Kinetic theory of small-amplitude fluctuations in astrophysical plasmas

Kolberg, U.; Schlickeiser, R.

doi:10.1016/j.physrep.2018.10.003

Cited by 8 publications

(10 citation statements)

References 36 publications

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“…Secondly, it is necessary to start initially at time t = 0 with at least one infected person, i.e., I(0) = f k (ε) > 0 (see the second initial condition ( 12)), as the dynamical equation ( 2) implies for the initial change İ(t = 0) = 0 if I(t = 0) = 0. Nothing can grow out of nothing; a situation very similar to kinetic plasma instabilities where seed electromagnetic fluctuations, often from spontaneous emission [27], are needed for starting the instability. Keeping exact tract of the initial conditions during the analysis will prove to be essential to avoid mathematical singularities in the analysis.…”

Section: Three Important Remarksmentioning

confidence: 99%

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Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor

Kröger¹,

Schlickeiser²

2020

J. Phys. A: Math. Theor.

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We revisit the susceptible-infectious-recovered/removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t = 0, where data is assumed to be available. We address the question of how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be relaxed in part B. Relaxing this assumption allows us to formulate initial conditions incompatible with the original SIR model.

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Section: Three Important Remarksmentioning

confidence: 99%

“…From the invariant J(t)dt = j(τ )dτ we obtain with equations ( 25), (27), and equation (17) in the form dτ/dt = a(t) for differential rate of newly infected persons (figure 3) (36). The simple approximant for j(τ ), equations ( 36) and (86), is shown in green.…”

Section: Differential and Cumulative Rates Of Newly Infected Personsmentioning

confidence: 99%

Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor

Kröger¹,

Schlickeiser²

2020

J. Phys. A: Math. Theor.

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show abstract

“…which apart from a different notation corresponds to Eqs. (23) and (27) of Harko et al 24 . The solution to the dimensional version of Eq.…”

Section: A Reduced Timementioning

confidence: 96%

“…Nothing can grow out of nothing; a situation very similar to kinetic plasma instabilities where seed electromagnetic fluctuations, often from spontaneous emission 23 , are needed for starting the instability. Keeping exact tract of the initial conditions during the analysis will prove to be essential to avoid mathematical singularities in the analysis.…”

Section: Three Important Remarksmentioning

confidence: 99%

Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

Kröger

Schlickeiser

2020

Preprint

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We revisit the Susceptible-Infectious-Recovered/Removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t=0, where data is assumed to be available. We address the question on how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be released in part B. Releasing this assumption allows to formulate initial conditions incompatible with the original SIR model.

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“…This is the so-called Klimontovich function, whose time evolution is obtained by applying the Liouville's theorem for conservation of phase-space. The resulting Klimontovich equation together with Maxwell's equations provide fundamental description of plasmas that accounts for a discrete nature of the particles [e.g., 153,334]. It allows rigorous treatment of binary collisions or spontaneously-emitted fluctuations, e.g., within the weak turbulence theory.…”

Section: Kinetic Description Of Collisionless Plasmamentioning

confidence: 99%

PIC simulation methods for cosmic radiation and plasma instabilities

Pohl

Hoshino

Niemiec

2020

Progress in Particle and Nuclear Physics

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Particle acceleration in collisionless plasma systems is a central question in astroplasma and astroparticle physics. The structure of the acceleration regions, electron-ion energy equilibration, preacceleration of particles at shocks to permit further energization by diffusive shock acceleration, require knowledge of the distribution function of particles besides the structure and dynamic of electromagnetic fields, and hence a kinetic description is desirable. Particle-in-cell simulations offer an appropriate, if computationally expensive method of essentially conducting numerical experiments that explore kinetic phenomena in collisionless plasma. We review recent results of PIC simulations of astrophysical plasma systems, particle acceleration, and the instabilities that shape them.

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Kinetic theory of small-amplitude fluctuations in astrophysical plasmas

Cited by 8 publications

References 36 publications

Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor

Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor

Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

PIC simulation methods for cosmic radiation and plasma instabilities

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