Waves of HantavirusWaves of HantavirusG. Abramson1,2 and V. M. Kenkre1

1Center for Advanced Studies and Department of Physics and Astronomy, University of New Mexico, Albuquerque, USA.2Centro Atómico Bariloche and CONICET, Bariloche, Argentina.

SinSin NombreNombrePeromyscus maniculatus

Rio SegundoRio SegundoReithrodontomys mexicanusReithrodontomys mexicanus

El Moro CanyonEl Moro CanyonReithrodontomys megalotisReithrodontomys megalotis

AndesAndesOligoryzomys longicaudatusOligoryzomys longicaudatus

BayouBayouOryzomys palustrisOryzomys palustris

Black Creek CanalBlack Creek CanalSigmodon hispidusSigmodon hispidus

RioRio MamoreMamoreOligoryzomys microtisOligoryzomys microtis

LagunaLaguna NegraNegraCalomys lauchaCalomys laucha

MuleshoeMuleshoeSigmodon hispidus

New YorkNew YorkPeromyscus leucopusPeromyscus leucopus

JuquitibaJuquitibaUnknown HostUnknown Host MacielMaciel

Necromys benefactusNecromys benefactusHu39694Hu39694Unknown HostUnknown HostLechiguanasLechiguanasOligoryzomys flavescensOligoryzomys flavescensPergaminoPergaminoAkodon azaraeAkodon azarae

OrOráánnOligoryzomys longicaudatusOligoryzomys longicaudatus

CCaañño Delgaditoo DelgaditoSigmodon alstoniSigmodon alstoni

IslaIsla VistaVistaMicrotus californicus

BloodlandBloodland LakeLakeMicrotus ochrogasterMicrotus ochrogaster

Prospect HillProspect HillMicrotus pennsylvanicusMicrotus pennsylvanicus

New WorldNew World HantavirusesHantaviruses

BermejoBermejoOligoryzomys chacoensisOligoryzomys chacoensis

Courtesy CDC

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)

•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

Chronically infected Chronically infected rodentrodent

Virus is present in aerosolized Virus is present in aerosolized excreta, particularly urineexcreta, particularly urine

Horizontal transmission of infection Horizontal transmission of infection by by intraspecific intraspecific aggressive behavioraggressive behavior

Virus also present in throat Virus also present in throat swab and fecesswab and feces

Secondary aerosols, mucous membrane Secondary aerosols, mucous membrane contact, and skin breaches are also a contact, and skin breaches are also a

considerationconsideration

Transmission ofTransmission of HantavirusesHantaviruses

Courtesy CDC

Two field observations and a simple model

• 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 two features appear as consequences of a bifurcation of the solutions.

Basic model

,

,

ISI

II

ISS

SS

MMaK

MMMcdt

dM

MMaK

MMMcMbdt

dM

+−−=

−−−=

Rationale behind each termBirths: bM → all mice contribute to reproductionDeaths: -cMS,I → infection does not affect death rateCompetition: -MS,IM/K → population limited by environmentalparameter Contagion: ±aMSMI → simple contact between pairs

MS (t) : Susceptible mice

MI (t) : Infected mice

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

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

The susceptible population is always positive.

)( cbabKc −

=

BIFURCATION

Temporal behavior

Temporal behavior

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

SPATIALLY EXTENDED PHENOMENA

,)(

),(

,)(

),(

2

2

IIISI

II

SSISS

SS

MDMMaxKMMMc

ttxM

MDMMaxKMMMcMb

ttxM

∇++−−=∂

∂

∇+−−−=∂

∂

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

A refugium in 1D.

SIMPLE REFUGIA

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

REALISTIC REFUGIA

Landscape Carrying capacity

TRAVELING WAVES

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

There exist solutions of this equations in the form of a front traveling at a constant speed

How does infection spread from the refugia?

MDKcb

MMcbt

txM 2

)(1)(),(

∇+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−=∂

∂

0),()()(

0),()()(

2

22

2

22

1

12

1

12

=++

=++

ISI

II

ISS

SS

MMgdz

zdMvdz

zMdD

MMfdz

zdMvdz

zMdD

tvxz

tvxz

I

S

−=

−=

2

1Traveling waves ansatz:

(different speeds?)

Equations for the traveling fronts:

Traveling waves of the complete system

0 200 400 600 8000

4

8

12

16

v

b)

z

MS ,

MI susceptible

infected

0 200 400 600 8000

4

8

12

16

v

a)

MS ,

MI susceptible

infected

Two interesting scenarios

•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

DELAYED FRONTS

Linear stability analysis of the unstable equilibria:

MS MI M’S M’I

a) 0 0 0 0

b) (b-c)K 0 0 0

DcbaKbDvv

DcbDvv

II

SS

2)]([4

μ

2)(4

μ

2

4,3

2

2,1

−−+±−=

−+±−=

DcbDvv

DDcvv

SS

II

2)(4

λ

24

λ

2

4,3

2

2,1

−−±−=

+±−=

[ ])(2

)(2

cbaKbDv

cbDv

I

S

−+−≥

−≥

Allowed speeds:

Does not depend on K or a

Depends on K and a

Two regimes of propagation:

)(2

0 cbacbK

−−=

0

0

if if

KKvvKKvv

SI

SI

>=<<

The delay Δ is also controlled by the carrying capacity

Digression: control of a propagating wave?

Reduce K below Kc

If not possible, or too drastical:

Reduce K below K0 > Kc

Piecewise linearization of the equations

⇒ Analytical expression for the front shapes

⇒ Analytical expression for Δ

)log()()( 21

0

wwKKacbi

D−−

=Δ

[ ] +− →−−≈Δ 02/1

0 when ,)()( KKKKacbD

The delay depends critically on the carrying capacity

SUMMARY

•Simple model of infection in the mouse population

•Important effects controlled by the environment

•Extinction of the infected population

•Spatial segregation of the infected population

•Propagation of infection fronts

•Delay of the infection with respect to the suceptibles

•Control of propagating waves?