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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
Oh, my God, Kenkre has told everything!
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
                 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).


            Population dynamics
                     +
                Contagion                  Mathematical model
                     +
             (Mice movement)


    Single control parameter in the model simulate environmental effects.

    The other two appear as consequences of a bifurcation of the solutions.
BASIC MODEL         (no mice movement yet!)

dM S                MSM                   MS (t) : Susceptible mice
      bM  c M S        aM S M I ,
 dt                  K                    MI (t) : Infected mice
dM I            MIM
      c M I        aM S M I ,         M(t)= MS (t)+MI (t): Total
 dt              K                        mouse population

                    carrying capacity
 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
BIFURCATION

          b
 Kc 
      a (b  c )




 The carrying capacity controls a bifurcation in the equilibrium
 value of the infected population.
 The susceptible population is always positive.
The same model, with vertical transmission
                   dM S                 M M
                         b S M  cM S  S  aM S M I ,
                    dt                   K
                   dM I                 M M
                         b I M  cM I  I  aM S M I ,
                    dt                   K




                                 a  0.1
                        14


                        12
                                 c  0.1
                        10
                                 bS  1.0
            MS and MI



                        8
                                 bI  0.0
                        6


                        4


                        2


                        0
                             0    2   4     6   8   10   12   14   16   18   20


                                                    K Kc
The same model, with vertical transmission
                   dM S                 M M
                         b S M  cM S  S  aM S M I ,
                    dt                   K
                   dM I                 M M
                         b I M  cM I  I  aM S M I ,
                    dt                   K




                                 a  0 .1
                        14


                        12
                                 c  0.1
                        10
                                 bS  0.99
            MS and MI



                        8
                                 bI  0.01
                        6


                        4


                        2


                        0
                             0    2   4   6   8   10   12   14   16   18   20


          Kc  0                                  K
The same model, with vertical transmission
                   dM S                 M M
                         b S M  cM S  S  aM S M I ,
                    dt                   K
                   dM I                 M M
                         b I M  cM I  I  aM S M I ,
                    dt                   K




                                 a  0.1
                        14


                        12
                                 c  0.1
                        10
                                 bS  0.9
            MS and MI



                        8
                                 bI  0.1
                        6


                        4


                        2


                        0
                             0    2   4     6   8   10   12   14   16   18   20


          Kc  0                                    K
The same model, with vertical transmission
                   dM S                 M M
                         b S M  cM S  S  aM S M I ,
                    dt                   K
                   dM I                 M M
                         b I M  cM I  I  aM S M I ,
                    dt                   K




                                 a  0.1
                        14


                        12
                                 c  0.1
                        10
                                 bS  0.8
            MS and MI



                        8
                                 bI  0.2
                        6


                        4


                        2


                        0
                             0    2   4     6   8   10   12   14   16   18   20


          Kc  0                                    K
The same model, with vertical transmission
                   dM S                 M M
                         b S M  cM S  S  aM S M I ,
                    dt                   K
                   dM I                 M M
                         b I M  cM I  I  aM S M I ,
                    dt                   K




                                 a  0.1
                        14


                        12
                                 c  0.1
                        10
                                 bS  0.5
            MS and MI



                        8
                                 bI  0.5
                        6


                        4


                        2


                        0
                             0    2   4     6   8   10   12   14   16   18   20


          Kc  0                                    K
                  Temporal behavior
K=K(t)




A “realistic” time dependent carrying capacity induces the
occurrence of extinctions and outbreaks as controlled by the
environment.
           Temporal behavior of real mice




                                                   critical
                                                   population
                                                   Nc=Kc(b-c)




Real populations of susceptible and infected deer mice at
Zuni site, NM. Nc=2 is the “critical population” derived from
approximate fits.
     THE DIFFUSION PARADIGM
          u ( x, t )
                       r u (1  u )  D  u
                                          2                           (Fisher, 1937)
             t
                                                          diffusion
                                          nonlinear “reaction”
                                          (logistic growth)
Epidemics of Hantavirus in P. maniculatus
                       Abramson, Kenkre, Parmenter, Yates (2001-2002)


