scholarly journals Forecast for the second Covid-19 wave based on the improved SIR model with a constant ratio of recovery to infection rate

2021 ◽  
Author(s):  
Martin Kröger ◽  
Reinhard Schlickeiser
Keyword(s):  
Initial Conditions ◽  
Dynamical Evolution ◽  
Sir Model ◽  
Constant Ratio ◽  
Exponential Tails ◽  
Analytic Expressions ◽  
Early Peak ◽  
Moderate Change ◽  
Final Amount ◽  
Epidemic Wave

We start out by deriving simple analytic expressions for all measurable amounts of cases and fatalities during a pandemic evolution exhibiting multiple waves, described by the semi-time SIR model. The approximant shares all relevant features with the exact solution, including time and position of the peak of daily new infections, as well as the asymptotic behaviors at small and large times. We derive exact analytic expressions for the early doubling time, late half decay time, and a half-early peak law, characterizing the dynamical evolution. We show, in particular, how the asymmetry of the first epidemic wave and its exponential tails are affected by the initial conditions; a feature that has no analogue in the all-time SIR model. We apply the approach to available data from different continents. Our analysis reveals that the immunity is very strongly increasing during the 2nd wave, while it was still at a very moderate level of a few percent in several countries at the end of the first wave. The wave-specific SIR parameters describing the infection and recovery rates we find to behave in a similar fashion, while their ratio k was decreasing only by a about 5% for most countries. Still, an apparently moderate change of k can have significant consequences for the relevant numbers like the final amount of infected or deceased population. As we show, the probability for an additional wave is however low in several countries due to the fraction of immune inhabitants at the end of the 2nd wave, irrespective the currently ongoing vaccination efforts. We compare with alternate approaches.

scholarly journals Verification of the accuracy of the SIR model in forecasting based on the improved SIR model with a constant ratio of recovery to infection rate by comparing with monitored second wave data

2021 ◽  
Vol 8 (9) ◽  
pp. 211379
Author(s):  
M. Kröger ◽  
R. Schlickeiser
Keyword(s):  
Temporal Evolution ◽  
Initial Conditions ◽  
Dynamical Evolution ◽  
Sir Model ◽  
Peak Time ◽  
Present Contribution ◽  
Analytic Expressions ◽  
Early Peak ◽  
Moderate Change ◽  
Second Wave

The temporal evolution of second and subsequent waves of epidemics such as Covid-19 is investigated. Analytic expressions for the peak time and asymptotic behaviours, early doubling time, late half decay time, and a half-early peak law, characterizing the dynamical evolution of number of cases and fatalities, are derived, where the pandemic evolution exhibiting multiple waves is described by the semi-time SIR model. The asymmetry of the epidemic wave and its exponential tail are affected by the initial conditions, a feature that has no analogue in the all-time SIR model. Our analysis reveals that the immunity is very strongly increasing in several countries during the second Covid-19 wave. Wave-specific SIR parameters describing infection and recovery rates we find to behave in a similar fashion. Still, an apparently moderate change of their ratio can have significant consequences. As we show, the probability of an additional wave is however low in several countries due to the fraction of immune inhabitants at the end of the second wave, irrespective of the ongoing vaccination efforts. We compare with alternate approaches and data available at the time of submission. Most recent data serves to demonstrate the successful forecast and high accuracy of the SIR model in predicting the evolution of pandemic outbreaks as long as the assumption underlying our analysis, an unchanged situation of the distribution of variants of concern and the fatality fraction, do not change dramatically during a wave. With the rise of the α variant at the time of submission the second wave did not terminate in some countries, giving rise to a superposition of waves that is not treated by the present contribution.

scholarly journals Mass determination of the 1:3:5 near-resonant planets transiting GJ 9827 (K2-135)

