137 results on '"Péter L. Simon"'
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2. Insights from exact social contagion dynamics on networks with higher-order structures.
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István Z. Kiss, Iacopo Iacopini, Péter L. Simon, and Nicos Georgiou
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- 2023
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3. The impact of spatial and social structure on an SIR epidemic on a weighted multilayer network.
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ágnes Backhausz, István Z. Kiss, and Péter L. Simon
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- 2022
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4. Learning the parameters of a differential equation from its trajectory via the adjoint equation.
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Imre Fekete, András Molnár, and Péter L. Simon
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- 2022
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5. Fast Variables Determine the Epidemic Threshold in the Pairwise Model with an Improved Closure.
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István Z. Kiss, Joel C. Miller, and Péter L. Simon
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- 2018
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6. On Parameter Identifiability in Network-Based Epidemic Models
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István Z. Kiss and Péter L. Simon
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Pharmacology ,Computational Theory and Mathematics ,General Mathematics ,General Neuroscience ,Immunology ,General Agricultural and Biological Sciences ,General Biochemistry, Genetics and Molecular Biology ,General Environmental Science - Abstract
Modelling epidemics on networks represents an important departure from classical compartmental models which assume random mixing. However, the resulting models are high-dimensional and their analysis is often out of reach. It turns out that mean-field models, low-dimensional systems of differential equations, whose variables are carefully chosen expected quantities from the exact model provide a good approximation and incorporate explicitly some network properties. Despite the emergence of such mean-field models, there has been limited work on investigating whether these can be used for inference purposes. In this paper, we consider network-based mean-field models and explore the problem of parameter identifiability when observations about an epidemic are available. Making use of the analytical tractability of most network-based mean-field models, e.g. explicit analytical expressions for leading eigenvalue and final epidemic size, we set up the parameter identifiability problem as finding the solution or solutions of a system of coupled equations. More precisely, subject to observing/measuring growth rate and final epidemic size, we seek to identify parameter values leading to these measurements. We are particularly concerned with disentangling transmission rate from the network density. To do this, we give a condition for practical identifiability and we find that except for the simplest model, parameters cannot be uniquely determined, that is, they are practically unidentifiable. This means that there exist multiple solutions (a manifold of infinite measure) which give rise to model output that is close to the data. Identifying, formalising and analytically describing this problem should lead to a better appreciation of the complexity involved in fitting models with many parameters to data.
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- 2023
7. Super compact pairwise model for SIS epidemic on heterogeneous networks.
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Péter L. Simon and Istvan Z. Kiss
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- 2016
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- View/download PDF
8. Dynamic Control of Modern, Network-Based Epidemic Models.
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Fanni Sélley, ádám Besenyei, Istvan Z. Kiss, and Péter L. Simon
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- 2015
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9. Large Matrices Arising in Traveling Wave Bifurcations.
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Péter L. Simon
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- 2007
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10. Numerical and analytical study of bifurcations in a model of electrochemical reactions in fuel cells.
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Gábor Csörgo and Péter L. Simon
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- 2013
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11. Gasless Combustion Fronts with Heat Loss.
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Anna Ghazaryan, Stephen Schecter, and Péter L. Simon
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- 2013
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12. Differential equation approximations of stochastic network processes: An operator semigroup approach.
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András Bátkai, Istvan Z. Kiss, Eszter Sikolya, and Péter L. Simon
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- 2012
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13. Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks
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Péter L. Simon and Noémi Nagy
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Combinatorics ,Physics ,Transcritical bifurcation ,Cover (topology) ,Degree (graph theory) ,Computer Science::Information Retrieval ,Applied Mathematics ,Discrete Mathematics and Combinatorics ,Second moment of area ,Mathematics - Dynamical Systems ,Degree distribution - Abstract
The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at \begin{document}$ \tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n} $\end{document} , where \begin{document}$ \tau $\end{document} and \begin{document}$ \gamma $\end{document} are infection and recovery rates, respectively, \begin{document}$ n $\end{document} is the average degree of the network and \begin{document}$ \langle n^{2}\rangle $\end{document} is the second moment of the degree distribution. For subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.
