Chaos in a spatial epidemic model (original) (raw)
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Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate
Journal of Statistical Mechanics: Theory and Experiment, 2008
Spatial epidemiology is the study of spatial variation in disease risk or incidence, including the spatial patterns of the population. The spread of diseases in human populations can exhibit large scale patterns, underlining the need for spatially explicit approaches. In this paper, the spatiotemporal complexity of a spatial epidemic model with nonlinear incidence rate, which includes the behavioral changes and crowding effect of the infective individuals, is investigated. Based on both theoretical analysis and computer simulations, we find out when, under the parameters which can guarantee a stable limit cycle in the non-spatial model, spiral and target waves can emerge. Moreover, two different kinds of breakup of waves are shown. Specifically, the breakup of spiral waves is from the core and the breakup of target waves is from the far-field, and both kinds of waves become irregular patterns at last. Our results reveal that the spatiotemporal chaos is induced by the breakup of waves. The results obtained confirm that diffusion can form spiral waves, target waves or spatial chaos of high population density, which enrich the findings of spatiotemporal dynamics in the epidemic model.
Formation of spatial patterns in an epidemic model with constant removal rate of the infectives
Journal of Statistical Mechanics: Theory and Experiment, 2007
This paper addresses the question of how population diffusion affects the formation of the spatial patterns in the spatial epidemic model by Turing mechanisms. In particular, we present theoretical analysis to results of the numerical simulations in two dimensions. Moreover, there is a critical value for the system within the linear regime. Below the critical value the spatial patterns are impermanent, whereas above it stationary spot and stripe patterns can coexist over time. We have observed the striking formation of spatial patterns during the evolution, but the isolated ordered spot patterns don't emerge in the space.
Spatial organization and evolution period of the epidemic model using cellular automata
Physical Review E, 2006
We investigate epidemic models with spatial structure based on the cellular automata method. The construction of the cellular automata is from the study by Weimar and Boon about the reaction-diffusion equations ͓Phys. Rev. E 49, 1749 ͑1994͔͒. Our results show that the spatial epidemic models exhibit the spontaneous formation of irregular spiral waves at large scales within the domain of chaos. Moreover, the irregular spiral waves grow stably. The system also shows a spatial period-2 structure at one dimension outside the domain of chaos. It is interesting that the spatial period-2 structure will break and transform into a spatial synchronous configuration in the domain of chaos. Our results confirm that populations embed and disperse more stably in space than they do in nonspatial counterparts.
Chaotic Behaviour of Population on a Square Lattice
2008
Coverage of occupied sites on a square lattice is allowed to evolve according to a set of rules The rules imply an attractive interaction for growth of new members, the original members 'die', and the new population' multiplies and redistributes randomly over the lattice We show that this scenario leads to a steady coverage, cycles with a finite number of points and ultimately chaos as model parameters vary. The calculated results are verified by computer simulation. An immobile situation, where migration or redistribution over the lattice is restricted is also simulated.
Extinction of epidemics in lattice models with quenched disorder
Physical Review E, 2005
The extinction of the contact process for epidemics in lattice models with quenched disorder is analysed in the limit of small density of infected sites. It is shown that the problem in such a regime can be mapped to the quantum-mechanical one characterized by the Anderson Hamiltonian for an electron in a random lattice. It is demonstrated both analytically (self-consistent meanfield) and numerically (by direct diagonalization of the Hamiltonian and by means of cellular automata simulations) that disorder enhances the contact process given the mean values of random parameters are not influenced by disorder.
Epidemic outbreaks on two-dimensional quasiperiodic lattices
Physics Letters A, 2019
We considered the Asynchronous SIR (susceptible-infected-removed) model on Penrose and Ammann-Beenker quasiperiodic lattices, and obtained its critical behavior by using Newman-Ziff algorithm to track cluster propagation by making a tree structure of clusters grown at the dynamics, allowing to simulate SIR model on non-periodic lattices and measure any observable related to percolation. We numerically calculated the order parameter, defined in a geographical fashion by distinguish between an epidemic state, characterized by a spanning cluster formed by the removed nodes and the endemic state, where there is no spanning cluster. We obtained the averaged mean cluster size which plays the role of a susceptibility, and a cumulant ratio defined for percolation to estimate the epidemic threshold. Our numerical results suggest that the system falls into twodimensional dynamic percolation universality class and the quasiperiodic order is irrelevant, in according to results for classical percolation.
Phase-space transport of stochastic chaos in population dynamics of virus spread
Physical review letters, 2002
A general way to classify stochastic chaos is presented and applied to population dynamics models. A stochastic dynamical theory is used to develop an algorithmic tool to measure the transport across basin boundaries and predict the most probable regions of transport created by noise. The results of this tool are illustrated on a model of virus spread in a large population, where transport regions reveal how noise completes the necessary manifold intersections for the creation of emerging stochastic chaos.
Relaxation dynamics of SIR-flocks with random epidemic states
Communications on Pure and Applied Analysis
We study the collective dynamics of a multi-particle system with three epidemic states as an internal state. For the collective modeling of active particle system, we adopt modeling spirits from the swarmalator model and the SIR epidemic model for the temporal evolution of particles' position and internal states. Under suitable assumptions on system parameters and non-collision property of initial spatial configuration, we show that the proposed model does not admit finite-time collisions so that the standard Cauchy-Lipschitz theory can be applied for the global well-posedness. For the relaxation dynamics, we provide several sufficient frameworks leading to the relaxation dynamics of the proposed model. The proposed sufficient frameworks are formulated in terms of system parameters and initial configuration. Under such sufficient frameworks, we show that the state configuration relaxes to the fixed constant configuration via the exponentially perturbed gradient system and explic...
Differential and Integral Equations
Equations with non-local dispersal have been used extensively as models in material science, ecology and neurology. We consider the scalar model ∂u ∂t(x,t)=ρ∫ Ω β (x,y) u (y,t) d y - u (x,t)+f(u(x,t)), where the integral term represents a general form of spatial dispersal and u(x,t) is the density at x∈Ω, the spatial region, and time t of the quantity undergoing dispersal. We discuss the asymptotic dynamics in the bistable case and contrast these with those for the corresponding reaction-diffusion model. First, we note that it is easy to show for large ρ that the behavior is similar to that of the reaction-diffusion system; in the case of the analogue of zero Neumann conditions, the dynamics are governed by the ODE u ˙=f(u). However, for small ρ, it is known that this is not the case, the set of equilibria being uncountably infinite and not compact in L p (1≤p≤∞). Our principal aim in this paper is to inquire whether every orbit converges to an equilibrium, regardless of the size of...
Population dynamics in a random environment
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