2018 ◽  
Vol 618 ◽  
pp. A116 ◽  
Author(s):  
J. Prieto-Arranz ◽  
E. Palle ◽  
D. Gandolfi ◽  
O. Barragán ◽  
E. W. Guenther ◽  
...  
Keyword(s):  
Planetary System ◽  
Initial Conditions ◽  
Dynamical Evolution ◽  
Space Mission ◽  
Mass Determination ◽  
Velocity Measurements ◽  
Orbital Periods ◽  
The Masses ◽  
Better Than

Context. Multiplanet systems are excellent laboratories to test planet formation models as all planets are formed under the same initial conditions. In this context, systems transiting bright stars can play a key role, since planetary masses, radii, and bulk densities can be measured. Aims. GJ 9827 (K2-135) has recently been found to host a tightly packed system consisting of three transiting small planets whose orbital periods of 1.2, 3.6, and 6.2 days are near the 1:3:5 ratio. GJ 9827 hosts the nearest planetary system (~30 pc) detected by NASA’s Kepler or K2 space mission. Its brightness (V = 10.35 mag) makes the star an ideal target for detailed studies of the properties of its planets. Methods. Combining the K2 photometry with high-precision radial-velocity measurements gathered with the FIES, HARPS, and HARPS-N spectrographs we revised the system parameters and derive the masses of the three planets. Results. We find that GJ 9827 b has a mass of Mb = 3.69−0.46+0.48 M⊕ and a radius of Rb = 1.58−0.13+0.14 R⊕, yielding a mean density of ρb = 5.11−1.27+1.74 g cm−3. GJ 9827 c has a mass of Mc = 1.45−0.57+0.58 M⊕, radius of Rc = 1.24−0.11+0.11 R⊕, and a mean density of ρc = 4.13−1.77+2.31 g cm−3. For GJ 9827 d, we derive Md = 1.45−0.57+0.58 M⊕, Rd = 1.24−0.11+0.11 R⊕, and ρd = 1.51−0.53+0.71 g cm−3. Conclusions. GJ 9827 is one of the few known transiting planetary systems for which the masses of all planets have been determined with a precision better than 30%. This system is particularly interesting because all three planets are close to the limit between super-Earths and sub-Neptunes. The planetary bulk compositions are compatible with a scenario where all three planets formed with similar core and atmosphere compositions, and we speculate that while GJ 9827 b and GJ 9827 c lost their atmospheric envelopes, GJ 9827 d maintained its primordial atmosphere, owing to the much lower stellarirradiation. This makes GJ 9827 one of the very few systems where the dynamical evolution and the atmosphericescape can be studied in detail for all planets, helping us to understand how compact systems form and evolve.

scholarly journals The Dynamical Evolution of Young Clusters and Associations

1986 ◽  
Vol 7 ◽  
pp. 481-488 ◽  
Author(s):  
Robert D. Mathieu
Keyword(s):  
Star Formation ◽  
Formation Process ◽  
Initial Conditions ◽  
Dynamical Evolution ◽  
Short Review ◽  
Stellar Systems ◽  
Galactic Field ◽  
Dynamical State ◽  
Orbital Mixing ◽  
Formation Efficiency

A young cluster or association bears the imprint of the conditions at its birth for perhaps ten million years, after which the initial conditions are lost to either dilution in the galactic field or erasure by orbital mixing and stellar encounters. In its youngest years, however, the dynamical state of the system can provide valuable information concerning the structure and energetics of the parent gas, the star-formation efficiency and the star-formation process itself. This short review discusses recent theoretical and observational progress in the study of the very youngest of stellar systems.

scholarly journals Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials

2015 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
Mohamed A. Agwa ◽  
António Pinto Da Costa
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Initial Conditions ◽  
Orthotropic Materials ◽  
Isotropic Case ◽  
Linear Elastic ◽  
Analytic Expressions ◽  
The Coefficient Of Friction ◽  
Unstable Solutions ◽  
Computer Algebra Software

