Diffusion and home ranges

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Diffusion and home ranges in mice
movement  Guillermo Abramson   Statistical
Physics Group, Centro Atómico Bariloche and
CONICET Bariloche, Argentina.
with L. Giuggioli and V.M. Kenkre
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Oh, my God, Kenkre has told everything!
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OUTLINE

• The basic model
• Implications of the bifurcation
• Lack of vertical transmission
• Temporal behavior
• Traveling waves
• The diffusion paradigm
• Analysis of actual mice transport
• Model of mice transport

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OUTLINE

• Presentation of the system
• Basic models
• Mean field and spatially extended
• The diffusion paradigm
• Analysis of actual mice transport
• Model of mice transport

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Deer mouse Peromyscus maniculatus Reservoir of
Sin Nombre Virus (N.A. Southwest)
Ratón colilargo chico Oligoryzomys
longicaudatus Reservoir of Andes Virus (Argentina
- Chile)
Pictures courtesy of CDC Centers for Disease
Control and Prevention (Atlanta)
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HANTA AND THE DEER MOUSE

• Most numerous mammal in North America
• Main host and reservoir of Sin Nombre Virus (SNV)
• Coevolved with SNV for millions of years
• SNV can produce a severe pulmonary syndrome (HPS)
  
• Very high mortality 50
• First outbreak in the Four Corners region in
  1993
• Strong influence by environmental conditions
  (already known to the Navajo people!)

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Transmission of Hantaviruses
Chronically infected rodent
Horizontal transmission of infection by
intraspecific aggressive behavior
Virus also present in throat swab and feces
Virus is present in aerosolized excreta,
particularly urine
Secondary aerosols, mucous membrane contact, and
skin breaches are also a consideration
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The El Niño connection
Sea surface temperature, precipitation, and Hanta
cases
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THREE FIELD OBSERVATIONS AND A SIMPLE MODEL
 

• Strong influence by environmental conditions.
• Sporadical dissapearance of the infection from a
  population.
• Spatial segregation of infected populations
  (refugia).

Single control parameter in the model simulate
environmental effects. The other two appear as
consequences of a bifurcation of the solutions.
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BASIC MODEL (no mice movement yet!)
MS (t) Susceptible mice MI (t) Infected
mice M(t) MS (t)MI (t) Total mouse population
Rationale behind each term Births bM ? only of
susceptibles, all mice contribute to it Deaths
-cMS,I ? infection does not affect death
rate Competition -MS,I M/K ? population limited
by environmental parameter Contagion ? aMS MI
? simple contact between pairs
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BIFURCATION
The carrying capacity controls a bifurcation in
the equilibrium value of the infected
population. The susceptible population is always
positive.
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The same model, with vertical transmission
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The same model, with vertical transmission
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The same model, with vertical transmission
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The same model, with vertical transmission
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The same model, with vertical transmission
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Temporal behavior
KK(t)
A realistic time dependent carrying capacity
induces the occurrence of extinctions and
outbreaks as controlled by the environment.
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Temporal behavior of real mice
Real populations of susceptible and infected deer
mice at Zuni site, NM. Nc2 is the critical
population derived from approximate fits.
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SPATIALLY EXTENDED PHENOMENA
 

• Diffusive transport
• Spatially varying K, following the diversity of
  the landscape

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SIMPLE REFUGIA
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REALISTIC REFUGIA
    An illustration from northern Patagonia. The
carrying capacity is supposed proportional to the
vegetation cover.
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TRAVELING WAVES
How does infection spread from the refugia?
The sum of the equations for MS and MI is
Fishers equation for the total population
(Fisher, 1937)
There exist solutions of this equations in the
form of a front wave traveling at a constant
speed.
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Traveling waves of the complete system
Traveling waves ansatz (different speeds?)
Equations for the traveling fronts
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Two interesting scenarios
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DELAYED FRONTS

• Initial system empty of mice
• K Kc
• In contact with a refugium to the left
• MS MS 0 and MI MI 0

