Tandar – Buenos Aires – 2008 Estimando la calidad de las ... · Estimando la calidad de las...
Transcript of Tandar – Buenos Aires – 2008 Estimando la calidad de las ... · Estimando la calidad de las...
Estimando la calidad de las “oscilaciones
estocásticas”Guillermo Abramson
Grupo de Física Estadística e Interdisciplinaria (CAB),CONICET e Instituto Balseiro, Bariloche, Argentina
(con Sebastián Risau-Guzmán)
Tandar – Buenos Aires – 2008
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”
xc3xc2xc1 xc/2-xc/2 G/2-G/2
L/2
U1U2 U3
P3(x)P2(x)P1(x)
L/2L/2
OUR HANTAVIRUS ROADMAP AS OF TODAY
Fluctuations
Master equation
Oscillations
Extinction (First Passage Time)
STOCHASTIC FORMULATION: WHY?
The “microscopic” processes are stochastic in nature.
• Role of fluctuations?
• New phenomena?
Departures from Mean Field (small systems?)Extinction by fluctuationsSuppression of prevalence by fluctuations
Departures from Mean Field (small systems?)Extinction by fluctuationsSuppression of prevalence by fluctuations
Oscillations! (Solari, McKane…)Oscillations! (Solari, McKane…)
Macroscopic model (deterministic)
Microscopic model II
Microscopic model I
LEVELS OF DESCRIPTION
(stochastic)
OSCILLATIONS IN BIOLOGICAL AND CHEMICAL SYSTEMS
• Predator-prey systems(lynx vs. snow hare, foxes vs. rabbits, etc.)
• Chemical reactions(Belousov-Zhabotinskii, etc.)
• Epidemic systems(measles, pertussis, syphilis, etc.)
measles cases in New York
non-seasonal oscillations!
OSCILLATIONS IN MODELS
Deterministic:
Limit cycles
Damped oscillations (→ fixed point)
Deterministic:
Limit cycles
Damped oscillations (→ fixed point)
OSCILLATIONS IN MODELS
Deterministic:
Limit cycles
Damped oscillations (→ fixed point)
Deterministic:
Limit cycles
Damped oscillations (→ fixed point)
Stochastic:
Sustained oscillations in the same regime!
Stochastic:
Sustained oscillations in the same regime!
0 100 200
0.3
0.4
0.5
φ(t)
t
macroscopic 10000 realizations 1 realization
STOCHASTIC “OSCILLATIONS”?
Constructive role of noise! Coherence from stochasticity!
OSCILLATIONS IN SI MODELS
0 200 400 6000.00
0.05
0.10
0.15
m/Ω
, n/Ω
, φA, φ
Β
t
susceptible infected
bA=bBA=0.1dA=0.13dB=0.49p=3.02
(All the examples from an SI epidemic model)
Deterministic:Damped oscillations
Stochastic:Sustained oscillations in the same regime!
State of the system: P(m, n, t)
DYNAMICS: MASTER EQUATION
∑∑ −=kj
kjkj
kjkj nmPnmnmTnmPnmnmTdt
nmdP,,
),(),|,(),(),|,(),(
transitions from the (m,n) statetransitions to the (m,n) state
transition probabilities
m m+1m-1
n -1
n+1
n
m m+1m-1
n -1
n+1
nspecies A(e.g. Susceptibles)
species B(e.g. Infected)
EXPANSION OF THE MASTER EQUATION(van Kampen’s Expansion)
BB
AA
n
m
ξφ
ξφ
Ω+Ω=
Ω+Ω=
…and expand the Master Equation in powers of 1/Ω
• leading order: equations for φA and φB(macroscopic equations)
• second order: FP equation for the PDF of ξA and ξB(broadening of P)
Ansatz:
n(0)=Ωφ(0)
P(n,t)P(n,0)
Ω1/2ξ
S(t)=Ωφ(t)n(t)
FIRST ORDER
),(),(
),(),(
BABBAij
ijB
BAABAij
ijA
CTjdt
d
CTidt
d
φφφφφ
φφφφφ
≡=
≡=
∑
∑
Deterministic “macroscopic” equations
Many choices of T’s give the samemacroscopic equations
A linearly stable equilibrium exists if:
BBAA
ABBABBAA
CCTCCCC
,,
,,,,
+=
−=Δ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=j
iji C
CC ,
⎩⎨⎧<>
Δ=spiral2node2
/Tε...damped oscillations
(Many microscopic models give the same macroscopic model)
FLUCTUATIONS
∑∑ Π∂∂
∂+Π
∂∂
=∂
Π∂
ij jiij
ijj
iij
BA bat
tξξ
ξξ
ξξ 2),,(
The following order is a linear Fokker-Planck eq. for the PDF of ξA and ξB :
…with coefficients that depend on time through the solutions of the macroscopic equations
)(
)(
,,
,,
tCC
tCC
BBBBAABB
ABBAAAAA
ηξξξ
ηξξξ
++=
++=&
&
BBB
ABBA
AAA
Dtttt
Dtttt
Dtttt
)'()'()(
)'()'()(
)'()'()(
−=
−=
−=
δηη
δηη
δηη
∑∑∑ ===ij
ijBij
ijABij
ijA TjDTijDTiD 22 ,, with
Equivalent Langevin equations:
Gaussian noises:
