Waves of Hantavirus
Transcript of Waves of Hantavirus
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?