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Diffusion and home ranges in mice movement 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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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 OUTLINE

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

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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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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!) A deer mouse. Is he S or I? HANTA AND THE DEER MOUSE

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Chronically infected rodent Virus is present in aerosolized excreta, particularly urine Horizontal transmission of infection by intraspecific aggressive behavior Virus also present in throat swab and feces Secondary aerosols, mucous membrane contact, and skin breaches are also a consideration Transmission of Hantaviruses

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Sea surface temperature, precipitation, and Hanta cases The El Niño connection

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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). Population dynamics + Contagion + (Mice movement) Mathematical model 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!) Rationale behind each term Births: bM only of susceptibles, all mice contribute to it Deaths: -cM S,I infection does not affect death rate Competition: -M S,I M/K population limited by environmental parameter Contagion: aM S M I simple contact between pairs M S (t) : Susceptible mice M I (t) : Infected mice M(t)= M S (t)+M I (t) : Total mouse population carrying capacity

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The carrying capacity controls a bifurcation in the equilibrium value of the infected population. The susceptible population is always positive. BIFURCATION

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The same model, with vertical transmission

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Temporal behavior A “realistic” time dependent carrying capacity induces the occurrence of extinctions and outbreaks as controlled by the environment. K=K(t)

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Temporal behavior of real mice Real populations of susceptible and infected deer mice at Zuni site, NM. Nc=2 is the “critical population” derived from approximate fits. critical population N c =K c (b-c)

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SPATIALLY EXTENDED PHENOMENA Diffusive transport Spatially varying K, following the diversity of the landscape

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A refugium in 1D. SIMPLE REFUGIA

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An illustration from northern Patagonia. The carrying capacity is supposed proportional to the vegetation cover. REALISTIC REFUGIA

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LandscapeCarrying capacity

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TRAVELING WAVES The sum of the equations for M S and M I is Fisher’s equation for the total population: There exist solutions of this equations in the form of a front wave traveling at a constant speed. How does infection spread from the refugia? (Fisher, 1937)

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Traveling waves ansatz: (different speeds?) Equations for the traveling fronts: Traveling waves of the complete system

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Two interesting scenarios

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Initial system empty of mice K >K c In contact with a refugium to the left: M S = M S * > 0 and M I = M I * > 0 Always an unstable equilibrium, but a biological possibility DELAYED FRONTS

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Allowed speeds: Depends on K and a Traveling waves of the complete system

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Two regimes of propagation: The delay 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

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The delay depends critically on the carrying capacity

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THE DIFFUSION PARADIGM Epidemics of Hantavirus in P. maniculatus Abramson, Kenkre, Parmenter, Yates ( ) diffusion nonlinear “reaction” (logistic growth) (Fisher, 1937)

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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. Three categories of wrongfulness Okubo & Levin, Diffusion and Ecological Problems

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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 17 P 11 P 12 P 13 P 14 P 15 P 16 P 27 PA 21 P 22 P 23 PA 24 PA 25 PA 26 PA 37 PA 31 P 32 P 33 PA 34 PA 35 PA 36 PA 47 B 41 B 42 B 43 B 44 B 45 B 46 B 57 B 51 B 52 B 53 B 54 B 55 B 56 B 67 B 61 B 62 B 63 B 64 B 65 B 66 B 77 B 71 B 72 B 73 B 74 B 75 B 76 B 60 m

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200 m Terry Yates, Bob Parmenter, Jerry Dragoo and many others, UNM Peromyscus maniculatus Host of Hantavirus Sin Nombre Ten years of field work in New Mexico (1994-) THE SOURCE OF THE DATA

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Zygodontomys brevicauda, 846 captures: 411 total mice, 188 captured more than once (2-10 times) JSAA F M Total Recapture probability: Recapture and age 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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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) JSAA Z. Brevicauda *0.49 P. maniculatus Recapture probability: Recapture and age 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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…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: 1 day intervals P. maniculatus

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Distribution of displacements P. maniculatus 1 day intervals

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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 Combination of both Capture grid

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A harmonic model for home ranges PDF of an animal

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Mean square displacement Steady state: dependence on G distribution of home ranges (let it be uniform…) initial positions within home range dependence on L

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Time dependent MSD L = L < G L > G L = G saturation initially diffusive ~ t box potential, concentric with the window Z. brevicauda (Panama) P. maniculatus (New Mexico)

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Mean square displacement Z. brevicauda (Panama) P. maniculatus (New Mexico)

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Saturation of the MSD There is a dependence on the potential, that results in slightly different saturation curves. home range linear sizeobservation window

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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 a ……

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intersection

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The polygon to measure home ranges

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measurement of msd measurement of the polygon

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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 The sum of the equations for M S and M I is Fisher’s equation for the total population: There exist solutions of this equations in the form of a front wave traveling at a constant speed. How does infection spread from the refugia? (Fisher, 1937)

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Allowed speeds: Depends on K and a Traveling waves of the complete system

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Two regimes of propagation: The delay 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) JSAATotal F M Total Captured once:

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JSAATotal F M Total Recaptured:

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1. Spatio-temporal patterns in the Hantavirus infection, by G. Abramson and V. M. Kenkre, Phys. Rev. E 66, (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, (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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