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- 2020
14. Network analysis of England’s single parent household COVID-19 control policy impact: a proof-of-concept study
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Péter L. Simon, Natalie Edelman, Jackie Cassell, and Istvan Z. Kiss
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2019-20 coronavirus outbreak ,Coronavirus disease 2019 (COVID-19) ,Proof of concept ,Control (management) ,Single parent ,Context (language use) ,Demographic economics ,Business ,Network analysis ,Population survey - Abstract
SummaryLockdowns have been a key infection control measure for many countries during the COVID-19 pandemic. In England’s first lockdown, children of single parent households (SPHs) were permitted to move between parental homes. By the second lockdown, SPH support bubbles between households were also permitted, enabling larger within-household networks. We investigated the combined impact of these approaches on household transmission dynamics, to inform policymaking for control and support mechanisms in a respiratory pandemic context.This network modelling study applied percolation theory to a base model of SPHs constructed with population survey estimates of SPH family size. To explore putative impact, varying estimates were applied regarding extent of bubbling and proportion of Different-parentage SPHs (DSPHs) (in which children do not share both the same parents). Results indicate that the formation of giant components (in which Covid-19 household transmission accelerates) are more contingent on DSPHs than on formation of bubbles between SPHs; and that bubbling with another SPH will accelerate giant component formation where one or both are DSPHs. Public health guidance should include supportive measures that mitigate the increased transmission risk afforded by support bubbling among DSPHs. Future network, mathematical and epidemiological studies should examine both independent and combined impact of policies.
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- 2021
15. Bounds for the expected value of one-step processes.
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Benjamin Armbruster, ádám Besenyei, and Péter L. Simon
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- 2015
16. Detailed Study of Limit Cycles and Global bifurcations in a Circadian Rhythm Model.
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Péter L. Simon and András Volford
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- 2006
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17. A Retarded Differential Equation Model of Wave Propagation in a Thin Ring.
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Péter L. Simon
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- 2001
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18. Epidemic threshold in pairwise models for clustered networks: closures and fast correlations
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Luc Berthouze, Istvan Z. Kiss, Péter L. Simon, and Rosanna C. Barnard
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Physics - Physics and Society ,Computer science ,FOS: Physical sciences ,Epidemic ,Network ,Physics and Society (physics.soc-ph) ,Dynamical Systems (math.DS) ,Pairwise model ,34D20 ,Communicable Diseases ,01 natural sciences ,Article ,Clustering ,010305 fluids & plasmas ,03 medical and health sciences ,Fast variables ,92D30 ,0103 physical sciences ,FOS: Mathematics ,Range (statistics) ,Humans ,Computer Simulation ,Statistical physics ,Mathematics - Dynamical Systems ,Epidemics ,Quantitative Biology - Populations and Evolution ,QA ,Cluster analysis ,030304 developmental biology ,Clustering coefficient ,0303 health sciences ,Degree (graph theory) ,Applied Mathematics ,Model selection ,Populations and Evolution (q-bio.PE) ,90B10 ,Models, Theoretical ,Agricultural and Biological Sciences (miscellaneous) ,Correlation ,34E10 ,FOS: Biological sciences ,Modeling and Simulation ,Pairwise comparison ,Perturbation theory (quantum mechanics) ,Focus (optics) - Abstract
The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.
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- 2019
19. LIAPUNOV FUNCTIONS FOR NEURAL NETWORK MODELS
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Márton Neogrády-Kiss and Péter L. Simon
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Liapunov function ,Artificial neural network ,Organic Chemistry ,Applied mathematics ,Biochemistry ,Mathematics - Abstract
The dynamical behaviour of continuous time recurrent neural network models is studied with emphasis on global stability of a unique equilibrium. First we show in a unified context two Liapunov functions that were introduced in the nineties by Hopfield, Grossberg, Matsouka and Forti. Then we introduce a class of networks for which the model becomes a special cooperative system with a unique globally stable steady state. Finally, we show that periodic orbits may occur when the sufficient conditions for the existence of Liapunov functions are violated.
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- 2021
20. Theoretical and Numerical Considerations of the Assumptions Behind Triple Closures in Epidemic Models on Networks
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Péter L. Simon, Istvan Z. Kiss, and Nicos Georgiou
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Structure (mathematical logic) ,Mathematical optimization ,education.field_of_study ,Degree (graph theory) ,Computer science ,Numerical analysis ,Population ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,symbols ,Multinomial distribution ,Pairwise comparison ,010306 general physics ,Link (knot theory) ,education - Abstract
Networks are widely used to model the contact structure within a population and in the resulting models of disease spread. While networks provide a high degree of realism, the analysis of the exact model is out of reach and even numerical methods fail for modest network size. Hence, mean-field models (e.g. pairwise) focusing on describing the evolution of some summary statistics from the exact model gained a lot of traction over the last few decades. In this paper we revisit the problem of deriving triple closures for pairwise models and we investigate in detail the assumptions behind some of the well-known closures as well as their validity. Using a top-down approach we start at the level of the entire graph and work down to the level of triples and combine this with information around nodes and pairs. We use our approach to derive many of the existing closures and propose new ones and theoretically connect the two well-studied models of multinomial link and Poisson link selection. The theoretical work is backed up by numerical examples to highlight where the commonly used assumptions may fail and provide some recommendations for how to choose the most appropriate closure when using graphs with no or modest degree heterogeneity.
