De ratones y virusDe ratones y virusGuillermo Abramson (IB86)

Grupo de Física Estadística, Centro Atómico Bariloche y CONICETBariloche, Argentina

con Kenkre, Peixoto, Giuggioli y otros

Jornadas de Física Bariloche 2005

Sin Sin NombreNombrePeromyscusPeromyscus maniculatusmaniculatus

Rio SegundoRio SegundoReithrodontomysReithrodontomys mexicanusmexicanus

El Moro CanyonEl Moro CanyonReithrodontomysReithrodontomys megalotismegalotis

AndesAndesOligoryzomysOligoryzomys longicaudatuslongicaudatus

BayouBayouOryzomysOryzomys palustrispalustris

Black Creek CanalBlack Creek CanalSigmodonSigmodon hispidushispidus

Rio Rio MamoreMamoreOligoryzomysOligoryzomys microtismicrotis

Laguna Laguna NegraNegraCalomysCalomys lauchalaucha

MuleshoeMuleshoeSigmodonSigmodon hispidushispidus

New YorkNew YorkPeromyscusPeromyscus leucopusleucopus

JuquitibaJuquitibaUnknown HostUnknown Host MacielMaciel

NecromysNecromys benefactusbenefactusHu39694Hu39694Unknown HostUnknown HostLechiguanasLechiguanasOligoryzomysOligoryzomys flavescensflavescensPergaminoPergaminoAkodonAkodon azaraeazarae

OrOráánnOligoryzomysOligoryzomys longicaudatuslongicaudatus

CCaaññoo DelgaditoDelgaditoSigmodonSigmodon alstonialstoni

IslaIsla VistaVistaMicrotusMicrotus californicuscalifornicus

BloodlandBloodland LakeLakeMicrotusMicrotus ochrogasterochrogaster

Prospect HillProspect HillMicrotusMicrotus pennsylvanicuspennsylvanicus

New World HantavirusesNew World Hantaviruses

BermejoBermejoOligoryzomysOligoryzomys chacoensischacoensis

CalabazoCalabazoSigmodontomisSigmodontomis brevicaudabrevicauda

THE SOURCES OF THE DATA

Gerardo Suzán & Erika Marcé, UNM

Six months of field work in Panamá (2003)

Terry Yates, Bob Parmenter, Jerry Dragoo and many others, UNM

Ten years of field work in New Mexico (1994-)

Zygodontomysbrevicauda(Hantavirus

Calabazo)

Peromyscusmaniculatus(Hantavirus

Sin Nombre)

Infinite complications!• Males and females, juvenile, sub-adults, adults• Different interactions• Adult mice diffusion within a home range• Sub-adult mice run away to establish a home range • Juvenile mice few excursions from nest• Males and females, explorations, “transient mice”…• Different species, other species…

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)+

(Non-host competitor)

Mathematical model

Single control parameter in the model simulate environmental effects.

The other two appear as consequences of a bifurcation of the solutions.

Simple model (mean field)

Spatially extended model

Waves

“Refugia”

Biodiversity

Metapopulations

Fluctuations

Master equation

Oscillations

Extinction (First Passage Time)

CompetitionPredation

Hierarchy

Other epidemic systems

Animal “transport”and “homes”

BASIC MODEL (1 species, no space, no movement!)

,)(

,)(

ISI

II

ISS

SS

MMatKMMMc

dtdM

MMatKMMMcMb

dtdM

+−−=

−−−=

Rationale behind each termBirths: bM → only of susceptibles, all mice contribute to itDeaths: -cMS,I → infection does not affect death rateCompetition: -MS,I M/K → population limited by environmentalparameter Contagion: ± aMS MI → simple contact between pairs

MS (t) : Susceptible mice

MI (t) : Infected mice

M(t)= MS (t)+MI (t): Total mouse population

environmental parameter, “carrying capacity”

The carrying capacity controls a bifurcation in the equilibrium value of the infected population.

The susceptible population is always positive.

)( cbabKc −

=

BIFURCATION

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.

predictedcriticalpopulationNc=Kc(b-c)

IISI

II

SISS

SS

MDMaMxKMMcM

tM

MDMaMxKMMcMbM

tM

2

2

)(

)(

∇++−−=∂

∂

∇+−−−=∂

∂

SPATIALLY EXTENDED PHENOMENA

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

The carrying capacity is supposed proportional to the vegetation cover.

REALISTIC REFUGIA

[ ])(2

)(2

cbaKbDv

cbDv

I

S

−+−≥

−≥

Allowed speeds:

Depends on K and a (different regimes of propagation)

TRAVELING WAVES OF INFECTION

How does infection spread from the refugia?

THE BIODIVERSITY MODEL

Coexistence equilibrium:

εεγβκ

εγβκ

qcbKz

abqqcbKm

abm

i

s

−−−−

=

−−

−−−=

=

1)()(

/1

)()(/

*

*

*

The effect of biodiversity in the Hantavirus epizootic, Peixoto and Abramson, Ecology (2005).

m: host species

z: non-host competitor species

Does a competitor regulate the infection?

Two mice species SI model

)()(

)(

)(

mzzzdtdz

mamqzmKmcm

dtdm

mamqzmKmcmbm

dtdm

isi

ii

iss

ss

εκ

γβ +−−=

++−−=

−+−−=

interspecific competition

A critical density of aliens drives the system completely to a non-infected state.

*1

*)(/1*

zcb

qK

KqzcbK

ab AKc

−−

−=−−

−=χ

aqbcbaKzc

−−=

)(

zcb

qKK AKcc )( −

+=

COMPETITIVE SUPPRESSION OF THE INFECTION

0 10 20 30 40 50 60 70 800

20

40

60

80

100

prevalence = 0

prevalence > 0

κ

K

0 10 20 30 40 50 60 70 800.0

0.2

0.4

0.6

χ*

K

0

20

40

60

80

100

0.0 0.2 0.4 0.6

κ

χ∗

alie

ns’c

. cap

acity

hosts’ c. capacity

prevalence > 0

prevalence = 0

Equilibrium prevalence:

Critical control parameters:

,)(

),(

,)(

),(

2

2

IISI

II

SISS

SS

MDMMaxKMMMc

ttxM

MDMMaxKMMMcMb

ttxM

∇++−−=∂

∂

∇+−−−=∂

∂

Parameters?...

… and validity of diffusion?...

A HARMONIC MODEL FOR HOME RANGES

xc3xc2xc1 xc/2-xc/2 G/2-G/2

L/2

U1U2 U3

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

L/2L/2

),(),()(),( 2 txPDtxPdx

xdUxt

txP∇+⎥⎦

⎤⎢⎣⎡

∂∂

=∂

∂

FPE MSD

0.0 0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

0.8

1.0

harmonic numerical harmonic analytical box numerical box analytical asymptotics

<<x2 >>

/(G

2 /6)

L/G

L2/6

Application of the use of the saturation curves to calculate the home range size of P. maniculatus (NM average)

Gracias!

Un teórico en el campo…

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, Ecology (in press, 2005).

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. 67, 1135 (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, Ecol. Complexity (in press, 2005).

8. Theory of home range estimation from mark-recapture measurements of animal populations, L. Giuggioli, G. Abramson and V.M. Kenkre, J. Theor. Biol. (in press, 2005).

9. The effect of biodiversity on the Hantavirus epizootic, I. Peixoto and G. Abramson, Ecology (in press, 2005)