          M S ( x, t )                  M M
                         b M  c M S  S  a M S M I  DS 2 M S ,
             t                           K ( x)
          M I ( x, t )            M M
                         c M I  I  a M S M I  DI 2 M I ,
             t                    K ( x)
 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.
THE SOURCE OF THE DATA

Gerardo Suzán & Erika Marcé, UNM
Six months of field work in Panamá (2003)
17   27   37   47     57   67   77
P    PA   PA   B      B    B    B

16   26   36   46     56   66   76
P    PA   PA   B      B    B    B

15   25   35   45     55   65   75
P    PA   PA   B      B    B    B

14   24   34   44     54   64   74
P    PA   PA   B      B    B    B

13   23   33   43     53   63   73
P    PA   PA   B      B    B    B

12   22   32   42     52   62   72
P    P    P    B      B    B    B

11
P
     21
     P
          31
          P
               41
               B
                      51
                      B
                           61
                           B
                                71
                                B
                                     Zygodontomys brevicauda
                                     Host of Hantavirus Calabazo
               60 m
       THE SOURCE OF THE DATA

Terry Yates, Bob Parmenter, Jerry Dragoo and many others, UNM
                                            Ten years of field work in
100                               N         New Mexico (1994-)

 50



  0



 -50



-100
                                            Peromyscus maniculatus
        -100   -50    0      50       100
                                            Host of Hantavirus Sin Nombre
                     200 m
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        SA         A
 Z. Brevicauda 0.13             0.37*     0.49          J: juvenile
                                                        SA: sub-adult
 P. maniculatus 0.32            0.48      0.58          A: adult



  *One mouse (SA, F) recaptured off-site, 200 m away
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…
                    The recaptures

         60
                                         Z. brevicauda
         40

         20
x (m)




                                                                    200
          0
                                                                                                        P. maniculatus
         -20                                                                                            all sites, all mice
                                                                    100
         -40




                                                           x (m)
         -60
                                                                      0
               0   20   40     60   80   100   120   140
                             t (days)
                                                                    -100



                                                                    -200
                                                                           0   30 60 90 120 150 180 210 240 270 300 330
                                                                                          time (days)
                 MOUSE WALKS

            60   Z. brevicauda captured ~10 times
            50                                                 30                       926
            40                                                                          799
            30                                                                  835       801
                                                                                                897               P.m. tag 3460
            20
                                                                0       898
            10
x(t) (m)




                                                                                        899
             0                                                                                        953
           -10                                                                927             745                744
           -20                                                 -30                      925
                                                     A117008
           -30                                       A117039                                                               954

           -40                                       A117075
                                                                        771
                                                     A117104
           -50                                       A117281           834
                                                               -60
           -60
                 0         30        60         90   120
                                                                 -40                0                       40               80
                                  t (days)
                                                                                                Julian date 2450xxx
                                                                                                (Sept. 97 to May 98)
An ensemble of displacements
An ensemble of displacements
An ensemble of displacements
                      …representing
                      the walk of an
                      “ideal mouse”
                PDF of individual displacements
As three ensembles, at three time scales:

        0.35     Z. brevicauda
        0.30                                        dt ~ 1day
                 247 steps                          dt ~ 1 month
        0.25     170 steps                          dt ~ 2 months
                 17 steps
        0.20
P(dx)




        0.15

        0.10
                                                                                                                    q(x)
        0.05                                                                0.016         p(x) (renormalized)
                                                                                          Gaussian fit
                                                                                                                    0.03

                                                                                                                    0.02

        0.00                                                                                                        0.01
                                                                            0.012      1 day intervals
                                                                                       P. maniculatus               0.00
                                                                                                                           -100 -50    0      50 100
        -0.05

                                                                     p(x)
                                                                                                                                      x (m)

                -80   -60    -40   -20   0    20   40    60     80          0.008


                                         dx                                 0.004


                                                                            0.000
                                                                                -150     -100     -50           0          50              100         150
                                                                                                          x (m)
                    Mean square displacement

                    600
<x >, <y > (m )




                                                                               3000
2




                    400
                                                                               2500
2




                                                           <x >, <y > (m )
                                                           2
                                                                               2000
                    200
2




