Download Modeling transmission of directly transmitted infectious diseases

Mathematical Biosciences 185 (2003) 1–13
www.elsevier.com/locate/mbs
Modeling transmission of directly transmitted
infectious diseases using colored stochastic Petri nets
Narges Bahi-Jaber *, Dominique Pontier
UMR C.N.R.S. 5558 ‘Biom
etrie et Biologie Evolutive’, Universit
e Claude Bernard Lyon-1,
43 Boul. 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Received 31 July 2002; received in revised form 7 February 2003; accepted 12 June 2003
Abstract
In order to improve our understanding of directly transmitted pathogens within host populations,
epidemic models should take into account individual heterogeneities as well as stochastic fluctuations in
individual parameters. The associated cost results in an increasing level of complexity of the mathematical
models which generally lack consistent formalisms. In this paper, we demonstrate that complex epidemic
models could be expressed as colored stochastic Petri nets (CSPN). CSPN is a mathematical tool developed
in computer science. The concept is based on the Markov Chain theory and on a standard well codified
graphical formalism. This approach presents an alternative to other computer simulation methods since it
offers both a theoretical formalism and a graphical representation that facilitate the implementation, the
understanding and thus the replication or modification of the model. We explain how common concepts of
epidemic models – such as the incidence function – can be easily translated into an individual based point of
view in the CSPN formalism. We then illustrate this approach by using the well documented susceptibleinfected model with recruitment and death.
Ó 2003 Elsevier Inc. All rights reserved.
Keywords: Epidemic models; Colored stochastic Petri nets; Incidence function; SI model
1. Introduction
One of the major challenges of epidemic modeling is to provide a valid explanation as why
some pathogen agents spread and persist, while others fail to establish [1]. Although simple
*
Corresponding author. Tel.: +33-4 72 43 13 37; fax: +33-4 78 89 27 19.
E-mail address: [email protected] (N. Bahi-Jaber).
0025-5564/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved.
doi:10.1016/S0025-5564(03)00088-9
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mathematical models are able to capture many aspects of the observed dynamics, it has
been increasingly recognized that many real host populations are far from being homogeneous from a demographic, behavioral, genetic and/or environmental point of view [2–5]. The
incidence of infectious diseases is highly sensitive to the contact rates that may differ among individuals according to their age, sex, genotype or social status [6]. Small changes in transmission
rates may affect the pattern of disease dynamics [7]. Furthermore, because in many host–parasite
systems either the disease prevalence is quite low and/or because very few individuals are
responsible for the major disease transmission within the host population (e.g., the Feline
Immunodeficiency Virus [6,8]), the introduction of stochastic fluctuations in individual parameters may also substantially change the spread of the disease, possibly leading to its extinction
[9].
Today, the majority of the models within the standard susceptible-infected-removed (SIR)
framework fails to take into account individual heterogeneities. On one hand, in the deterministic
approach, the host population may be structured into classes with different characteristics but all
individuals of the same class are strictly identical and have, in particular, the same contact rate
and demographic parameters [10]. On another hand, stochastic models based on Markov Chain
theory include stochastic individual heterogeneity [11]. But extending the latter to realistic setup
allowing for different classes of hosts and/or variable population size could rapidly make them
intractable in an analytical sense [12]. Furthermore, in both approaches, the rate of production of
new infected is often reduced to a simple mathematical function, the incidence function, representing the flow between susceptible and infected subpopulations [13]. Consequently, computer
simulation models using, for example, the individual based approach have been developed these
last years [14–17]. However, there generally consist in a case-by-case study for which a correct
description goes through the program itself, making them not well suited for exchange and thus
difficult to replicate or to modify [18].
In this paper, we investigate a new methodological approach to represent complex stochastic epidemic models using colored stochastic Petri nets (CSPN) [19–22]. Petri nets is a powerful mathematical and graphical modeling tool well known in engineering sciences which has
been proven useful in modeling and analyzing discrete event systems in several areas like
communications, computers, dataflow analysis, manufacturing and process design [23–25].
The similarity of all these systems lies in the modeling of material flow (e.g. data, object)
in a discrete event system. In this sense, the objectives are similar to those of biological modeling. However, the applications of Petri nets to biological system remain quite rare and
mainly concern molecular biology [26–28] and to a less extent ecology [18,29]. Our purpose
is to show that CSPN formalism represents a powerful tool for modeling host–parasite systems.
