From Symmetric Nets to Differential Equations exploiting Model Symmetries

Beccuti, Marco; Fornari, Chiara; Franceschinis, Giuliana; Halawani, Sami M.; Barukab, Omar; Ahmad, Ab Rahman; Balbo, Gianfranco

doi:10.1093/comjnl/bxt111

Cited by 19 publications

(10 citation statements)

References 48 publications

(54 reference statements)

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“…In the literature, different techniques are proposed to solve (or analyse) the CTMC underlying an SSN; in particular, in case of very complex models, the so-called deterministic approach [15] can be efficiently exploited. According to this, in [16] we described how to derive a deterministic process, represented through a system of ODEs, which well approximates the stochastic behavior of an SSN model. In particular for each place p and possible color tuple c ∈ c d ( p ) we have the following ODE: …”

Section: Methodsmentioning

confidence: 99%

A computational approach based on the colored Petri net formalism for studying multiple sclerosis

et al. 2019

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Background: Multiple Sclerosis (MS) is an immune-mediated inflammatory disease of the Central Nervous System (CNS) which damages the myelin sheath enveloping nerve cells thus causing severe physical disability in patients. Relapsing Remitting Multiple Sclerosis (RRMS) is one of the most common form of MS in adults and is characterized by a series of neurologic symptoms, followed by periods of remission. Recently, many treatments were proposed and studied to contrast the RRMS progression. Among these drugs, daclizumab (commercial name Zinbryta), an antibody tailored against the Interleukin-2 receptor of T cells, exhibited promising results, but its efficacy was accompanied by an increased frequency of serious adverse events. Manifested side effects consisted of infections, encephalitis, and liver damages. Therefore daclizumab has been withdrawn from the market worldwide. Another interesting case of RRMS regards its progression in pregnant women where a smaller incidence of relapses until the delivery has been observed. Results: In this paper we propose a new methodology for studying RRMS, which we implemented in GreatSPN, a state-of-the-art open-source suite for modelling and analyzing complex systems through the Petri Net (PN) formalism. This methodology exploits: (a) an extended Colored PN formalism to provide a compact graphical description of the system and to automatically derive a set of ODEs encoding the system dynamics and (b) the Latin Hypercube Sampling with PRCC index to calibrate ODE parameters for reproducing the real behaviours in healthy and MS subjects. To show the effectiveness of such methodology a model of RRMS has been constructed and studied. Two different scenarios of RRMS were thus considered. In the former scenario the effect of the daclizumab administration is investigated, while in the latter one RRMS was studied in pregnant women. Conclusions: We propose a new computational methodology to study RRMS disease. Moreover, we show that model generated and calibrated according to this methodology is able to reproduce the expected behaviours.

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Section: Methodsmentioning

confidence: 99%

A computational approach based on the colored Petri net formalism for studying multiple sclerosis

et al. 2019

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“…The first model depicted in Fig. 1(a) is inspired by the epidemiological Susceptible-Infectious-Recovered (SIR) mathematical representation of this problem originally introduced in [10] and discussed with respect to some of its variants in [4,5]. It describes the diffusion of an epidemic on a large population, and assumes that the population members are part of three sub-populations according to their health status: (a) susceptible members (represented in the model by tokens in place S) that are not ill, but that are susceptible to the disease; (b) infected members (i.e., tokens in place I) that are subject to the disease and can spread it among susceptible members; (c) recovered members (i.e., tokens in place R) that were previously ill and are now immune.…”

Section: Resultsmentioning

confidence: 99%

“…This choice is done to avoid the initial barrier effect due to empty places 5. The generation of a CTMC from a SPN model and its solution are obtained using the GSPN solvers available in the GreatSPN suite.…”

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confidence: 99%

Analysis of Petri Net Models through Stochastic Differential Equations

Beccuti

Bibbona

Horváth

et al. 2014

Application and Theory of Petri Nets and Concurrency

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It is well known, mainly because of the work of Kurtz, that density dependent Markov chains can be approximated by sets of ordinary differential equations (ODEs) when their indexing parameter grows very large. This approximation cannot capture the stochastic nature of the process and, consequently, it can provide an erroneous view of the behavior of the Markov chain if the indexing parameter is not sufficiently high. Important phenomena that cannot be revealed include non-negligible variance and bi-modal population distributions. A lessknown approximation proposed by Kurtz applies stochastic differential equations (SDEs) and provides information about the stochastic nature of the process. In this paper we apply and extend this diffusion approximation to study stochastic Petri nets. We identify a class of nets whose underlying stochastic process is a density dependent Markov chain whose indexing parameter is a multiplicative constant which identifies the population level expressed by the initial marking and we provide means to automatically construct the associated set of SDEs. Since the diffusion approximation of Kurtz considers the process only up to the time when it first exits an open interval, we extend the approximation by a machinery that mimics the behavior of the Markov chain at the boundary and allows thus to apply the approach to a wider set of problems. The resulting process is of the jumpdiffusion type. We illustrate by examples that the jump-diffusion approximation which extends to bounded domains can be much more informative than that based on ODEs as it can provide accurate quantity distributions even when they are multi-modal and even for relatively small population levels. Moreover, we show that the method is faster than simulating the original Markov chain.

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“…Equivalent analysis method is a kind of generalised symmetry analysis method, allowing more dynamic / generalised equivalent concept, so it can be used when system lacks symmetry in the system. For High-Level SPN state reduction, Marco Beccuti (Beccuti et al, 2015) automatically derived a reduced set of ordinary differential equations (ODEs) which mimic the system behaviour. Exploiting symmetrical properties of SSN representations is not only at the construction level, but also at that of the solution.…”

Section: State Space Construction and Reduction Methods Base On Petri mentioning

confidence: 99%