Fourier transforming and averaging:
frequency of thestochasticoscillation
POWER SPECTRUM
Δ⎥⎦
⎤⎢⎣
⎡+−
+= AA
ADFS 2222
2
ˆ)ˆ1(ˆ
)(εωω
ωω
A
ABBBBAABBBBAA D
DCCDCDCF
Δ−+
=Δ= ,,2
,2
,22 2,/ˆ with ωω
is either monotonically decreasing or ithas a single maximum at
)(ωAS
AAAA FFF 222 )1(ˆ εω −++−=
BOUNDS FOR THE FREQUENCIES
1ˆ2/1 22 <<− Aωε
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
^
^ω
2 Α ,
ω2 B
ε
susceptible infected
4/1ˆ 2 εω −=Deterministic
It can be proved that:
In all possible stochastic models that lead to the same macroscopic model, the position of the maximum can vary within a finite range
STOCHASTIC “OSCILLATIONS”
Characterization: through the power spectrum of the fluctuations
Frequency: maximum of the peak
Quality:
( )( )
( )peak peak
peak
SQ
S d
ω ωω
ω ω=
∫
1500 1600 1700 1800 1900 20000.01
0.02
(b)
Qi = 2.11, ε = 0.30bA=bBA=0.1dA=0.13, dB=0.49p=3.02
n/Ω
t(a)
0.00 0.03 0.06 0.0902468
10
S( f )
f
500 600 700 800 900 1000
0.085
0.090
0.095Qi = 0.25, ε = 1.4bA=bBA=0.1dA=0.05, dB=0.25p=2.2
n/Ω
t
0.00 0.03 0.06 0.090
1
2
3
4
S( f )
f
Q = 6.6Q = 6.6
Q = 0.79Q = 0.79
BOUNDS FOR THE QUALITIES
πεωε 2)()( << AQf
It can be proved that:
Quality can belarge if and only ifε is small
0 1 2 3 4
0.1
1πε2
QA, Q
B
ε
susceptible infected
f(ε)
0.0 0.2 0.4 0.6 0.8 1.010-4
10-3
10-2
10-1
δQ
ε
relative differerence between bounds
Analytically, from QAconsidered as a function of FA for fixed ε
and when it is large, it is so for all
the models
Notice that:
For small ε, the evolution will look oscillatory, and one can be sure that the frequencies of the stochastic oscillations will be very close to the deterministic ones.
For large ε, the quality will be small, therefore it does not really matter if the frequencies of both populations are close (they do not really correspond to oscillations!)
1500 1600 1700 1800 1900 20000.01
0.02
(b)
Qi = 2.11, ε = 0.30bA=bBA=0.1dA=0.13, dB=0.49p=3.02
n/Ω
t(a)
0.00 0.03 0.06 0.0902468
10
S( f )
f
500 600 700 800 900 1000
0.085
0.090
0.095Qi = 0.25, ε = 1.4bA=bBA=0.1dA=0.05, dB=0.25p=2.2
n/Ω
t
0.00 0.03 0.06 0.090
1
2
3
4
S( f )
f
Q = 6.6Q = 6.6
Q = 0.79Q = 0.79
FINITE SIZE SYSTEMS
bs=bis=0.1bii=0.0ds=0.01di=0.4a=0.5
0.00 0.02 0.04 0.06 0.080
1
2
3
P(f)
Ω
f
Ω=5000 Ω=2000 Ω=1000 Ω=500 Ω=300 Ω=200 Ω=150
analytic (vK)analytic (vK)
The oscillations are apparent even for rather small systems
How good is the approximation for not so large systems?
• Quantification of the oscillatory appearance of a time series.
• Good stochastic oscillations are present if the damping of the macroscopic model is small (ε < ~0.5).
• No stochastic model will show oscillations if the damping of the macroscopic model is large (ε > ~0.5).
• There is a smooth transition between these two regimes.• Given a macroscopic deterministic model, one can know
whether any microscopic stochastic model that has it as a limit, will display good sustained stochastic oscillations.
CONCLUSIONS
Example: predator-prey systemModel with logistic growth of the prey
Deterministic:
Stochastic 1: Stochastic 2:
Both St1 and St2 give rise to the same deterministic equationswhen averaged, for large populations
φA= a φA φB − μφΑφB =rφB (1−φB/K)-βφA φB
T(m+1,n | m,n)=a φA φBT(m-1,n | m,n)=μ φAT(m,n+1 | m,n)=r φBT(m+1,n-1 | m,n)=r (φB)2 /K +b φAφB
T(m+1,n | m,n-1)=a φA φBT(m-1,n | m,n)=μ φAT(m,n+1 | m,n)=r φB(1-φA-φB )/KT(m+1,n-1 | m,n)=(b-a-r/K)φAφB+
r(1/K-1)φB
ME: P(m,n;t+1)=Σ' T(m,n | m',n') P(m',n') - P(m,n) Σ' T(m,n | m,n)
•
•
Birth (indirect competition)
S + E → S + S
I + E → I + S
I + E → I + I
Death
S → E
I → EContagion
S + I → I + I
Direct competition
S + S → S + E
I + I → I + E
I + I → S + E
S + I → S + E
S: susceptible individualsI: infected individualsE: shared resource (“empty slots”)
REACTION (MICROSCOPIC) MODEL
Vertical transmission:
rddbbb Kisiiiss ,,,,,, :Rates 1
(Ω = S+I+E)