? Always an unstable equilibrium, but a
biological possibility
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Traveling waves of the complete system
Allowed speeds
Depends on K and a
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Two regimes of propagation
The delay D is also controlled by the carrying
capacity
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Piecewise linearization of the equations ?
Analytical expression for the front shapes ?
Analytical expression for D
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The delay depends critically on the carrying
capacity
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THE DIFFUSION PARADIGM
(Fisher, 1937)
diffusion
nonlinear reaction (logistic growth)
Epidemics of Hantavirus in P. maniculatus Abramson
, Kenkre, Parmenter, Yates (2001-2002)
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Three categories of wrongfulness
Okubo Levin, Diffusion and Ecological Problems
Wrong but useful the simplest diffusion models
cannot possibly be exactly right for any organism
in the real world (because of behavior,
environment, etc). But they provide a
standardized framework for estimating one of
ecology most neglected parameters the diffusion
coefficient.
Not necessarily so wrong diffusion models are
approximations of much more complicated
mechanisms, the net displacements being often
described by Gaussians.
Woefully wrong for animals interacting socially,
or navigating according to some external cue, or
moving towards a particular place.
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THE SOURCE OF THE DATA
Gerardo Suzán Erika Marcé, UNM Six months of
field work in Panamá (2003)
Zygodontomys brevicauda Host of Hantavirus
Calabazo
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THE SOURCE OF THE DATA
Terry Yates, Bob Parmenter, Jerry Dragoo and many
others, UNM
Ten years of field work in New Mexico (1994-)
Peromyscus maniculatus Host of Hantavirus Sin
Nombre
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Recapture and age
Zygodontomys brevicauda, 846 captures 411 total
mice, 188 captured more than once (2-10 times)
Recapture probability
J juvenile SA sub-adult A adult F female M
male
One mouse (SA, F) recaptured off-site, 200 m away
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Recapture and age
Zygodontomys brevicauda, 846 captures 411 total
mice, 188 captured more than once (2-10 times) P.
maniculatus 3826 captures 1589 total mice, 849
captured more than once (2-20 times)
Recapture probability
J juvenile SA sub-adult A adult
One mouse (SA, F) recaptured off-site, 200 m away
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Recapture and weight
Distribution of weight ?
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Different types of movement
Adult mice ? diffusion within a home range
Sub-adult mice ? run away to establish a
home range
Juvenile mice ? excursions from nest
Males and females
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The recaptures
Z. brevicauda
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MOUSE WALKS
Julian date 2450xxx (Sept. 97 to May 98)
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Mean square displacement?
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An ensemble of displacements
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An ensemble of displacements
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An ensemble of displacements
representing the walk of an ideal mouse
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PDF of individual displacements
The population as an ensemble of walkers
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PDF of individual displacements
As three ensembles, at three time scales
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Distribution of displacements
P. maniculatus
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Ideal mouse walks
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Mean square displacement
Z. brevicauda (Panama)
P. maniculatus (New Mexico)
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Confinements to diffusive motion

• Home ranges

• Capture grid

• Combination of both

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A harmonic model for home ranges
MSD
FPE
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Mean square displacement
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Time dependent MSD
saturation
box potential, concentric with the window
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Mean square displacement
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Saturation of the MSD
There is a dependence on the potential, that
results in slightly different saturation curves.
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Saturation of the MSD
Application of the use of the saturation curves
to calculate the home range size of P.
maniculatus (NM average)
from measurements
resulting value
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Periodic arrangement of home ranges
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Periodic arrangement of home ranges
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The polygon to measure home ranges
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• SUMMARY
• Mouse transport is more complex than diffusion
• Different subpopulations with different
  mechanisms
• Existence of home ranges
• Existence of transient mice
• Limited data sets can be used to derive some
  statistically sensible parameters
• Diffusion coefficient
• Home range size
• Overlap, or exclusivity, of home ranges
• Possibility of analytical models

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• SUMMARY
• Simple model of infection in the mouse population
• Important effects controlled by the environment
• Extinction and spatial segregation of the
  infected population
• Propagation of infection fronts
• Delay of the infection with respect to the
  suceptibles
• Mouse transport is more complex than diffusion
• Different subpopulations with different
  mechanisms
• Existence of home ranges
• Existence of transient mice
• Limited data sets can be used to derive some
  statistically sensible parameters D, L, a
• Possibility of analytical models

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Thank you!
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TRAVELING WAVES
How does infection spread from the refugia?
The sum of the equations for MS and MI is
Fishers equation for the total population
(Fisher, 1937)
There exist solutions of this equations in the
form of a front wave traveling at a constant
speed.
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Traveling waves of the complete system
Allowed speeds
Depends on K and a
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Two regimes of propagation
The delay D is also controlled by the carrying
capacity
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Ideal mouse walks
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Zygodontomys brevicauda 846 captures 411 total
mice, 188 captured more than once (2-10 times)
Captured once
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Recaptured
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1. Spatio-temporal patterns in the Hantavirus
infection, by G. Abramson and V. M. Kenkre, Phys.
Rev. E 66, 011912 (2002). 2. Simulations in the
mathematical modeling of the spread of the
Hantavirus, by M. A. Aguirre, G. Abramson, A. R.
Bishop and V. M. Kenkre, Phys. Rev. E 66, 041908
(2002). 3. Traveling waves of infection in the
Hantavirus epidemics, by G. Abramson, V. M.
Kenkre, T. Yates and B. Parmenter, Bulletin of
Mathematical Biology 65, 519 (2003). 4. The
criticality of the Hantavirus infected phase at
Zuni, G. Abramson (preprint, 2004). 5. The
effect of biodiversity on the Hantavirus
epizootic, I. D. Peixoto and G. Abramson
(preprint, 2004). 6. Diffusion and home range
parameters from rodent population measurements in
Panama, L. Giuggioli, G. Abramson, V.M. Kenkre,
G. Suzán, E. Marcé and T. L. Yates, Bull. of
Math. Biol (accepted, 2005). 7. Diffusion and
home range parameters for rodents II. Peromyscus
maniculatus in New Mexico, G. Abramson, L.
Giuggioli, V.M. Kenkre, J.W. Dragoo, R.R.
Parmenter, C.A. Parmenter and T.L. Yates
(preprint, 2005). 8. Theory of home range
estimation from mark-recapture measurements of
animal populations, L. Giuggioli, G. Abramson and
V.M. Kenkre (preprint, 2005).