                                                           2
                                                 2
                                              <x >                            1500
                                                 2
                                              <y >
                                                                               1000



                                                           2
                      0
                          0   30          60          90                                        East-West direction
                                                                                500
                                                                                                North-South direction
                                   t (days)
                                                                                  0
                                                                                      0   30   60      90      120   150   180
                    Z. brevicauda (Panama)                                                          t (days)


                                                               P. maniculatus (New Mexico)
Confinements to diffusive motion

         • Home ranges




         • Capture grid

                                 L

         • Combination of both
                             G

                             G       L
A harmonic model for home ranges
                U1                  U2                 U3




                     P1(x)               P2(x)               P3(x)



                             L/2                 L/2                 L/2




                              xc1                xc2                 xc3
        -xc/2                                                              xc/2
                                -G/2                   G/2



  P( x, t )   dU ( x )          
                         P( x, t )  D2 P( x, t )
    t        x  dx
                                                   PDF of an animal
                    Time dependent MSD

                               L=G
                    600

                                                           L>G                              L=
<x >, <y > (m )




                                                                                 3000
2




                    400
                                                                                 2500
                                                                                             saturation
2




                                                             <x >, <y > (m )
                                                             2
                                                                                 2000
                    200
2




                                                             2
                                                 2
                                              <x >                              1500
                                                 2
                                              <y >
                                                                                 1000



                                                             2
                      0                                                             L<G            East-Westpotential,
                          0   30          60          90
                                                                                  500
                                                                                                      box direction
                                                                                                   North-South direction
                                   t (days)                                                             concentric with
                                     initially diffusive ~t0                            0    30   60    the window 180
                                                                                                         90  120 150
                    Z. brevicauda (Panama)                                                             t (days)


                                                                 P. maniculatus (New Mexico)
  Saturation of the MSD

                           1.0
                                          2
                                                                                         Application of
                                         L /6
                           0.8                                                           the use of the
          <<x >> /(G /6)




                                                                                         saturation
          2




                           0.6                                                           curves to
                                                             harmonic numerical          calculate the
          2




                           0.4                               harmonic analytical
from                                                         box numerical               home range
                                                             box analytical
measurements 0.2                                             asymptotics                 size of P.
                                                                                         maniculatus
                           0.0                                                           (NM average)
                             0.0   0.5          1.0         1.5         2.0        2.5
                                                      L/G

                                         resulting value
Periodic arrangement of home ranges




                                 …                               …
      2.0



      1.5                                                    a
      1.0                              2   2
                                   x /(G /6)
a/G




                                               1.0
                                               0.9
                                               0.8
                                               0.7
                                               0.6
                                               0.5
      0.5                                      0.4
                                               0.3
                                               0.2
                                               0.1
                                               0


      0.0
         0.0   0.5   1.0         1.5             2.0   2.5

                           L/G
Periodic arrangement of home ranges
    1.0
                              Measurement 1 (G1 = 1)
                              Measurement 2 (G2 = 0.5)
    0.8
                              Measurement 3 (G3 = 0.75)



    0.6
                                                               x12  f L / G1 , a / G1 
a                            intersection                      x2  f L / G2 , a / G2 
                                                                 2


    0.4
                                                               x3  f L / G3 , a / G3 
                                                                 2



    0.2


    0.0
       0.0   0.2   0.4       0.6         0.8             1.0
                         L
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
Thank you!
TRAVELING WAVES

How does infection spread from the refugia?


The sum of the equations for MS and MI is Fisher’s
equation for the total population:


      M ( x, t )                 M 
                   (b  c) M 1           D 2 M
                               (b  c) K 
        t                               
                                          (Fisher, 1937)


There exist solutions of this equations in the form
of a front wave traveling at a constant speed.
Traveling
waves of
the
complete
system




Allowed speeds:

 vS  2 D(b  c)
 vI  2 D b  aK (b  c)
                              Depends on K and a
Two regimes of propagation:

v I  vS if K  K 0
v I  vS if K  K 0



          2b  c
    K0 
         a (b  c )



The delay    is also controlled by the carrying capacity
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).

				
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