This paper is organized as follows: we first give the background materials and methods of
CSPN and classical models of disease transmission. Then, we show that epidemiological models
can be easily defined and interpreted into a CSPN formalism and give a correspondence between
CSPN and classical concepts of epidemic modeling. To illustrate the use of the method, we apply
it to the well-studied SI model with recruitment and death. Its deterministic and stochastic behavior being well documented [30] allows for an effective validation of simulation results obtained
from the CSPN approach.
N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
3
2. Material
2.1. Colored stochastic Petri nets
A CSPN [20–22] consists of (1) a finite set of places, P; (2) a finite set of transitions, T; (3) a
probability distribution associated to each transition, (4) input and output arcs connecting places
to transitions, (5) input and output functions associated to arcs. Places may contain tokens
carrying data values (named colors) that define and characterize tokens. Tokens can move from
one place to another by crossing enabled transitions, after a random sojourn time (defined by the
probability distributions of transitions and by input functions) in the input place. The color of
the tokens may be modified after the crossing of a transition; these modifications are defined by
the output functions.
The methods developed for analyzing stochastic petri nets (SPN) are numerous and differ in
their techniques and applications. SPN are structurally isomorphic to continuous time Markov
chains (CTMC). Thus, it is theoretically possible to explicitly generate the CTMC associated with
the SPN. CSPN can be treated in the same way after having ÔdevelopedÕ the net. However, the
limit of this method lies in the size of the state space of the Markov chain. Nevertheless, even if the
size of the state space makes the explicitly generation of the CTMC impossible, methods that are
based on very specific techniques like algebraic theory or operation analysis can be applied [31].
All these methods provide the transient states at a given time and the steady state distribution.
When analytical resolution is not possible, CSPN becomes a powerful Monte Carlo simulation
tool. Many algorithms exist for simulating the behavior of the system. They are utilized in
software packages publicly available that also integrate graphical representation of stochastic and/
or colored Petri nets along with their specific analysis tools [32]. The development of software
packages devoted to PN modeling is an important area of research in computer science. The
choice of the package depends on the type of the analysis to be performed. Some software
packages are more appropriate for analytical resolution, others for simulation approaches. Most
of them provide a graphical user interface to edit the net corresponding to the model and then
select the type of analysis to run on.
2.2. Concepts from classical models of disease transmission
Classical epidemiological models are compartmental models. The host population is partitioned into compartments with labels such as S (susceptible), I (infected), R (removed) according
to the serological state of hosts [10]. The key process in a host–parasite system is the transmission
of the disease from an infectious to a susceptible host occurring during at-risk contacts. Compartmental approaches model the flow between susceptible and infected subpopulations by a
mathematical expression called the Ôincidence functionÕ [10,13]. This function takes the generic
form bðNÞSI=N where S and I are the average number of susceptible and infected individuals
respectively, N the average host population size and bðNÞ the transmission rate [33,34]. The
transmission rate is the product of the number of at-risk contacts made by one infectious individual during a time interval times the probability that the pathogen will be transmitted during
this contact. In most models of host–parasite systems, disease transmission between infected and
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N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
susceptible hosts takes two alternative forms [13]. The Ôdensity-dependentÕ transmission, also
called Ôpseudomass actionÕ or simply Ômass actionÕ, assumes that the number of at-risk contacts
rises in proportion to the size of the population and the incidence function takes the form bSI. The
alternative is the Ôfrequency-dependentÕ transmission that assumes that each host makes a fixed
number of at-risk contacts during a time interval, independent of the size of the population. This
incidence function, also called Ôtrue mass actionÕ or Ôproportionate mixingÕ takes the form bSI=N .
One of the most studied epidemic model is the well-known SI model with recruitment and death
for which many analytical and simulation results exist [30]. In this model, the population is divided into two serological classes: susceptible ðSÞ and infected ðIÞ individuals with initial conditions N0 ¼ S0 þ I0 . Infected individuals are assumed to be instantaneously infectious and suffer an
additional mortality due to the disease. Individual life-lengths are exponentially distributed with
parameters m and m þ a respectively for susceptible and infected individuals. Encounters occur at
the points of a Poisson process of intensity k. If an encounter involves one susceptible and one
infected individual, the susceptible one has a probability p to be infected. Finally, recruitment of
new susceptible individuals occurs at the points of a Poisson process of intensity mN0 in order to
keep population size constant in the absence of infection.