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- 2020
21. The Effect of Inhibitory Neurons on a Class of Neural Networks
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Márton Neogrády-Kiss and Péter L. Simon
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Monotone polygon ,Artificial neural network ,Computer science ,Step function ,Bounded function ,Activation function ,Function (mathematics) ,Special case ,Topology ,Stability (probability) - Abstract
The understanding of the effect of inhibitory neurons on neural networks’ dynamics is crucial to gain more insight into the biological process. Here we examine the dynamics of a special excitatory-inhibitory neural network where the network is complete. In this special case the dynamics has an order preserving property if the activation function is a positive bounded monotone increasing function. With a special choice of activation functions such as step functions we are able to analyse the whole dynamics. We do this in the case of two- and three-valued step functions. The three-valued case can exhibit stable limit cycles, so it would be worthwhile to analyse the dynamics on more complicated networks.
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- 2020
22. Stochastic simulation control of epidemic propagation on networks
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Péter L. Simon and Ágnes Bodó
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0301 basic medicine ,03 medical and health sciences ,030104 developmental biology ,Control theory ,Applied Mathematics ,0103 physical sciences ,Stochastic simulation ,Control (linguistics) ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Published
- 2018
23. Analytic Study of Bifurcations of the Pairwise Model for SIS Epidemic Propagation on an Adaptive Network
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Ágnes Bodó and Péter L. Simon
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0301 basic medicine ,Steady state (electronics) ,Applied Mathematics ,Mathematical analysis ,Ode ,Saddle-node bifurcation ,Bifurcation diagram ,01 natural sciences ,010305 fluids & plasmas ,03 medical and health sciences ,030104 developmental biology ,Bifurcation theory ,Transcritical bifurcation ,Control theory ,0103 physical sciences ,Homoclinic bifurcation ,Analysis ,Bifurcation ,Mathematics - Abstract
The pairwise ODE model for SIS epidemic propagation on an adaptive network with link number preserving rewiring is studied. The model, introduced by Gross et al. (Phys Rev Lett 96:208701, 2006), consists of four ODEs and contains three parameters, the infection rate $$\tau $$ , the recovery rate $$\gamma $$ and the rewiring rate w. It is proved that transcritical, saddle-node and Andronov–Hopf bifurcations may occur. These bifurcation curves are determined analytically in the $$(\tau , w)$$ parameter plane by using the parametric representation method, together with the two co-dimensional Takens–Bogdanov bifurcation point. It is shown that this parameter plane is divided into four regions by the above bifurcation curves. The possible behaviours are as follows: (a) globally stable disease-free steady state, (b) stable disease-free steady state with two unstable endemic equilibria and a stable periodic orbit, (c) stable disease-free steady state with a stable and an unstable endemic equilibrium and (d) a globally stable endemic equilibrium. Numerical evidence is shown that homoclinic bifurcation, giving rise to an unstable periodic orbit, and cycle-fold bifurcation also occur.
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- 2017
24. Solvability of implicit final size equations for SIR epidemic models
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Subekshya Bidari, Dylanger Pittman, Péter L. Simon, Daniel Peters, and Xinying Chen
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0301 basic medicine ,Statistics and Probability ,General Immunology and Microbiology ,Implicit function ,Applied Mathematics ,Mathematical analysis ,Complete graph ,General Medicine ,Star (graph theory) ,General Biochemistry, Genetics and Molecular Biology ,Mathematical modelling of infectious disease ,03 medical and health sciences ,030104 developmental biology ,Distribution (mathematics) ,Modeling and Simulation ,Line (geometry) ,Pairwise comparison ,General Agricultural and Biological Sciences ,Epidemic model ,Mathematics - Abstract
Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.
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- 2016
25. Progress in Industrial Mathematics at ECMI 2018
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István Faragó, Ferenc Izsák, Péter L. Simon, István Faragó, Ferenc Izsák, and Péter L. Simon
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- Mathematical physics, Mathematical optimization, Probabilities, Discrete mathematics, Data structures (Computer science), Information theory, Image processing—Digital techniques, Computer vision
- Abstract
This book explores mathematics in a wide variety of applications, ranging from problems in electronics, energy and the environment, to mechanics and mechatronics. The book gathers 81 contributions submitted to the 20th European Conference on Mathematics for Industry, ECMI 2018, which was held in Budapest, Hungary in June 2018. The application areas include: Applied Physics, Biology and Medicine, Cybersecurity, Data Science, Economics, Finance and Insurance, Energy, Production Systems, Social Challenges, and Vehicles and Transportation. In turn, the mathematical technologies discussed include: Combinatorial Optimization, Cooperative Games, Delay Differential Equations, Finite Elements, Hamilton-Jacobi Equations, Impulsive Control, Information Theory and Statistics, Inverse Problems, Machine Learning, Point Processes, Reaction-Diffusion Equations, Risk Processes, Scheduling Theory, Semidefinite Programming, Stochastic Approximation, Spatial Processes, System Identification, and Wavelets. The goal of the European Consortium for Mathematics in Industry (ECMI) conference series is to promote interaction between academia and industry, leading to innovations in both fields. These events have attracted leading experts from business, science and academia, and have promoted the application of novel mathematical technologies to industry. They have also encouraged industrial sectors to share challenging problems where mathematicians can provide fresh insights and perspectives. Lastly, the ECMI conferences are one of the main forums in which significant advances in industrial mathematics are presented, bringing together prominent figures from business, science and academia to promote the use of innovative mathematics in industry.
