675 research outputs found
Reaction-diffusion spatial modeling of COVID-19: Greece and Andalusia as case examples
We examine the spatial modeling of the outbreak of COVID-19 in two regions:
the autonomous community of Andalusia in Spain and the mainland of Greece. We
start with a 0D compartmental epidemiological model consisting of Susceptible,
Exposed, Asymptomatic, (symptomatically) Infected, Hospitalized, Recovered, and
deceased populations. We emphasize the importance of the viral latent period
and the key role of an asymptomatic population. We optimize model parameters
for both regions by comparing predictions to the cumulative number of infected
and total number of deaths via minimizing the norm of the difference
between predictions and observed data. We consider the sensitivity of model
predictions on reasonable variations of model parameters and initial
conditions, addressing issues of parameter identifiability. We model both
pre-quarantine and post-quarantine evolution of the epidemic by a
time-dependent change of the viral transmission rates that arises in response
to containment measures. Subsequently, a spatially distributed version of the
0D model in the form of reaction-diffusion equations is developed. We consider
that, after an initial localized seeding of the infection, its spread is
governed by the diffusion (and 0D model "reactions") of the asymptomatic and
symptomatically infected populations, which decrease with the imposed
restrictive measures. We inserted the maps of the two regions, and we imported
population-density data into COMSOL, which was subsequently used to solve
numerically the model PDEs. Upon discussing how to adapt the 0D model to this
spatial setting, we show that these models bear significant potential towards
capturing both the well-mixed, 0D description and the spatial expansion of the
pandemic in the two regions. Veins of potential refinement of the model
assumptions towards future work are also explored.Comment: 28 pages, 16 figures and 2 movie
General tooth boundary conditions for equation free modelling
We are developing a framework for multiscale computation which enables models
at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte
Carlo or Molecular Dynamics simulators, to perform modelling tasks at
``macroscopic'' length scales of interest. The plan is to use the microscopic
rules restricted to small "patches" of the domain, the "teeth'', using
interpolation to bridge the "gaps". Here we explore general boundary conditions
coupling the widely separated ``teeth'' of the microscopic simulation that
achieve high order accuracy over the macroscale. We present the simplest case
when the microscopic simulator is the quintessential example of a partial
differential equation. We argue that classic high-order interpolation of the
macroscopic field provides the correct forcing in whatever boundary condition
is required by the microsimulator. Such interpolation leads to Tooth Boundary
Conditions which achieve arbitrarily high-order consistency. The high-order
consistency is demonstrated on a class of linear partial differential equations
in two ways: firstly through the eigenvalues of the scheme for selected
numerical problems; and secondly using the dynamical systems approach of
holistic discretisation on a general class of linear \textsc{pde}s. Analytic
modelling shows that, for a wide class of microscopic systems, the subgrid
fields and the effective macroscopic model are largely independent of the tooth
size and the particular tooth boundary conditions. When applied to patches of
microscopic simulations these tooth boundary conditions promise efficient
macroscale simulation. We expect the same approach will also accurately couple
patch simulations in higher spatial dimensions.Comment: 22 page
Bose-Einstein Condensates in Superlattices
We consider the Gross--Pitaevskii (GP) equation in the presence of periodic and quasi-periodic superlattices to study cigar-shaped Bose--Einstein condensates (BECs) in such potentials. We examine spatially extended wavefunctions in the form of modulated amplitude waves (MAWs). With a coherent structure ansatz, we derive amplitude equations describing the evolution of spatially modulated states of the BEC. We then apply second-order multiple scale perturbation theory to study harmonic resonances with respect to a single lattice substructure as well as ultrasubharmonic resonances that result from interactions of both substructures of the superlattice. In each case, we determine the resulting system's equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding wavefunctions by direct simulations of the GP equation, identifying them as typically stable solutions of the model. We then study subharmonic resonances using Hamiltonian perturbation theory, tracing robust spatio-temporally periodic patterns
Dynamical Superfluid-Insulator Transition in a Chain of Weakly Coupled Bose-Einstein Condensates
We predict a dynammical classical superfluid-insulator transition (CSIT) in a
Bose-Einstein condensate (BEC) trapped in an optical and a magnetic potential.
In the tight-binding limit, this system realizes an array of weakly-coupled
condensates driven by an external harmonic field. For small displacements of
the parabolic trap about the equilibrium position, the BEC center of mass
oscillates with the relative phases of neighbouring condensates locked at the
same (oscillating) value. For large displacements, the BEC remains localized on
the side of the harmonic trap. This is caused by a randomization of the
relative phases, while the coherence of each individual condensate in the array
is preserved. The CSIT is attributed to a discrete modulational instability,
occurring when the BEC center of mass velocity is larger than a critical value,
proportional to the tunneling rate between adjacent sites.Comment: 5 pages, 4 figures, to appear in Phys. Rev. Let
A discrete nonlinear model with substrate feedback
We consider a prototypical model in which a nonlinear field (continuum or
discrete) evolves on a flexible substrate which feeds back to the evolution of
the main field. We identify the underlying physics and potential applications
of such a model and examine its simplest one-dimensional Hamiltonian form,
which turns out to be a modified Frenkel-Kontorova model coupled to an extra
linear equation. We find static kink solutions and study their stability, and
then examine moving kinks (the continuum limit of the model is studied too). We
observe how the substrate effectively renormalizes properties of the kinks. In
particular, a nontrivial finding is that branches of stable and unstable kink
solutions may be extended beyond a critical point at which an effective
intersite coupling vanishes; passing this critical point does not destabilize
the kink. Kink-antikink collisions are also studied, demonstrating alternation
between merger and transmission cases.Comment: a revtex text file and 6 ps files with figures. Physical Review E, in
pres
Compactons in Nonlinear Schr\"odinger Lattices with Strong Nonlinearity Management
The existence of compactons in the discrete nonlinear Schr\"odinger equation
in the presence of fast periodic time modulations of the nonlinearity is
demonstrated. In the averaged DNLS equation the resulting effective inter-well
tunneling depends on modulation parameters {\it and} on the field amplitude.
This introduces nonlinear dispersion in the system and can lead to a
prototypical realization of single- or multi-site stable discrete compactons in
nonlinear optical waveguide and BEC arrays. These structures can dynamically
arise out of Gaussian or compactly supported initial data.Comment: 4 pages, 4 figure
Coarse Molecular-Dynamics Determination of the Onset of Structural Transitions: Melting of Crystalline Solids
Using a coarse molecular-dynamics (CMD) approach with an appropriate choice
of coarse variable (order parameter), we map the underlying effective
free-energy landscape for the melting of a crystalline solid. Implementation of
this approach provides a means for constructing effective free-energy
landscapes of structural transitions in condensed matter. The predictions of
the approach for the thermodynamic melting point of a model silicon system are
in excellent agreement with those of ''traditional'' techniques for
melting-point calculations, as well as with literature values
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