Abstract The present work addresses the problem of determining under what conditions the impending slip state or the steady sliding of a linear elastic orthotropic layer or half space with respect to a rigid flat obstacle is dynamically unstable. In other words, we search the conditions for the occurrence of smooth exponentially growing dynamic solutions with perturbed initial conditions arbitrarily close to the steady sliding state, taking the system away from the equilibrium state or the steady sliding state. Previously authors have shown that a linear elastic isotropic half space compressed against and sliding with respect to a rigid flat surface may get unstable by flutter when the coefficient of friction μ and Poisson’s ratio ν are sufficiently large. In the isotropic case they have been able to derive closed form analytic expressions for the exponentially growing unstable solutions as well as for the borders of the stability regions in the space of parameters, because in the isotropic case there are only two dimensionless parameters (μ and ν). Already for the simplest version of orthotropy (an orthotropic transversally isotropic material) there are seven governing parameters (μ, five independent material constants and the orientation of the principal directions of orthotropy) and the expressions become very lengthy and literally impossible to manipulate manually. The orthotropic case addressed here is impossible to solve with simple closed form expressions, and therefore the use of computer algebra software is required, the main commands being indicated in the text.

scholarly journals Chemo-dynamical evolution of dwarf galaxies: from flat to cuspy dark matter density profiles

2009 ◽  
Vol 5 (H15) ◽  
pp. 78-78
Author(s):  
S. Pasetto ◽  
E. K. Grebel ◽  
P. Berczik ◽  
R. Spurzem
Keyword(s):  
Dark Matter ◽  
Dwarf Galaxies ◽  
Initial Conditions ◽  
Smooth Particle Hydrodynamic ◽  
Dynamical Evolution ◽  
Mass Range ◽  
Density Profiles ◽  
Hydrodynamic Approach ◽  
The Stability ◽  
Evolution With Time

A model of an isolated dwarf spherical galaxy (dSph) is considered in its chemo-dynamical evolution with time. The system is composed by 3 γ-model density profiles: gas, stellar and dark matter, and it is realized in a spherical symmetric equilibrium configuration. The total masses used in our simulations are covering the dwarf galaxies mass range. The stability of this configuration is first tested for the system evolving under the gravity effect alone and then evolved taking into account for the most relevant stellar astrophysical processes implemented with a Smooth Particle Hydrodynamic approach. The two different kinds of evolution are compared. The dark matter evolves naturally from a centrally cuspy density profile into a flatter one within a timescale of several Gyr. The effect manifests itself naturally, without any tuned initial conditions, as soon as few standard criteria on star formation are assumedand the SN feedback on the ISM has been adopted the prescription in (Cioffi & Shull 1991) and (Bradamante et al. 1998). This result is expected to be a possible natural explanation for the discrepancy between observations that want flatter dark matter profiles (e.g. de Block 2005), and N-body simulations that predict cuspy dark matter profiles (Navarro et al. 1997). Chemical considerations are presented as a tool to follow with observational parameters the theory predictions.

EXACT ANALYTICAL EXPRESSIONS FOR THE FINAL EPIDEMIC SIZE OF AN SIR MODEL ON SMALL NETWORKS

2016 ◽  
Vol 57 (4) ◽  
pp. 429-444 ◽  
Author(s):  
K. MCCULLOCH ◽  
M. G. ROBERTS ◽  
C. R. LAING
Keyword(s):  
Initial Conditions ◽  
Transmission Rate ◽  
Sir Model ◽  
Contact Network ◽  
Sir Epidemic Model ◽  
Node Number ◽  
Sir Epidemic ◽  
Epidemic Size ◽  
Final Epidemic Size ◽  
Analytical Expressions

We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of$N$nodes, there are$3^{N}$configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