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- 2019
26. Mean-field approximation of counting processes from a differential equation perspective
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Dávid Kunszenti-Kovács and Péter L. Simon
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exact master equation ,Partial differential equation ,Semigroup ,Differential equation ,Applied Mathematics ,Linear system ,mean-field model ,fokker–planck equation ,Integro-differential equation ,Ordinary differential equation ,Master equation ,QA1-939 ,Applied mathematics ,Fokker–Planck equation ,Statistical physics ,Mathematics - Abstract
Deterministic limit of a class of continuous time Markov chains is considered based purely on differential equation techniques. Starting from the linear system of master equations, ordinary differential equations for the moments and a partial differential equation, called Fokker–Planck equation, for the distribution is derived. Introducing closures at the level of the second and third moments, mean-field approximations are introduced. The accuracy of the mean-field approximations and the Fokker–Planck equation is investigated by using two differential equation-based and an operator semigroup-based approach.
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- 2016
27. Transcritical bifurcation yielding global stability for network processes
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Ágnes Bodó and Péter L. Simon
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Transcritical bifurcation ,Property (philosophy) ,Steady state (electronics) ,Dynamical systems theory ,Simple (abstract algebra) ,Applied Mathematics ,Ode ,Applied mathematics ,Stability (probability) ,Analysis ,Mathematics ,Network model - Abstract
Several network processes exhibiting transcritical bifurcation have globally stable steady states. Their dynamical behaviour is captured by a simple property of the right hand side of the corresponding system of ODEs. Based on this property, a class of dynamical systems is introduced, for which the local stability of the trivial steady state determines the global behaviour of the system. It is shown that this condition is satisfied by three network models, namely the individual-based and degree-based ODE approximations of SIS epidemic propagation on networks and the Hopfield model with non-negative weights. The general result enables us to describe the global behaviour of these systems that was not available for the first and third models and was proved in a significantly more complicated way for the second.
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- 2020
28. Super compact pairwise model for SIS epidemic on heterogeneous networks
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Péter L. Simon and Istvan Z. Kiss
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0301 basic medicine ,Discrete mathematics ,Control and Optimization ,Degree (graph theory) ,Computer Networks and Communications ,Differential equation ,Applied Mathematics ,Closure (topology) ,Context (language use) ,Management Science and Operations Research ,Degree distribution ,03 medical and health sciences ,Computational Mathematics ,030104 developmental biology ,A priori and a posteriori ,Pairwise comparison ,Statistical physics ,Mathematics - Dynamical Systems ,QA ,Quantitative Biology - Populations and Evolution ,Heterogeneous network ,Mathematics - Abstract
In this paper we provide the derivation of a super compact pairwise model with only 4 equations in the context of describing susceptible-infected-susceptible (SIS) epidemic dynamics on heterogenous networks. The super compact model is based on a new closure relation that involves not only the average degree but also the second and third moments of the degree distribution. Its derivation uses an a priori approximation of the degree distribution of susceptible nodes in terms of the degree distribution of the network. The new closure gives excellent agreement with heterogeneous pairwise models that contain significantly more differential equations.
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- 2015
29. Dynamic Control of Modern, Network-Based Epidemic Models
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Ádám Besenyei, Istvan Z. Kiss, Péter L. Simon, and Fanni M. Sélley
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education.field_of_study ,Population ,Control (management) ,Ode ,Outcome (game theory) ,Controllability ,Model predictive control ,Mathematics - Classical Analysis and ODEs ,Control theory ,Modeling and Simulation ,Pairwise comparison ,Quantitative Biology - Populations and Evolution ,34H20, 05C82, 37N25, 92D30 ,QA ,education ,Analysis ,Mathematics - Abstract
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., Comment: 20 pages, 8 figures
- Published
- 2015
30. Mathematics of Epidemics on Networks : From Exact to Approximate Models
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István Z. Kiss, Joel C. Miller, Péter L. Simon, István Z. Kiss, Joel C. Miller, and Péter L. Simon
- Subjects
- Mathematical statistics, Mathematical models
- Abstract
This textbook provides an exciting new addition to the area of network science featuring a stronger and more methodical link of models to their mathematical origin and explains how these relate to each other with special focus on epidemic spread on networks. The content of the book is at the interface of graph theory, stochastic processes and dynamical systems. The authors set out to make a significant contribution to closing the gap between model development and the supporting mathematics. This is done by:Summarising and presenting the state-of-the-art in modeling epidemics on networks with results and readily usable models signposted throughout the book;Presenting different mathematical approaches to formulate exact and solvable models;Identifying the concrete links between approximate models and their rigorous mathematical representation;Presenting a model hierarchy and clearly highlighting the links between model assumptions and model complexity;Providing a reference source for advanced undergraduate students, as well as doctoral students, postdoctoral researchers and academic experts who are engaged in modeling stochastic processes on networks;Providing software that can solve differential equation models or directly simulate epidemics on networks.Replete with numerous diagrams, examples, instructive exercises, and online access to simulation algorithms and readily usable code, this book will appeal to a wide spectrum of readers from different backgrounds and academic levels. Appropriate for students with or without a strong background in mathematics, this textbook can form the basis of an advanced undergraduate or graduate course in both mathematics and other departments alike.
