arXiv subjects

Péter L. Simon

Publications and source records attributed to Péter L. Simon.

23 records · Page 2Linked to original sources

arXiv2014

Dynamic control of modern, network-based epidemic models

In this paper we make the first steps to bridge the gap between classic control theory and modern, network-based epidemic models. In particular, we apply nonlinear model predictive control (NMPC) to a pairwise ODE model which we use to model a susceptible-infectious-susceptible (SIS) epidemic on non-trivial contact structures. While classic control of epidemics concentrates on aspects such as vaccination, quarantine and fast diagnosis, our novel setup allows us to deliver control by altering the contact network within the population. Moreover, the ideal outcome of control is to eradicate the disease while keeping the network well connected. The paper gives a thorough and detailed numerical investigation of the impact and interaction of system and control parameters on the controllability of the system. The analysis reveals, that for certain set parameters it is possible to identify critical control bounds above which the system is controllable. We foresee, that our approach can be extended to even more realistic or simulation-based models with the aim to apply these to real-world situations.

math.CA↗

arXiv2013

Exact deterministic representation of Markovian SIR epidemics on networks with and without loops

In a previous paper Sharkey et al. [13] proved the exactness of closures at the level of triples for Markovian SIR (susceptible-infected-removed) dynamics on tree-like networks. This resulted in a deterministic representation of the epidemic dynamics on the network that can be numerically evaluated. In this paper, we extend this modelling framework to certain classes of networks exhibiting loops. We show that closures where the loops are kept intact are exact, and lead to a simplified and numerically solvable system of ODEs (ordinary-differential-equations). The findings of the paper lead us to a generalisation of closures that are based on partitioning the network around nodes that are cut-vertices (i.e. the removal of such a node leads to the network breaking down into at least two disjointed components or subnetworks). Exploiting this structural property of the network yields some natural closures, where the evolution of a particular state can typically be exactly given in terms of the corresponding or projected sates on the subnetworks and the cut-vertex. A byproduct of this analysis is an alternative probabilistic proof of the exactness of the closures for tree-like networks presented in Sharkey et al. [13]. In this paper we also elaborate on how the main result can be applied to more realistic networks, for which we write down the ODEs explicitly and compare output from these to results from simulation. Furthermore, we give a general, recipe-like method of how to apply the reduction by closures technique for arbitrary networks, and give an upper bound on the maximum number of equations needed for an exact representation.

math.PR↗

arXiv2012

Identification of criticality in neuronal avalanches: I. A theoretical investigation of the non-driven case

In this paper we study a simple model of a purely excitatory neural network that, by construction, operates at a critical point. This model allows us to consider various markers of criticality and illustrate how they should perform in a finite-size system. By calculating the exact distribution of avalanche sizes we are able to show that, over a limited range of avalanche sizes which we precisely identify, the distribution has scale free properties but is not a power law. This suggests that it would be inappropriate to dismiss a system as not being critical purely based on an inability to rigorously fit a power law distribution as has been recently advocated. In assessing whether a system, especially a finite-size one, is critical it is thus important to consider other possible markers. We illustrate one of these by showing the divergence of susceptibility as the critical point of the system is approached. Finally, we provide evidence that power laws may underlie other observables of the system, that may be more amenable to robust experimental assessment.

q-bio.NC↗

arXiv2011

Exact and approximate epidemic models on networks: a new, improved closure relation

Recently, research that focuses on the rigorous understanding of the relation between simulation and/or exact models on graphs and approximate counterparts has gained lots of momentum. This includes revisiting the performance of classic pairwise models with closures at the level of pairs and/or triples as well as effective-degree-type models and those based on the probability generating function formalism. In this paper, for a fully connected graph and the simple $SIS$ (susceptible-infected-susceptible) epidemic model, a novel closure is introduced. This is done via using the equations for the moments of the distribution describing the number of infecteds at all times combined with the empirical observations that this is well described/approximated by a binomial distribution with time dependent parameters. This assumption allows us to express higher order moments in terms of lower order ones and this leads to a new closure. The significant feature of the new closure is that the difference of the exact system, given by the Kolmogorov equations, from the solution of the newly defined approximate system is of order $1/N^2$. This is in contrast with the $\mathcal{O}(1/N)$ difference corresponding to the approximate system obtained via the classic triple closure.

math.DS↗

arXiv2011

Differential equation approximations of stochastic network processes: an operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE.

math.DS↗