Thomas M Michelitsch

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French National Centre for Scientific Research | CNRS · Institut Jean le Rond d'Alembert, CNRS UMR 7190

We focus on the propagation of vector-transmitted diseases in complex networks such as Barab\'asi-Albert (BA) and Watts-Strogatz (WS) types. The class of such diseases includes Malaria, Dengue (vectors are mosquitos), Pestilence (vectors are fleas), and many others. There is no direct transmission of the disease among individuals. Individuals are m...

We consider a generalized SEIS (susceptible, exposed, infectious, and susceptible) model where individuals are divided into three compartments: S (healthy and susceptible), E (infected but not just infectious, or exposed), and I (infectious). Finite waiting times in the compartments yield a system of delay-differential or memory equations and may e...

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-M...

In this work, the elastic 4×4 Green tensor of one-dimensional quasicrystals is given and has phonon, phason and phonon–phason coupling components. Using the residue method, a closed-form expression of the elastic 4×4 Green tensor for one-dimensional hexagonal quasicrystals of Laue class 10, which possess 10 independent material constants, is derive...

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-M...

Message from the Guest Editors In a wide range of phenomena in nature, complexity emerges through an interplay between nonlinearly interacting agents and stochastic effects. Examples include spreading phenomena such as the propagation of epidemics, forest fires, air pollution, chemical reactions, the foraging of animals, and anomalous transport pr...

Citation: Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. A Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks. Entropy 2024, 26, 362. https://

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on Barabási-Albert (BA), Erdös-Rényi (ER) and Watts-Strogatz (WS) types. Both, walkers and nodes can be either susceptible (S) or infecte...

We consider three kinds of discrete-time arrival processes: transient, intermediate and recurrent, characterized by a finite, possibly finite and infinite number of events, respectively. In this context, we study renewal processes which are stopped at the first event of a further independent renewal process whose inter-arrival time distribution can...

One dominant aspect of cities is transport and massive passenger mobilization which remains a challenge with the increasing demand on the public as cities grow. In addition, public transport infrastructure suffers from traffic congestion and deterioration, reducing its efficiency. In this paper, we study the capacity of transport in 33 worldwide me...

Our study is devoted to a four-compartment epidemic model of a constant population of independent random walkers. Each walker is in one of four compartments (S-susceptible, C-infected but not infectious (period of incubation), I-infected and infectious, R-recovered and immune) characterizing the states of health. The walkers navigate independently...

One dominant aspect of cities is transport and massive passenger mobilization which remains a challenge with the increasing demand on the public as cities grow. In addition, public transport infrastructure suffers from traffic congestion and deterioration, reducing its efficiency. In this paper, we study the capacity of transport in 33 worldwide me...

Safe and resistant infrastructure is an essential component of public safety. However, existing structures are vulnerable to damage resulting from excessive ground movement due to seismic activity or underground explosions. The aim of this paper, which is part of an extensive study, is to develop an isolation system based on periodic materials with...

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected (I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compa...

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is vi...

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected(I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compar...

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remain...

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred this random walk to as squirrel random walk (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unch...

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We deriv...

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the compartments susceptible (S); incubated - infected yet not infectious (C), infected and infectious (I), and recovered - immune (R). An infection is 'visible' only wh...

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We deriv...

We introduce a compartment model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states S=susceptible, I=infected, or R=recovered (SIR model). In state R an individual is assumed to stay immune within a finite-time interval. In the first part, we introduce a random lifetim...

In recent years a huge interdisciplinary field has emerged which is devoted to the ‘complex dynamics’ of anomalous transport with long-time memory and non-markovian features. It was found that the framework of fractional calculus and its generalizations are able to capture these phenomena. Many of the classical models are based on continuous-time r...

We introduce a modified SIR model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states susceptible (${\bf S}$), infected (${\bf I}$) or recovered (${\bf R}$). In the state ${\bf R}$ an individual is assumed to stay immune within a finite time interval. In the first part,...

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the ‘generator process’ of the walk. We refer the so defined walk...

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the `it generator process' of the walk. We refer the so defined wa...

We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time Markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for...