- Published
- 2017
31. On bounding exact models of epidemic spread on networks
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Péter L. Simon and Istvan Z. Kiss
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0209 industrial biotechnology ,Mathematical optimization ,Stochastic process ,Applied Mathematics ,Ode ,Markov process ,Context (language use) ,02 engineering and technology ,Stability (probability) ,symbols.namesake ,020901 industrial engineering & automation ,Bounding overwatch ,Mathematics - Classical Analysis and ODEs ,symbols ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Enhanced Data Rates for GSM Evolution ,Mathematics - Dynamical Systems ,QA ,Independence (probability theory) ,Mathematics - Abstract
In this paper we use comparison theorems from classical ODE theory in order to rigorously show that the N-Intertwined Mean-Field Approximation (NIMFA) model provides an upper estimate on the exact stochastic process. The proof of the results relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively), which we also prove rigorously. The results in the paper hold for arbitrary weighted and directed networks. We cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models., Comment: 18 pages
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- 2017
32. Dynamic and adaptive networks
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Joel C. Miller, Péter L. Simon, and Istvan Z. Kiss
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Structure (mathematical logic) ,Relation (database) ,Computer science ,Distributed computing ,Feature (machine learning) ,Terminology - Abstract
An important feature of many real-world networks is the transient nature of some interactions. Thus far, our models have explicitly assumed that the network is static. That is, we assume that the rate of partner turnover is so slow that we can ignore its impact on epidemic dynamics.Over the past decade, there has been tremendous progress in modelling and analysing disease spread in non-static networks, i.e. networks whose structure changes due to endogenous or exogenous factors, or because of the disease dynamics unfolding on the network. The terminology used to describe such networks is not standardised. We summarise some common terminology and link it to the relation between the time scales of the dynamics on the network and the dynamics of the network. Ordered from static to networks that change quickly, we have
- Published
- 2017
33. Introduction to networks and diseases
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Péter L. Simon, Istvan Z. Kiss, and Joel C. Miller
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Mathematical model ,Order (exchange) ,Computer science ,Real systems ,Process (engineering) ,Management science ,0103 physical sciences ,010306 general physics ,All models are wrong ,01 natural sciences ,010305 fluids & plasmas - Abstract
Mathematical models are caricatures of real systems that aim to capture the fundamental mechanisms of some process in order to explain observations or predict outcomes. No model — no matter how complicated — is perfect, or in the words of George Box [46]: “All models are wrong; some models are useful”. A useful model can provide valuable insights which improve our understanding of a system, and ultimately informs our decision-making.
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- 2017
34. Mean-field approximations for heterogeneous networks
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István Z. Kiss, Joel C. Miller, and Péter L. Simon
- Published
- 2017
35. Propagation models on networks: bottom-up
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Péter L. Simon, Istvan Z. Kiss, and Joel C. Miller
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Closed system (control theory) ,Computer science ,Order (ring theory) ,Applied mathematics ,Node (circuits) ,State (functional analysis) ,Top-down and bottom-up design ,Representation (mathematics) - Abstract
In this chapter, we present a different approach to deriving exact models. In Chapter 2, we began with equations for every possible state of the system and then aggregated them into a simpler form. Here, we begin by deriving separate equations for the status of each node. These typically depend on the states of pairs of nodes, so we introduce equations for the pairs, which in turn depend on triples. We build up equations at each level. For a typical network, the number of equations we obtain is too large to be tractable. Consequently, we introduce “closures”, whereby terms corresponding to larger structures are represented in terms of smaller structures, in order to create a closed system of equations. In most cases, this representation involves an approximation, but in the case of SIR dynamics on trees or networks with cut-vertices, it is possible to reduce the number of equations considerably while keeping the model exact.
