TY - GEN A1 - Bestehorn, Michael A1 - Bestehorn, Michael A1 - Michelitsch, Thomas M. A1 - Collet, Bernard A. A1 - Riascos, Alejandro P. A1 - Nowakowski, Andrzej. F. T1 - Simple model of epidemic dynamics with memory effects T2 - Physical Review E N2 - 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 lifetime or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and not captured by the classical standard SIR model. KW - Bifurcations KW - Diseases KW - Dynamics of networks KW - Patterns in complex systems KW - Nonlinear Dynamics KW - Interdisciplinary Physics KW - Biological Physics Y1 - 2022 UR - https://journals.aps.org/pre/abstract/10.1103/PhysRevE.105.024205 U6 - https://doi.org/10.1103/PhysRevE.105.024205 SN - 2470-0045 VL - 105 IS - 2 SP - 024205-1 EP - 024205-10 ER - TY - JOUR A1 - Bestehorn, Michael A1 - Fantz, M. A1 - Friedrich, R. A1 - Haken, H. A1 - Pérez-García, C. T1 - Spiral Patterns in Thermal Convection KW - 2D Order Parameter Models Y1 - 1992 ER - TY - JOUR A1 - Bestehorn, Michael A1 - Fantz, M. A1 - Friedrich, R. A1 - Haken, H. T1 - Pattern formation in rotating Bénard convection KW - 2D Order Parameter Models Y1 - 1992 ER - TY - CHAP A1 - Vaerenbergh, S. van A1 - Legros, J. C. A1 - Colinet, P. A1 - Velarde, M. G. A1 - Bestehorn, Michael A1 - Lebon, G. A1 - Thess, A. A1 - Karcher, C. T1 - Processes With Double Diffusive Instabilities Studied in Microgravity T2 - The 2nd European Symposium on Utilisation of the International Space Station, 16 - 18 November 1998, ESTEC, Noordwijk, The Netherlands Y1 - 1999 SN - 92-9092-732-1 SP - 241 EP - 248 PB - ESA Publ. CY - Noordwijk ER - TY - JOUR A1 - Bestehorn, Michael A1 - Fantz, M. A1 - Friedrich, R. A1 - Haken, H. T1 - Hexagonal and Spiral Patterns of Thermal Convection KW - 2D Order Parameter Models Y1 - 1993 ER - TY - JOUR A1 - Bestehorn, Michael A1 - Neufeld, M. A1 - Friedrich, R. A1 - Haken, H. T1 - Pattern formation in rotating Bénard convection KW - 2D Order Parameter Models Y1 - 1994 ER - TY - JOUR A1 - Thiele, Uwe A1 - Velarde, M. G. A1 - Neuffer, Kai A1 - Bestehorn, Michael A1 - Pomeau, Y. T1 - Sliding drops in the diffusive interface model coupled to hydrodynamics JF - Physical Review E Y1 - 2001 SN - 1550-2376 VL - 64 IS - 4 SP - 061601 ER - TY - JOUR A1 - Thiele, Uwe A1 - Neuffer, Kai A1 - Bestehorn, Michael A1 - Pomeau, Y. A1 - Velarde, M. G. T1 - Sliding drops on an inclined plane JF - Colloids and Surfaces A Y1 - 2002 UR - 0927-7757 VL - 206 IS - 1-3 SP - 87 EP - 104 ER - TY - JOUR A1 - Millán-Rodriguez, J. A1 - Pérez-García, C. A1 - Bestehorn, Michael A1 - Fantz, M. A1 - Friedrich, R. T1 - Pattern Formation in Convection of Rotating Fluids with Broken Vertical Symmetry Y1 - 1992 ER - TY - JOUR A1 - Millán-Rodriguez, J. A1 - Bestehorn, Michael A1 - Pérez-García, C. A1 - Friedrich, R. A1 - Neufeld, M. T1 - Defects Motion in Rotating Fluids Y1 - 1995 ER - TY - JOUR A1 - Millán-Rodriguez, J. A1 - Bestehorn, Michael A1 - Pérez-García, C. A1 - Neufeld, M. A1 - Friedrich, R. T1 - Motion of Defects in Rotating Fluids Y1 - 1994 ER - TY - JOUR A1 - Bestehorn, Michael A1 - Neufeld, M. A1 - Friedrich, R. A1 - Haken, H. T1 - Comment on Spiral pattern formation in Rayleigh-Bénard Convection Y1 - 1994 ER - TY - GEN A1 - Bestehorn, Michael A1 - Tyvand, Peder A. A1 - Michelitsch, Thomas M. T1 - Dimension-Reduced Model for Deep-Water Waves T2 - Journal of Applied Mathematics and Physics N2 - Starting from the 2D Euler equations for an incompressible potential flow, a dimension-reduced model describing deep-water surface waves is derived. Similar to the Shallow-Water case, the z-dependence of the dependent variables is found explicitly from the Laplace equation and a set of two one- dimensional equations in x for the surface velocity and the surface elevation remains. The model is nonlocal and can be formulated in conservative form, describing waves over an infinitely deep layer. Finally, numerical solutions are presented for several initial conditions. The side-band instability of Stokes waves and stable envelope solitons are obtained in agreement with other work. The conservation of the total energy is checked. KW - Hydrodynamics KW - Ocean Waves KW - DeepWater Waves KW - Numerical Solutions KW - Fractal Derivatives Y1 - 2019 UR - https://www.scirp.org/journal/paperabs.aspx?paperid=89888 U6 - https://doi.org/10.4236/jamp.2019.71007 SN - 2327-4379 SN - 2327-4352 VL - 7 IS - 1 SP - 72 EP - 92 ER - TY - GEN A1 - Granger, Téo A1 - Michelitsch, Thomas M. A1 - Bestehorn, Michael A1 - Riascos, Alejandro P. A1 - Collet, Bernard A. T1 - Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks T2 - Entropy N2 - 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 the Barabási–Albert (BA), Erdös–Rényi (ER), and Watts–Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0with and without mortality, respectively, and prove that RM1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others. KW - epidemic spreading KW - compartment model with mortality KW - memory effects KW - random walks KW - random graphs Y1 - 2024 U6 - https://doi.org/10.3390/e26050362 SN - 1099-4300 VL - 26 IS - 5 ER - TY - GEN A1 - Granger, Téo A1 - Michelitsch, Thomas M. A1 - Bestehorn, Michael A1 - Riascos, Alejandro P. A1 - Collet, Bernard A. T1 - Four-compartment epidemic model with retarded transition rates T2 - Physical Review E N2 - 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 visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time tC, tI, and tR, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S−C−I−R−S model. We derive memory evolution equations involving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S−C−I−R−S evolution equations take the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S−C−I−R−S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S−C−I−R−S–type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior. KW - Nonlinear time-delay systems KW - Techniques Brownian dynamics KW - Diffusion & random walks KW - Epidemic spreading KW - Non-Markovian processes KW - Interdisciplinary Physics KW - Biological Physics KW - Statistical Physics Y1 - 2023 UR - https://journals.aps.org/pre/abstract/10.1103/PhysRevE.107.044207 U6 - https://doi.org/10.1103/PhysRevE.107.044207 SN - 2470-0045 VL - 107 IS - 4 SP - 044207-1 EP - 044207-15 ER - TY - GEN A1 - Bestehorn, Michael A1 - Riascos, Alejandro P. A1 - Michelitsch, Thomas M. A1 - Collet, Bernard A. T1 - A Markovian random walk model of epidemic spreading T2 - Continuum Mechanics and Thermodynamics N2 - 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 the reproduction numbers. Then, we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on the same node, there is a certain probability for the susceptible walker to get infected. By implementing this hypothesis in computer simulations, we study the space-time evolution of the emerging infection patterns. Generally, random walk approaches seem to have a large potential to study epidemic spreading and to identify the pertinent parameters in epidemic dynamics. KW - Markovian random walks KW - Ergodic networks KW - Epidemic spreading Y1 - 2021 UR - https://link.springer.com/article/10.1007/s00161-021-00970-z U6 - https://doi.org/10.1007/s00161-021-00970-z SN - 1432-0959 SN - 0935-1175 VL - 33 IS - 4 SP - 1207 EP - 1221 ER - TY - GEN A1 - Bestehorn, Michael A1 - Michelitsch, Thomas M. T1 - Oscillating Behavior of a Compartmental Model with Retarded Noisy Dynamic Infection Rate T2 - International Journal of Bifurcation and Chaos N2 - 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 compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) to an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes. KW - Epidemic models KW - delay-differential equation KW - bifurcation theory KW - numerical simulation KW - sstability analysis Y1 - 2023 UR - https://www.worldscientific.com/doi/epdf/10.1142/S0218127423500566 U6 - https://doi.org/10.1142/S0218127423500566 SN - 1793-6551 VL - 33 IS - 5 ER - TY - GEN A1 - Harlander, U. A1 - Schön, F.-T. A1 - Borcia, I. D. A1 - Richter, S. A1 - Borcia, R. A1 - Bestehorn, M. T1 - Resonant water-waves in ducts with different geometries: forced KdV solutions T2 - European Journal of Mechanics - B/Fluids Y1 - 2024 U6 - https://doi.org/10.1016/j.euromechflu.2024.03.008 SN - 0997-7546 VL - 106 IS - July–August 2024 SP - 107 EP - 115 ER - TY - GEN A1 - Borcia, I. D. A1 - Bestehorn, M. A1 - Borcia, R. A1 - Schön, F.-T. A1 - Harlander, U. A1 - Richter, S. T1 - Mean flow generated by asymmetric periodic excitation in an annular channel T2 - The European Physical Journal Special Topics Y1 - 2024 U6 - https://doi.org/10.1140/epjs/s11734-024-01181-8 VL - 233 (2024) SP - 1665 EP - 1672 PB - Springer ER - TY - GEN A1 - Schön, F.-T. A1 - Borcia, I. D. A1 - Harlander, U. A1 - Borcia, R. A1 - Richter, S. A1 - Bestehorn, M. T1 - Resonant surface waves in an oscillating periodic tank with a submerged hill T2 - Journal of Fluid Mechanics Y1 - 2024 U6 - https://doi.org/10.1017/jfm.2024.885 VL - 999 (2024) SP - 1 EP - 18 ER -