In this paper, we explore the reduction of functionality in a complex system as a consequence of cumulative random damage and imperfect reparation, a phenomenon modeled as a dynamical process on networks. We analyze the global characteristics of the diffusive movement of random walkers on networks that hop considering the capacity of transport of e...

In recent years a huge interdisciplinary field has emerged which is devoted to the complex dynamics of anomalous transport with long-time memory and non-markovian features. It was found that the framework of fractional calculus and its generalizations are able to capture these phenomena. Many of the classical models are based on continuous-time ren...

In this paper, we explore the reduction of functionality in a complex system as a consequence of cumulative random damage and imperfect reparation, a phenomenon modeled as a dynamical process on networks. We analyze the global characteristics of the diffusive movement of random walkers on networks that hop considering the capacity of transport of e...

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Pois...

We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for...

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Pois...

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Pois...

In this paper, we study nonlocal random walk strategies generated with the fractional Laplacian matrix of directed networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete-time Markovian process with transition probabilities between nodes expressed in terms of powers of the Laplacian matrix. We an...

We survey the `generalized fractional Poisson process' (GFPP). The GFPP is a renewal process generalizing Laskin's fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges $0<\beta\leq 1$, $\alpha >0$ and a parameter characterizing the time scale. The GFPP involv...

Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with intege...

Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with intege...

We analyze generalized space–time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effe...

Sur une généralisation de l'opérateur fractionnaire Thomas M. Michelitsch, Gérard A. Maugin, Shahram Derogar, Andrzej F. Nowakowski, Franck C. G. A. Nicolleau arXiv: 1111.1898

In this paper, we study non-local random walk strategies generated with the fractional Laplacian matrix of directed networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete-time Markovian process with transition probabilities between nodes expressed in terms of powers of the Laplacian matrix. We a...

We analyze generalized space-time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effe...

In this paper we explore the evolution of transport capacity on networks with stochastic incidence of damage and accumulation of faults in their connections. For each damaged configuration of the network, we analyze a Markovian random walker that hops over weighted links that quantify the capacity of transport of each connection. The weights of the...

In this paper we explore the evolution of transport capacity on networks with stochastic incidence of damage and accumulation of faults in their connections. For each damaged configuration of the network, we analyze a Markovian random walker that hops over weighted links that quantify the capacity of transport of each connection. The weights of the...

A non-Markovian counting process, the `generalized fractional Poisson process' (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP contains two index parameters $0<\beta\leq 1$, $\alpha >0$ and a time scale parameter. Generalizations to Laskin's fractional Poisson distribution and to the fractional Kolmogorov-Feller equation are der...

We introduce a new stochastic process, the `generalized fractional Poisson process' (GFPP). The GFPP generalizes Laskin's fractional Poisson process [{\it N. Laskin, Communications in Nonlinear Science and Numerical Simulation 8 (2003) 201-213}]. The GFPP contains two index parameters $0<\beta\leq 1$ and $\alpha >0$ and a parameter characterizing t...

This chapter describes explicit representations for the fractional Laplacian matrix on finite and infinite rings. It defines continuum limits and obtains distributional representations that take the forms of Riesz fractional derivatives on the periodic and infinite embedding 1D space. The chapter discusses the properties of the periodic string cont...

This chapter uniquely considers homogeneous random walks and their asymptotic behaviors. It analyzes continuum limits of Markovian walks on undirected connected networks, which are generated by admissible Laplacian matrix functions where essentially the two classes type (i) and type (ii) lead to distinct behaviors. The chapter explains the general...

This chapter analyzes some aspects of the remarkably rich dynamics of a “fractional” random walk strategy, which referred to as “fractional random walk” (FRW). The FRW appears to be particularly interesting when applied as random walk based search strategy due to its relationship to anomalous transport and diffusion phenomena and Levy flights. The...

This chapter explores different random walk strategies on networks with transition probabilities defined in terms of a family of functions of the Laplacian matrix that describes the network. It introduces general formalism, which allows us to generalize different results and techniques developed in the context of the fractional Laplacian of a graph...

This chapter describes classical and quantum dynamics on networks with continuous time and an evolution defined in terms of the fractional Laplacian. It explores the fractional diffusion that allows defining continuous‐time random walks with a long‐range dynamics providing a general framework for anomalous diffusion and navigation in networks. The...