The deterministic version of this model leads to the following set of differential equations:
dS
SI
¼ mN0 mS kp
dt
SþI
dI
SI
¼ ðm þ aÞI þ kp
:
dt
SþI
ð1Þ
The behavior of the system (1) is driven by a unique threshold parameter R0 ¼ kp=ðm þ aÞ, the
basic reproduction number, defined as the number of secondary cases generated by an infected
individual introduced in a population of susceptible individuals [35]. If R0 < 1 the disease cannot
persist among the host population whereas if R0 > 1 the disease persists and reaches a constant
positive prevalence [30]
I
1
¼1 :
R0
ðS þ I Þ
3. Methods
3.1. Modeling the host population dynamics using CSPN
To model a host–parasite system as a CSPN, we first need to represent the host population with
all the heterogeneities identified as crucial from observations. Indeed, all the individuals of a same
host population are not identical from an epidemiological point of view. The probability that an
individual will transmit the infection or will be infected by a pathogen may strongly differ from
one individual to another. In the case of a directly transmitted pathogen, infections occur during
at-risk contacts whose frequencies determine the probability per individual to transmit the infection or to be infected. The host population can be divided into at-risk classes in which the
members have similar characteristics. The division into classes can be based on transmission
mode, contact patterns, infectious period, genetic susceptibility or resistance, but also on social,
N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
5
demographic or geographic factors. We will call these characteristics Ôstructural variablesÕ in
opposition to serological characteristics because they divide the population into at-risk classes
according to the implication of the individuals in the epidemiological process.
In terms of CSPN, each at-risk class is represented by a specific place, named Ôstructural placeÕ.
Individuals are represented by tokens. All tokens present in a same structural place have the same
structural variables, i.e. they represent individuals having on average the same probability to
transmit or to be infected by the pathogen. Furthermore, introducing the disease into the model
implies the distinction of individuals according to their serological state with respect to the disease
(e.g. susceptible (S), infected (I), removed (R)). In the CSPN, the serological state of individuals is
characterized by the color of the tokens. Each structural place may obviously contain individuals
of different serological states, i.e. tokens of different colors. The number of tokens of each color in
a place is its marking. The state of the system is given by a vector M, the global marking, which
summarizes the number and the color of the tokens in each place of the net.
The dynamical part of the system is modeled using transitions. The firing of a transition corresponds to an event concerning one (or more) token in the input places connected to the transition. Those events may be demographic (birth, death. . .), behavioral (encounter, mating. . .) or
epidemiological (infection, recovery. . .). The frequencies at which events occur for each individual
are defined by the probability distribution of the transitions.
3.2. Modeling the incidence function in the CSPN approach
In contrast to classical models that use a mathematical incidence function to model disease
transmission, an individual-based approach requires to model in detail the processes that determine how, when and which individuals would be in contact and the probability of disease
transmission between individuals during at-risk contacts. Using CSPN models allows us to define
which individuals may be in contact (by determining the input structural places that are connected
to the transition corresponding to the encounter), the frequency of at-risk contacts for each individual (by specifying the probability distribution of the transition) and the probability that an
at-risk contact may lead to a new infection (by defining the rules by which token colors are
modified).
