scholarly journals Asymptotic analysis of the SIR model. Applications to COVID-19 modelling

2021 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Real Line ◽  
Sir Model ◽  
Parametric Estimation ◽  
European Countries ◽  
Numerical Inversion ◽  
Parametric Solution ◽  
Preliminary Step ◽  
Entire Real Line

Abstract The SIR (Susceptible-Infected-Removed) model can be very useful in modelling epidemic outbreaks. The present paper derives the parametric solution of the model in terms of quadratures. The paper demonstrates a simple analytical asymptotic solution for the I-variable, which is valid on the entire real line. Moreover, the solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the parametric solution. The approach is applied to the ongoing coronavirus disease 2019 (COVID-19) pandemic in three European countries --Belgium, Italy and Sweden.

Validity of the Finite Fracture Mechanics Based Asymptotic Analysis for Predictions of Crack Deflection in Thin Layers of Ceramic Laminates

2014 ◽  
Vol 627 ◽  
pp. 237-240 ◽  
Author(s):  
Oldřich Ševeček ◽  
Dominique Leguillon ◽  
Tomáš Profant ◽  
Michal Kotoul
Keyword(s):  
Potential Energy ◽  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Crack Extension ◽  
Crack Deflection ◽  
Thin Layers ◽  
Reference Solution ◽  
Finite Fracture Mechanics ◽  
Higher Order Terms ◽  
Good Agreement

The work studies and compares different approaches suitable for predictions of the crack deflection (bifurcation) in ceramic laminates containing thin layers under high residual stresses and discuss a suitability and limits of using of the asymptotic analysis for such problems. The thickness of the thin compressive layers where the crack deflection occurs is only one order higher than the crack extension lengths considered within the solution. A purely FEM based calculation of the energy and stress conditions, necessary for the crack propagation, serves as the reference solution to the problem. The asymptotic analysis is used after for calculations of the same quantities (especially of energy release rate – ERR). This concept enables semi-analytical calculations of ERR or changes in potential energy induced by the crack extensions of different lengths and directions. Such approach can save a large amount of simulations and time compared with the pure FEM based calculations. It was found that the asymptotic analysis provides a good agreement for investigations of the crack increments enough far from the adjacent interfaces but for longer extensions (of length above 1/5-1/10 of the distance from the interface) starts more significantly to deviate from the correct solution. Involvement of the higher order terms in the asymptotic solution or other improvement of the model is thus advisable.

scholarly journals Effects of small boundary perturbation on the MHD duct flow

2017 ◽  
Vol 44 (1) ◽  
pp. 83-101 ◽  
Author(s):  
Ulavathi Mahabaleshwar ◽  
Igor Pazanin ◽  
Marko Radulovic ◽  
Francisco Suárez-Grau
Keyword(s):  
Magnetic Field ◽  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Cross Section ◽  
Transverse Magnetic ◽  
Boundary Perturbation ◽  
Duct Flow ◽  
Rectangular Duct ◽  
Induced Magnetic Field ◽  
Explicit Formulae

In this paper, we investigate the effects of small boundary perturbation on the laminar motion of a conducting fluid in a rectangular duct under applied transverse magnetic field. A small boundary perturbation of magnitude ? is applied on cross-section of the duct. Using the asymptotic analysis with respect to ?, we derive the effective model given by the explicit formulae for the velocity and induced magnetic field. Numerical results are provided confirming that the considered perturbation has nonlocal impact on the asymptotic solution.

Asymptotic analysis of the corner-behaviour of a rigid punch bonded to an elastic substrate

2007 ◽  
Vol 42 (5) ◽  
pp. 415-422
Author(s):  
L Bohórquez ◽  
D. A Hills
Keyword(s):  
Stress Field ◽  
Plastic Zone ◽  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Half Plane ◽  
Poisson's Ratio ◽  
Rigid Block ◽  
Rigid Punch ◽  
Oscillatory Behaviour ◽  
Elastic Substrate

The contact between a flat-faced rigid block and an elastic half-plane has been studied, showing that an asymptotic solution correctly captures the stress field adjacent to the contact corners for all values of Poisson's ratio. It is shown that, in practical cases, the plastic zone, which is inevitably present at the contact corners, envelopes the oscillatory behaviour implied locally but is surrounded by an elastic hinterland correctly represented by the asymptote.

scholarly journals On the Distribution of the Nearly Unstable AR(1) Process with Heavy Tails