This chapter presents general properties of the fractional Laplacian and how the formalism is associated with long‐range correlations in networks. It discusses the non‐local character of the fractional Laplacian of a network, and describes some representations and general properties of the type (i) and type (ii) functions. The chapter explores the...

Dynamic processes that can be described in a stochastic manner by random walk strategies are ubiquitous. This chapter introduces some basic features of “Markovian” random walks on connected undirected graphs. It explains the basic properties of Markov chains. A random walk that is performed on a network is crucially determined by the network featur...

This chapter presents an introduction to several definitions in the context of the study of undirected connected networks. It describes graph theory and concepts related to the connectivity of networks, in particular, the concept of distance in networks and the average of this quantity that gives a global characterization of the network connectivit...

Description This book analyzes stochastic processes on networks and regular structures such as lattices by employing the Markovian random walk approach. Part 1 is devoted to the study of local and non-local random walks. It shows how non-local random walk strategies can be defined by functions of the Laplacian matrix that maintain the stochastici...

This book analyzes stochastic processes on networks and regular structures such as lattices by employing the Markovian random walk approach.

Correction to: H. Altenbach et al. (eds.), Generalized Models and Non-classical Approaches in Complex Materials 2, Advanced Structured Materials 90, https://doi.org/10.1007/978-3-319-77504-3

Reliable monitoring of solution crystallization processes is important to provide further insight into process dynamics and to improve process control in the regimen of Process Analytical Technology (PAT), e.g. as the case studied here: detection of crystallization of the anhydrous and monohydrate forms of Citric Acid (CA). To set up the relationsh...

Acute myeloid leukemia (AML) and chronic myeloid leukemia (CML) are two major formsof leukemia developed from myeloid cells (MCs). To understand why AML and CML occurin children, we analyzed the causes and the mechanism of cell transformation of a MC. I. Forthe MCs in marrow cavity, repeated bone-remodeling during bone-growth may be a source ofcell...

Acute lymphoblastic leukemia (ALL) and chronic lymphocytic leukemia (CLL) are two major forms of leukemia that arise from lymphoid cells (LCs). ALL occurs mostly in children and CLL occurs mainly in old people. However, the Philadelphia-chromosome-positive ALL (Ph+-ALL) and the Ph-like ALL occur in both children and adults. To understand childhood...

Lymphomas are a large group of neoplasms developed from lymphoid cells (LCs) in lymph nodes (LNs) or lymphoid tissues (LTs). Some forms of lymphomas, including Burkitt lymphoma (BL), ALK+ anaplastic large cell lymphoma (ALK+-ALCL), and T-cell lymphoblastic lymphoma/leukemia (T-LBL), occur mainly in children and teenagers. Hodgkin's lymphoma (HL) ha...

Lymphoid leukemia (LL) and lymphoma are neoplasms developed from lymphoid cells (LCs). To understand why different forms of LL/lymphoma occur at different ages, we analyzed the effects of different types of DNA changes on a LC and the cellular characteristics of LCs. Point DNA mutations (PDMs) and chromosome changes (CCs) are the two major types of...

Lymphoid leukemia (LL) and lymphoma are blood cancers developed from lymphoid cells (LCs). To understand the cause and the mechanism of cell transformation of a LC, we studied the potential sources of cell injuries of LCs and analyzed how DNA changes are generated and accumulate in LCs. I. The DNA changes that contribute to cell transformation of a...

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamics determined by functions of...

The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search strategies a key issue are first passage quantities: How long does it take a walker starting from a site p 0 to reach ‘b...

The algorithm is developed to model two-dimensional dynamic processes in a nonlocal square lattice on the basis of the shift operators. The governing discrete equations are obtained for local and nonlocal models. Their dispersion analysis reveals important differences in the dispersion curve and in the sign of the group velocity caused by nonlocali...

This book is the 2nd special volume dedicated to the memory of Gérard Maugin. Over 30 leading scientists present their contribution to reflect the vast field of scientific activity of Gérard Maugin. The topics of contributions employing often non-standard methods (generalized model) in this volume show the wide range of subjects that were covered b...