Reaction-diffusion theory in the presence of an attractive harmonic potential

Spendier, Kathrin; Sugaya, Satomi; Kenkre, V. M.

doi:10.1103/physreve.88.062142

Cited by 28 publications

(29 citation statements)

References 37 publications

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“…When infection is present, we write, as a consequence of the linearity of the equations, Defect technique procedures [16][17][18][19][20][21][22][23] along the lines originated in [15], proceed by Laplace transforming Eq. (3), setting r 1 = r 2 , and integrating over r 1 in the appropriate space of s dimensions.…”

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confidence: 99%

Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and Without Confinement

Kenkre

Sugaya

2014

Bull Math Biol

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A theory of the spread of epidemics is formulated on the basis of pairwise interactions in a dilute system of random walkers (infected and susceptible animals) moving in n dimensions. The motion of an animal pair is taken to obey a Smoluchowski equation in 2n-dimensional space that combines diffusion with confinement of each animal to its particular home range. An additional (reaction) term that comes into play when the animals are in close proximity describes the process of infection. Analytic solutions are obtained, confirmed by numerical procedures, and shown to predict a surprising effect of confinement. The effect is that infection spread has a non-monotonic dependence on the diffusion constant and/or the extent of the attachment of the animals to the home ranges. Optimum values of these parameters exist for any given distance between the attractive centers. Any change from those values, involving faster/slower diffusion or shallower/steeper confinement, hinders the transmission of infection. A physical explanation is provided by the theory. Reduction to the simpler case of no home ranges is demonstrated. Effective infection rates are calculated and it is shown how to use them in complex systems consisting of dense populations. The purpose of the following is to construct an analytic theory of the transmission of infection in epidemics spread on the basis of a simple but exactly soluble model of interacting random walkers representing animals moving about on the terrain and infecting one another on encounter. Seminal contributions by Anderson and May and others [1][2][3][4], involving concepts such as mass action, SIR, and the basic reproductive rate R 0 , launched this field of research which derives its importance from human relevance as well as intellectual challenge. Spatial considerations were introduced into the investigations independently by various authors [2,[5][6][7][8][9][10][11][12] giving the studies a kinetic equation flavor. Missing from some of these studies were confinement features that arise in animal motion from home ranges and yet are clear and compelling in the light of field observations [12][13][14]. These and other issues have made it essential to undertake a fundamental study of the transmission of infection in terms of interacting random walks specially under confinement.Model and Method of Analysis-Our model starts with just two animals, one initially infected and the other initially uninfected (susceptible), respectively denoted by 1 and 2, performing random walks around respective attractive centers at R 1 and R 2 , with a diffusion constant D, there being the possibility of the uninfected individual getting infected at a rate proportional to C when the two occupy the same position. The central quantity that serves as the focus of our calculation is the joint proba- * Electronic address: kenkre@unm.edu † Electronic address: satomi@unm.edu bility density P (r 1 , r 2 , t) that the infected animal is at r 1 and the susceptible animal is at r 2 . Given this definition, P (r 1 , r...

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Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and Without Confinement

Kenkre

Sugaya

2014

Bull Math Biol

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“…The curve denoted by MM and MCB are the results by Marechal and Moreau (1984) and the simulations by Montgomery et al (1979), respectively. There are many reactive systems and diffusion processes that are modelled with the Smoluchowski equation (Szabo et al 1980;Bagchi et al 1983;Chavanis 2006;Felderhof 2008) including protein folding (Bicout and Szabo 2000), dielectric relaxation (Coffey et al 2009) and a Smoluchowski equation with a capture term (Spendier et al 2013) that overlaps in some respects the studies of the nonequilibrium reactive system in Sect. 5.4.4.…”

Section: Fokker-planck or Smoluchowski Equation For Bistable Potentialsmentioning

confidence: 99%

Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

Shizgal

2015

Scientific Computation

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“…At long times, the MSD for the algebraic case diverges when ν → −0.5 from the right and when ν → 1 from the left. The divergence at the lower limit occurs due to the long-time algebraic dependence of the Green functions, (18).…”

Section: Spatial Extent In the Steady Statementioning

confidence: 99%

“…that we have derived. We use it along with its asymptotic form, (18), which is well known in the literature [34].…”

Section: Similarities and Dissimilarities In The Consequences Of The mentioning

confidence: 99%

“…We do this because it is easy to introduce our task in that manner and also because there has been a lot of recent activity on that equation. Thus, an analysis has shown [18] how the Smoluchowski equation may be applied to trapping situations and a recent application to the spread of epidemics has been made [19] leading to an interesting description of the transmission of infection in diseases such as the Hantavirus [20].…”

Section: Introductionmentioning

confidence: 99%

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Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

et al. 2016

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Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steadystate depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times.

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Reaction-diffusion theory in the presence of an attractive harmonic potential

Cited by 28 publications

References 37 publications

Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and Without Confinement

Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and Without Confinement

Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions

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