2010 ◽  
Vol 42 (01) ◽  
pp. 106-136 ◽  
Author(s):  
Mariana Olvera-Cravioto
Keyword(s):  
Real Line ◽  
Uniform Approximation ◽  
Unit Root ◽  
Heavy Tails ◽  
Gaussian Approximation ◽  
Heavy Tail ◽  
Tail Asymptotics ◽  
Precise Description ◽  
Entire Real Line ◽  
Near Unit Root

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

scholarly journals On the control of a truncated general immigration process through the introduction of a predator

2006 ◽  
Vol 2006 ◽  
pp. 1-12
Author(s):  
E. G. Kyriakidis
Keyword(s):  
Real Line ◽  
Optimal Policy ◽  
Average Cost ◽  
Control Limit ◽  
Decision Algorithm ◽  
Mild Conditions ◽  
Entire Real Line ◽  
Immigration Process ◽  
Markov Decision

This paper is concerned with the problem of controlling a truncated general immigration process, which represents a population of harmful individuals, by the introduction of a predator. If the parameters of the model satisfy some mild conditions, the existence of a control-limit policy that is average-cost optimal is proved. The proof is based on the uniformization technique and on the variation of a fictitious parameter over the entire real line. Furthermore, an efficient Markov decision algorithm is developed that generates a sequence of improving control-limit policies converging to the optimal policy.

scholarly journals Modeling the role of clusters and diffusion in the evolution of COVID-19 infections during lock-down

2020 ◽  
Author(s):  
W. J. T. Bos ◽  
J.-P. Bertoglio ◽  
L. Gostiaux
Keyword(s):  
Compartmental Model ◽  
Reproduction Number ◽  
Sir Model ◽  
Compartmental Models ◽  
European Countries ◽  
Epidemic Spreading ◽  
Decay Exponent ◽  
Local Reproduction ◽  
And Diffusion

Epidemics such as the spreading of the SARS-CoV-2 virus are highly non linear, and therefore difficult to predict. In the present pandemic as time evolves, it appears more and more clearly that a clustered dynamics is a key element of description. This means that the disease rapidly evolves within spatially localized networks, that diffuse and eventually create new clusters. We improve upon the simplest possible compartmental model, the SIR model, by adding an additional compartment associated with the clustered individuals. This sophistication is compatible with more advanced compartmental models and allows, at the lowest level of complexity, to leverage the well-mixedness assumption. The so-obtained SBIR model takes into account the effect of inhomogeneity on epidemic spreading, and compares satisfactorily with results on the pandemic propagation in a number of European countries, during and immediately after lock-down. Especially, the decay exponent of the number of new cases after the first peak of the epidemic is captured without the need to vary the coefficients of the model with time. We show that this decay exponent is directly determined by the diffusion of the ensemble of clustered individuals and can be related to a global reproduction number, that overrides the classical, local reproduction number.

Waves created against a vertical cliff – a uniform asymptotic solution

1978 ◽  
Vol 83 (2) ◽  
pp. 321-328 ◽  
Author(s):  
A. K. Pramanik
Keyword(s):  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Surface Pressure ◽  
Initial Value Problem ◽  
Unsteady State ◽  
Initial Value

SummaryThe initial-value problem of waves generated by a moving oscillatory surface pressure against a vertical cliff is solved and a uniform asymptotic analysis of the unsteady state is given. The same problem with no cliff is solved by Kaplan, and by Debnath and Rosenblat but their solutions are not uniform.

Crack Propagation in a Brittle Elastic Material With Defects

1999 ◽  
Vol 66 (1) ◽  
pp. 79-86 ◽  
Author(s):  
M. Valentini ◽  
S. K. Serkov ◽  
D. Bigoni ◽  
A. B. Movchan
Keyword(s):  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Isotropic Medium ◽  
Numerical Solutions ◽  
Porous Ceramics ◽  
Qualitative Description ◽  
Mode I ◽  
Pure Mode ◽  
Two Dimensional

A two-dimensional asymptotic solution is presented for determination of the trajectory of a crack propagating in a brittle-elastic, isotropic medium containing small defects. Brittleness of the material is characterized by the assumption of the pure Mode I propagation criterion. The defects are described by Po´lya-Szego¨ matrices, and examples for small elliptical cavities and circular inclusions are given. The results of the asymptotic analysis, which agree well with existing numerical solutions, give qualitative description of crack trajectories observed in brittle materials with defects, such as porous ceramics.

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