By this way, it is possible to model all kinds of transmission processes. In particular, densitydependent vs. frequency-dependent transmission can easily be retrieved. For sake of brevity, let us
assume that there is a unique structural place in the system containing S tokens of one color
(corresponding to S susceptible individuals) and I tokens of another color (corresponding to I
infected individuals) with N ¼ S þ I. In a first case (Fig. 1(A)), a stochastic transition with an
exponential distribution of parameter k=2 randomly takes a first token and instantaneously a
second token is chosen by the firing of a second transition. By this way, during an interval of time
Dt, each individual has on average kDt contacts: he ÔprovokesÕ half and ÔundergoesÕ half. The total
average number of contacts during Dt is k=2NDt. Thus, the average number of contacts involving
one susceptible and one infected individual is
SI
IS
þ
k=2N Dt
N ðN 1Þ N ðN 1Þ
¼k
SI
Dt:
N 1
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N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
A
<x>
N
C={s,i}
Expo(λ /2)
<x>
pre-encounter
single
C={s,i}
<x>
<x>
δ (0)
<x> U <x>
δ (0)
encounter
if one <s> and one <i>
then probability p that
<s> => <i>
<x> U <x>
<x> U <x>
B
N
C={s,i}
<x> U <x>
Expo(λ )
<x> U <x>
pair
C={s,i}
pair
C={s,i}
<x> U <x>
<x> U <x>
δ (0)
if one <s> and one <i>
then probability p that <s> => <i>
Fig. 1. (A) Modeling the proportionate-mixing incidence function using CSPN. The CSPN has a unique structural
place (named N ) containing two types of tokens (susceptible <s> or infected <i>). A stochastic transition ÔpreencounterÕ exponentially distributed with parameter k=2 determines at which frequency an individual provokes an
encounter. After the firing of this transition, the place ÔsingleÕ contains one token corresponding to an individual that is
ready to encounter a congener. This individual can be susceptible or infected (noted <x>). The firing of the instantaneous transition ÔencounterÕ selects the second individual that will encounter the individual in the place ÔsingleÕ. As for
the first token, the second one can be susceptible or infected. The two tokens are now in place ÔpairÕ. If the pair is
constituted by two susceptible or two infected individuals, nothing happens and the two individuals could return to
place N . If the pair contains one susceptible and one infected individual, nothing happens to the infected one but
susceptible one may be infected with a probability p before returning to the place N . (B) Modeling the mass-action
incidence function using CSPN. The model is similar to the one in A. However, the two tokens in the place ÔpairÕ are
now chosen simultaneously among all the tokens in the place N after the firing of the stochastic transition ÔencounterÕ.
Finally, if we note p the probability of disease transmission during a contact, the average number
of new infections during Dt is
SI
SI
Dt ¼ b
Dt with b ¼ kp;
N 1
N 1
that is equivalent to the proportionate-mixing incidence function.
In a second case (Fig. 1(B)), a stochastic transition with an exponential distribution of parameter k takes simultaneously a couple of individuals. The total average number of contacts during
Dt is kN ðN 1ÞDt; thus the average number of new infections during Dt is now
kp
kpN ðN 1ÞDt
SI
¼ bSIDt;
N ðN 1Þ
that is equivalent to the mass-action incidence function.
It is possible to model vertical transmission (i.e. from mother to offspring) in this way. This
requires to determine which female may mate and to model the process (pregnancy and/or
N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
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Fig. 2. Modeling the vertical transmission of a disease using CSPN in the particular case where each female give birth
to only one young after each period of gestation (see text for explanations).
maternal care period) during which the pathogen may be transmitted (Fig. 2). We assume that
only females in estrus can mate. After weaning, a female enters the estrus period again after a
random time delay. Mating occurs if a male is available during the receptive period. We assume
that the duration of mating is distributed according to a given distribution probability and that
the male is again available for other receptive females after a random time (e.g. during which
males search for other receptive females). The female becomes pregnant after P
mating. We assume
that a mother gives birth to i young ði 2 f1 . . . ngÞ with probability qi (with i qi ¼ 1) and that
offspring have a probability p1 to be infected if their mother is infected. Finally, the duration of the
parental care is random and during this period, an infected mother may infect its susceptible
offspring with probability p2 .
It is straightforward to verify that if there is at least one male in the population, if the duration
of mating, pregnancy and maternal care periods are assumed to be null, if i ¼ 1 and if two
consecutive estrus of a female are separated by a random delay exponentially distributed (with
parameter c), then the average number of new vertically infected individuals in the population is
given by cDtðp1 þ ð1 p1 Þp2 ÞIf with If being the number of infected females in the population.
3.3. Modeling the SI model using a CSPN approach
In the SI model, the host population is homogeneous thus all individuals are assumed to have
the same structural variables. Consequently, the corresponding CSPN model (see Fig. 3) is quite
simple and has only one structural place named N . The part of the model corresponding to the
transmission of the disease is similar to the model in Fig. 1(A). Recruitment is modeled by a
stochastic transition (named ÔrecruitmentÕ) with an exponential distribution of parameter mN0 .
