIR models used in
mathematical Epidemiology into colored Petri net models. Applying CPN models for
specifying both behavior patterns of individual cells and virus dynamics within real-life
scale organs results in a uniform concept for model-checking and simulation.
Acknowledgement
We would like to thank the EU COST Association for bringing the authors together
through funding events and STSM-grants within the framework of IMAAC (COST
Action CA 16227).
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