- Published
- 2017
36. Hierarchies of SIR models
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Istvan Z. Kiss, Joel C. Miller, and Péter L. Simon
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Computer science ,Mathematical economics - Abstract
This chapter focuses on the relationships between the continuous-time SIR models we have previously derived and identifying conditions under which they are appropriate. Unless otherwise noted, the models discussed in this chapter are SIR models. Each of these models involves some assumptions, and to understand their limitations, we need to understand whether the true spread “respects” these assumptions.
- Published
- 2017
37. Non-Markovian epidemics
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Istvan Z. Kiss, Joel C. Miller, and Péter L. Simon
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education.field_of_study ,Queueing theory ,Distribution (number theory) ,Population ,Markov process ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mixing (mathematics) ,Probability theory ,0103 physical sciences ,symbols ,Quantitative Biology::Populations and Evolution ,Statistical physics ,Limit (mathematics) ,0101 mathematics ,010306 general physics ,education ,Branching process ,Mathematics - Abstract
Early studies of non-Markovian epidemics focused on SIR dynamics on fully connected networks, or homogeneously mixing populations, with the infection process being Markovian but with the infectious period taken from a general distribution [8, 278, 292, 293]. These approaches use probability theory arguments and typically focus on characterising the distribution of final epidemic sizes for finite populations, or on the average size in the infinite population limit. Similarly, the quasi-stationary distribution in a stochastic SIS model, again in a fully connected network, has been the subject of many studies [66, 230, 231]. More recently, it has been shown that one can readily apply results from queueing [19] or branching process [233] theory, or use martingales [65] to cast the same questions within a different framework and obtain results more readily.
- Published
- 2017
38. Disease spread in networks with large-scale structure
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Joel C. Miller, Péter L. Simon, and Istvan Z. Kiss
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Degree correlation ,Theoretical computer science ,Scale (ratio) ,Process (engineering) ,Computer science ,Scale structure ,Closure (topology) ,Degree distribution ,Preferential attachment - Abstract
This book has developed analytic models of disease spread on networks. All of our tractable models require closure assumptions. The closure process assumes that we can explain the dynamics at the network scale by understanding the dynamics locally at the level of small units with random connections between these units.
- Published
- 2017
39. PDE limits for large networks
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Péter L. Simon, Istvan Z. Kiss, and Joel C. Miller
- Subjects
Markov chain ,Master equation ,Ode ,Applied mathematics ,Probability distribution ,State space ,State (functional analysis) ,Expected value ,Random variable ,Mathematics - Abstract
In previous chapters, it was shown that dynamics on networks can be described by continuous-time Markov chains, where probabilities of states are determined by master equations. While limiting mean-field ODE models can provide a good approximation of the expected value of certain random variables, a PDE-based approach is needed if information on the probability distribution of these is desired. The aim of this chapter is to develop and study mathematical methods to estimate the accuracy of mean-field ODE and PDE approximations. This will be carried out for continuous-time Markov chains with state space {0, 1, …, N}, for which transition from state k is possible only to states k − 1 and k + 1. These processes are called birth–death or one-step processes.
- Published
- 2017
40. Mapping Out Emerging Network Structures in Dynamic Network Models Coupled with Epidemics
- Author
-
Istvan Z. Kiss, Joel C. Miller, Péter L. Simon, and Luc Berthouze
- Subjects
Structure (mathematical logic) ,Bifurcation theory ,Steady state (electronics) ,Dynamic network analysis ,Degree (graph theory) ,Node (networking) ,Econometrics ,Network structure ,Pairwise comparison ,Topology ,Mathematics - Abstract
We consider the susceptible – infected – susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classical bifurcation theory to map out the spectrum of all possible epidemic behaviours. However, the major focus of the chapter is on the evolution and possible equilibria of the resulting networks. Whereas most studies are driven by determining system-level outcomes, e.g., whether the epidemic dies out or becomes endemic, with little regard for the emerging network structure, here, we want to buck this trend by augmenting the system-level results with mapping out of the structure and properties of the resulting networks. We find that depending on parameter values the network can become disconnected and show bistable-like behaviour whereas the endemic steady state sees the emergence of networks with qualitatively different degree distributions. In particular, we observe de-phased oscillations of both prevalence and network degree during which there is role reversal between the level and nature of the connectivity of susceptible and infected nodes. We conclude with an attempt at describing what a potential bifurcation theory for networks would look like.
- Published
- 2017
41. Percolation-based approaches for disease modelling
- Author
-
Péter L. Simon, Istvan Z. Kiss, and Joel C. Miller
- Subjects
Disease modelling ,Percolation (cognitive psychology) ,Computer science ,Node (networking) ,Complete theory ,Topology ,humanities ,Giant component - Abstract
The methods introduced thus far are applicable to both SIS and SIR diseases. This chapter focuses primarily on SIR disease. Once a node u becomes infected with an SIR disease, no other node affects the timing of any other action of u. This permits simplifications leading to a more complete theory of SIR disease.