This transition will fire after a random delay by creating a new <s> token in the place N . Death of
a susceptible individual will be represented by a stochastic transition (called Ôdeath of susceptibleÕ)
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N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
Expo(m+α )
Expo(m)
death of susceptible
death of infected
<i>
<s>
Expo(mN0)
recruitment
<s>
<x>
N
C={s,i}
<x>
single
C={s,i}
<x>
<x>
<x> U <x>
if one <s> and one <i>
then probability p that
<s> => <i>
Expo(λ /2)
pre-encounter
δ (0)
δ (0)
encounter
<x> U <x>
<x> U <x>
pair
C={s,i}
Fig. 3. Modeling the SI system with recruitment and death and proportionate-mixing incidence using CSPN.
with an exponential distribution of parameter m. This transition removes a <s> token from place
N after it had stay a random exponentially distributed time in place N . The transition Ôdeath
of infectedÕ is similar but concerns infected individuals and has an exponential distribution of
parameter m þ a.
The analysis of the model can be done using one of the available software packages [32]. Since
the Markov chain associated with the SI model with recruitment and death may be intractable as
soon as the population exceeds few individuals, the analysis of the system was made using simulations. We used the simulation software package MissRdPÓ by IXI [36] and all simulations
were carried out on a personal computer. In the Petri net language, a history is one possible
realization of the stochastic model, in other words, one run of the model. We present in the
following paragraph the results of 1000 histories for each set of parameters and compare our
results with those in the literature.
4. Results
Fig. 4 gives the probability of disease extinction for different values of R0 and different initial
conditions (I0 and N0 ). As expected, if R0 < 1 the disease does not persist in the host population.
In contrast, if R0 > 1 the behavior of the system is more complex. If R0 is small and I0 ¼ 1, the
probability of disease extinction ðPext Þ rises dramatically to one even for large population size.
However, for higher values of R0 or for I0 > 1, the probability of disease extinction reaches a
constant value for large population size. This probability increases as the population size decreases leading to extinction of the disease for small population size. Those results are in agreement with previous studies [30,37–39]. Indeed, an important general result of stochastic epidemic
models is that the probability of extinction Pext eventually goes to one [37,39]. However, the time
to extinction of the disease increases with the population size, the contact rate and the probability
of disease transmission during an at-risk contact [38,40,41], and thus may be very long even astronomic in view of the population life-length [37,42]. Furthermore, if the time to extinction is
a
long, Pext rises to a plateau that appears as an asymptotic value near ð1=R0 Þ , with a being the
initial number of infected individuals introduced into the population [1,30,43].
N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
9
Fig. 4. Probability of disease extinction for different R0 (see text for definition) and initial conditions N0 and I0
(population size and number of infected individuals at t ¼ 0 respectively).
Fig. 5. Probability distribution of the prevalence from the 1000 histories of the CSPN in Fig. 5 and deterministic
prevalence obtained from Eq. (1) (––––) for R0 ¼ 2, N0 ¼ 100 and I0 ¼ 1.
The probability distribution of the prevalence from t ¼ 0 to 300 and for R0 ¼ 2 is plotted in Fig.
5 for large host populations ðN0 ¼ 100Þ. The distribution is bimodal with one mass on zero and
one near the deterministic equilibrium prevalence. Hence, if the transmission rate is high, in large
populations, two patterns are possible in the middle run: either early in the epidemic, the first
infected individuals die before having infected as many susceptible individuals as required for the
disease to persist after the first year; or the disease persists for a long time in the host population.
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Furthermore, if we look at the average prevalence conditioned on the non-extinction of the
disease (i.e. 8t, the mean of the prevalence in all the histories in which IðtÞ > 0), we can see that it
is very close to the deterministic equilibrium prevalence. Once more, this result is not surprising:
a
during the time the probability of extinction remains constant and equal to ð1=R0 Þ , the epidemic
process under consideration admits a quasi-stationary state [39] which is very similar to the deterministic endemic equilibrium [42].
5. Discussion
Incorporating individual heterogeneities into epidemic models is now recognized as an
important but difficult task because of the complexity of the models developed. In this
paper, we have proposed to use the colored stochastic Petri net formalism to model directly transmitted infectious diseases. Colored stochastic Petri nets belong to the class of individual-based models in the sense that they consider individuals as entities with their own
characteristics. Therefore, individual interactions within the population can be modeled by describing individual behaviors rather than flow between classes as in the classical compartmental
approach. In the epidemiological context, hosts have to be characterized by at-risk classes and by
their serological states. At-risk contacts during which the disease may be transmitted are modeled
by processes implying two hosts and describing in details the behavior of both individuals
during this contact according to their serological state. This approach has allowed us to give an
interpretation of the incidence function at the individual level. This mathematical function is
replaced by modeling actions that define the frequency of at-risk contacts per individual according
to their characteristics, the dependency of this frequency on the host population size and the
probability of disease transmission during these contacts according to individual serological
states.