ON THE DYNAMICAL EVOLUTION OF THE FRACTAL STRUCTURE OF INTERSTELLAR MEDIUM

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 908-916 ◽  
Author(s):  
Z.Y. YUE ◽  
B. ZHANG ◽  
G. WINNEWISSER ◽  
J. STUTZKI
Keyword(s):  
Fractal Dimension ◽  
Interstellar Medium ◽  
Density Distribution ◽  
Fractal Structure ◽  
Initial Conditions ◽  
Initial Density ◽  
Dynamical Evolution ◽  
Fractal Dimensions ◽  
Compressible Turbulence ◽  
Initial States

Two-dimensional compressible turbulence in a self-gravitating, magnetic interstellar medium is calculated as an initial value problem. It is shown that even if the initial density distribution is homogeneous and the initial velocity distribution contains only a few Fourier components, the nonlinear interaction among the Fourier components will generate more and more Fourier components and lead to a turbulent and fractal structure in the interstellar medium. The calculations are carried out for three different initial states. In order to see the time evolution, detailed density distributions and fractal dimensions of the density contours are calculated at three moments of time for each of the initial states. The results show that the fractal dimension remains almost the same (~1.4–1.5), although the detailed density distribution has changed considerably. The insensibility of the fractal dimension of density contours to both the initial conditions and the evolution time is in good agreement with observations of molecular clouds in the interstellar medium.

scholarly journals The collision between the Milky Way and Andromeda and the fate of their Supermassive Black Holes

2019 ◽  
Vol 14 (S351) ◽  
pp. 161-164 ◽  
Author(s):  
Riccardo Schiavi ◽  
Roberto Capuzzo-Dolcetta ◽  
Manuel Arca Sedda ◽  
Mario Spera
Keyword(s):  
Black Holes ◽  
Velocity Vector ◽  
Milky Way ◽  
Orbital Motion ◽  
Initial Conditions ◽  
Dynamical Evolution ◽  
Close Approach ◽  
Supermassive Black Holes ◽  
Andromeda Galaxy ◽  
Proper Resolution

AbstractOur Galaxy and the nearby Andromeda Galaxy (M31) form a bound system, even though the relative velocity vector of M31 is currently not well constrained. Their orbital motion is highly dependent on the initial conditions, but all the reliable scenarios imply a first close approach in the next 3–5 Gyrs. In our study, we simulate this interaction via direct N-body integration, using the HiGPUs code. Our aim is to investigate the dependence of the time of the merger on the physical and dynamical properties of the system. Finally, we study the dynamical evolution of the two Supermassive Black Holes placed in the two galactic centers, with the future aim to achieve a proper resolution to follow their motion until they form a tight binary system.

scholarly journals Modelling Individual Globular Clusters

2007 ◽  
Vol 3 (S246) ◽  
pp. 121-130
Author(s):  
Douglas C. Heggie ◽  
Mirek Giersz
Keyword(s):  
Monte Carlo ◽  
Binary Stars ◽  
Globular Clusters ◽  
Initial Conditions ◽  
Dynamical Evolution ◽  
Major Axis ◽  
Monte Carlo Code ◽  
Evolutionary Models ◽  
Semi Major Axis ◽  
Static Models

AbstractAstronomers have constructed models of globular clusters for over 100 years. These models mainly fall into two categories: (i) static models, such as King's model and its variants, and (ii) evolutionary models. Most attention has been given to static models, which are used to estimate mass-to-light ratios and mass segregation, and to combine data from proper motions and radial velocities. Evolutionary models have been developed for a few objects using the gaseous model, the Fokker-Planck model, Monte Carlo models and N-body models. These models have had a significant role in the search for massive black holes in globular clusters, for example.In this presentation the problems associated with these various techniques will be summarised, and then we shall describe new work with Giersz's Monte Carlo code, which has been enhanced recently to include the stellar evolution of single and binary stars. We describe in particular recent attempts to model the nearby globular cluster M4, including predictions on the spatial distribution of binary stars and their semi-major axis distribution, to illustrate the effects of about 12 Gyr of dynamical evolution. We also discuss work on an approximate way of predicting the “initial” conditions for such modelling.

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