- Published
- 2017
42. The Effect of Graph Structure on Epidemic Spread in a Class of Modified Cycle Graphs
- Author
-
A. Szabó-Solticzky and Péter L. Simon
- Subjects
Block graph ,Discrete mathematics ,Applied Mathematics ,Butterfly graph ,law.invention ,Combinatorics ,Indifference graph ,Pathwidth ,law ,Modeling and Simulation ,Outerplanar graph ,Line graph ,Split graph ,Mathematics ,Distance-hereditary graph - Abstract
In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation is studied on a special class of 3-regular graphs, called modified cycle graphs. The modified cycle graph is constructed from a cycle graph with N nodes by connecting node i to the node i + d in a way that every node has exactly three links. Monte-Carlo simulations show that the propagation process depends on the value of d in a non-monotone way. A new theoretical model is developed to explain this phenomenon. This reveals a new relation between the spreading process and the average path length in the graph.
- Published
- 2014
43. Approximate Master Equations for Dynamical Processes on Graphs
- Author
-
Péter L. Simon, N. Nagy, and Istvan Z. Kiss
- Subjects
Modeling and Simulation ,Applied Mathematics ,Ordinary differential equation ,Mathematical analysis ,Master equation ,Epidemic dynamics ,Ode ,Network structure ,Numerical tests ,Graph property ,Graph ,Mathematics - Abstract
We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.
- Published
- 2014
44. Numerical and analytical study of bifurcations in a model of electrochemical reactions in fuel cells
- Author
-
GáBor CsöRg and Péter L. Simon
- Subjects
Ode ,Saddle-node bifurcation ,Mechanics ,Bifurcation diagram ,Stationary point ,Biological applications of bifurcation theory ,Computational Mathematics ,Transcritical bifurcation ,Pitchfork bifurcation ,Computational Theory and Mathematics ,Control theory ,Modelling and Simulation ,Modeling and Simulation ,Nonlinear Sciences::Pattern Formation and Solitons ,Bifurcation ,Mathematics - Abstract
The bifurcations in a three-variable ODE model describing the oxygen reduction reaction on a platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The goal of this paper is to determine the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, and to find the possible bifurcations. It is shown that a non-trivial steady state can appear through a transcritical bifurcation, or a stable and an unstable steady state can arise as a result of saddle-node bifurcation. The saddle-node bifurcation curve is determined by using the parametric representation method, and this enables us to determine numerically the parameter domain where bistability occurs, which is important from the chemical point of view.
- Published
- 2013
45. Gasless Combustion Fronts with Heat Loss
- Author
-
Stephen Schecter, Péter L. Simon, and Anna Ghazaryan
- Subjects
Singular perturbation ,Applied Mathematics ,Mathematical analysis ,Heat losses ,Sense (electronics) ,Function (mathematics) ,Physics::Chemical Physics ,Combustion ,Mathematics - Abstract
For a model of gasless combustion with heat loss, we use geometric singular perturbation theory to show existence of traveling combustion fronts. We show that the fronts are nonlinearly stable in an appropriate sense if an Evans function criterion, which can be verified numerically, is satisfied. For a solid reactant and exothermicity parameter that is not too large, we verify numerically that the criterion is satisfied.
- Published
- 2013
46. The impact of information transmission on epidemic outbreaks
- Author
-
Jackie Cassell, Istvan Z. Kiss, Péter L. Simon, and Mario Recker
- Subjects
Male ,Statistics and Probability ,Population ,Basic Reproduction Number ,Sexually Transmitted Diseases ,Information Dissemination ,Prevalence ,Disease ,Biology ,General Biochemistry, Genetics and Molecular Biology ,Disease Outbreaks ,Disease Transmission, Infectious ,Humans ,Computer Simulation ,education ,Dissemination ,education.field_of_study ,Actuarial science ,General Immunology and Microbiology ,Applied Mathematics ,General Medicine ,Models, Theoretical ,Alertness ,Action (philosophy) ,Modeling and Simulation ,Immunology ,Female ,General Agricultural and Biological Sciences ,Basic reproduction number - Abstract
For many diseases (e.g., sexually transmitted infections, STIs), most individuals are aware of the potential risks of becoming infected, but choose not to take action ('respond') despite the information that aims to raise awareness and to increases the responsiveness or alertness of the population. We propose a simple mathematical model that accounts for the diffusion of health information disseminated as a result of the presence of a disease and an 'active' host population that can respond to it by taking measures to avoid infection or if infected by seeking treatment early. In this model, we assume that the whole population is potentially aware of the risk but only a certain proportion chooses to respond appropriately by trying to limit their probability of becoming infectious or seeking treatment early. The model also incorporates a level of responsiveness that decays over time. We show that if the dissemination of information is fast enough, infection can be eradicated. When this is not possible, information transmission has an important effect in reducing the prevalence of the infection. We derive the full characterisation of the global behaviour of the model, and we show that the parameter space can be divided into three parts according to the global attractor of the system which is one of the two disease-free steady states or the endemic equilibrium.