To illustrate the application of colored stochastic Petri net, we have investigated the simplest
stochastic SI model with recruitment and deaths. As previously underlined [38], we have shown
that the deterministic model is a very poor approximation of the behavior of the host–parasite
system, even when the host population size is large. Ignoring the stochastic variation in the individual parameters leads to underestimate the probability of disease extinction: even for R0 > 1,
the disease may not develop by chance; if the disease becomes endemic it ends by chance in the
long run.
One of the major advantages of CSPN over other computer simulation methods lies in its
graphical formalism. The graphical representation is standard and well codified. It allows an easy
definition and interpretation of models that is very similar to standard representations in epidemiology and more generally in population biology. The modification and adaptation of a
CSPN can generally be made by suppressing or adding some places or transitions. Thus, CSPN
models can more easily be replicated and extended than computer simulation models that require
modifying the implemented code. Using CSPN, the biologist can concentrate on the design of his
model and the analysis of the results rather than the computer program. Furthermore, unlike
many computer models, CSPN are continuous time models. Individual parameters can be
modified at random times rather than at each time step. Consequently, rather than modeling the
probability that an event occurs during an interval of time, CSPN model the probability distri-
N. Bahi-Jaber, D. Pontier / Mathematical Biosciences 185 (2003) 1–13
T2
P1
11
T1
Fig. 6. The net consists of one place (P1) and two transitions (T1, T2). The transition T1 is assumed to fire on average
many times more frequently than the transition T2 (i.e. d e). For example, T1 may correspond to a contact with a
congener and T2 to the death. We can assume that each individual makes a lot of contacts during its life. If the
transition T2 is not exponentially distributed, the life expectancy of an individual is reinitialized each time it makes a
contact. This may increase drastically the hostÕs life expectancy leading to totally false models.
bution of the time of occurrence of this event. This approach is very close to the data obtained
from cohort studies or capture–recapture methods and is well adapted to modeling individual
histories. Finally, results obtained by simulating a CSPN are composed by a set of histories, giving
access to both steady and transient states. For each history, we know exactly all the changes in the
number of tokens in each place, the moment of each change and its consequences. It is thus
possible to extract some histories with a particular behavior, or those that lead to a particular
outcome for example.
The mathematical background underlying most of the stochastic epidemic models is continuous
time Markov chains. The consequence is that all the delays are exponentially distributed. However, many biological events occur at frequencies that are far from being exponentially distributed. In particular, many epidemiological studies show that latent and/or infectious periods are
very badly approximated by exponential laws [44–47]. In contrast, the application of CSPN is not
limited to exponential distributions. All transitions in the net may obey any distribution. For
instance, it is possible to model non-exponential infectious periods by defining a stochastic
transition with a Weibull distribution corresponding to the removal of an infected host. Although
the Markovian structure of the model is lost, an analytical study remains possible in some specific
cases [31] but in most cases, the analysis will become essentially based upon simulations. However,
the use of non-exponential distributions requires some cares. When all the transitions are exponentially distributed, the memoryless property of the exponential law implies that the probability
that a transition fire after a time s is the same whatever the time spent by the token in the place. If
we want to introduce a non-exponentially distributed sojourn time in a place, a problem may arise
(see example Fig. 6). Some ad-hoc methods exist to overcome this problem. They require adding
some places and transitions to the net but it is a case to case approach. Our future work is to
propose a method that permits to move from an exponential distribution to any other distribution
by only changing the law of the transition.
Acknowledgements
This work was supported by the Ôprogramme inter-EPST, action BioInformatiqueÕ (Programme
ÔModelisation par les Reseaux de Petri de la propagation de virus au sein de populations dÕh^
otes
structureesÕ). We thank J.M. Couvreur, P. Moreaux, E. Niel, F. Sauvage, Y. Bahi and two
anonymous reviewers for helpful discussion on a previous draft of the manuscript.
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