- Published
- 2016
47. New Moment Closures Based on A Priori Distributions with Applications to Epidemic Dynamics
- Author
-
Péter L. Simon and Istvan Z. Kiss
- Subjects
General Mathematics ,Immunology ,Epidemic dynamics ,Models, Biological ,General Biochemistry, Genetics and Molecular Biology ,Humans ,Applied mathematics ,Computer Simulation ,Epidemics ,General Environmental Science ,Mathematics ,Pharmacology ,Discrete mathematics ,Models, Statistical ,Markov chain ,General Neuroscience ,Complete graph ,Binomial distribution ,Binomial Distribution ,Computational Theory and Mathematics ,Closure (computer programming) ,A priori and a posteriori ,Pairwise comparison ,Disease Susceptibility ,General Agricultural and Biological Sciences ,Epidemic model - Abstract
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. The fully connected nature of the graph also allows us to interpret pairwise equations in terms of the moments and thus treat closures and the two approximate models within the same framework. Finally, the applicability and limitations of the new methodology is discussed in detail.
- Published
- 2012
48. From exact stochastic to mean-field ODE models: a new approach to prove convergence results
- Author
-
Istvan Z. Kiss and Péter L. Simon
- Subjects
Random graph ,Algebra ,Partial differential equation ,Markov chain ,Stochastic modelling ,Applied Mathematics ,Ordinary differential equation ,Ode ,Martingale (probability theory) ,Mathematical proof ,Mathematics - Abstract
In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.
- Published
- 2012
49. Modelling approaches for simple dynamic networks and applications to disease transmission models
- Author
-
Luc Berthouze, Tim Taylor, Istvan Z. Kiss, and Péter L. Simon
- Subjects
Steady state (electronics) ,Dynamic network analysis ,Markov chain ,business.industry ,Computer science ,General Mathematics ,General Engineering ,General Physics and Astronomy ,Statistical mechanics ,Complex network ,Network dynamics ,Benchmark (computing) ,Pairwise comparison ,Statistical physics ,Artificial intelligence ,business - Abstract
In this paper a random link activation–deletion (RLAD) model is proposed that gives rise to a stochastically evolving network. This dynamic network is then coupled to a simple susceptible-infectious-suceptible ( SIS ) dynamics on the network, and the resulting spectrum of model behaviour is explored via simulation and a novel pairwise model for dynamic networks. First, the dynamic network model is systematically analysed by considering link-type independent and dependent network dynamics coupled with globally constrained link creation. This is done rigorously with some analytical results and we highlight where such analysis can be performed and how these simpler models provide a benchmark to test and validate full simulations. The pairwise model is used to study the interplay between SIS -type dynamics on the network and link-type-dependent activation–deletion. Assumptions of the pairwise model are identified and their implications interpreted in a way that complements our current understanding. Furthermore, we also discuss how the strong assumptions of the closure relations can lead to disagreement between the simulation and pairwise model. Unlike on a static network, the resulting spectrum of behaviour is more complex with the prevalence of infections exhibiting not only a single steady state, but also bistability and oscillations.
- Published
- 2012
50. Detailed study of bifurcations in an epidemic model on a dynamic network
- Author
-
Istvan Z. Kiss, Péter L. Simon, and András Szabó
- Subjects
Hopf bifurcation ,Period-doubling bifurcation ,Steady state (electronics) ,Organic Chemistry ,Saddle-node bifurcation ,Bifurcation diagram ,Biochemistry ,Biological applications of bifurcation theory ,symbols.namesake ,Transcritical bifurcation ,Control theory ,symbols ,Applied mathematics ,Infinite-period bifurcation ,Mathematics - Abstract
The bifurcations in a four-variable ODE model of an SIS type epidemic on an adaptive network are studied. The model describes the propagation of the epidemic on a network where links (or edges) of different type (i.e. SI;II and SS ) can be activated or deleted according to a simple rule consisting of random link activation and deletion. In the case when II links cannot be neither deleted nor created it is proved that the system can have at most three steady states with the trivial, disease-free steady state being one of them. It is shown that a stable endemic steady state can appear through a transcritical bifurcation, or a stable and an unstable endemic steady state arise as a result of saddle-node bifurcation. Moreover, at the endemic steady state a Hopf bifurcation may occur giving rise to stable oscillation. The bifurcation curves in the parameter space are determined analytically using the parametric representation method. For certain parameter regimes or bifurcation types, analytical results based on the ODE model show good agreement when compared to results based on individual- based network simulations. When agreement between the two modelling approaches holds, the ODE-based model provides a faster and more reliable tool that can be used to explore full spectrum of model behaviour.
- Published
- 2012
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