Spatial extent of an outbreak in animal epidemics In collaboration - - PowerPoint PPT Presentation

Spatial extent of an outbreak in animal epidemics

In collaboration with E. Dumonteil, S. N. Majumdar, A. Zoia

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PNAS 110, 4239 (2013)

SIR model for epidemics

dS dt = −β I S dI dt = β I S − γI dR dt = γI

• mean field fully connected model
• β rate of infection transmission
• γ rate at which an infected recovers

Three species : susceptibles (S), infected (I), recovered (R) I(t) + S(t) + R(t) = N N being the total population

Outbreak of an epidemic

dS dt = −β I S dI dt = β I S − γI dR dt = γI Initial condition : I(0) = 1, S(0) = N − 1 ≈ N, R(0) = 0 dI dt ' (β N γ) I t ≈ 0, S ≈ N Outbreak regime Reproduction rate: R0 = βN

γ

Deterministic and stochastic models

Reproduction rate: R0 = βN

γ

• R0 < 1 epidemics extinction
• R0 > 1 epidemics invasion
• R0 = 1 critical case

SIR is a deterministic model. In the outbreak fluctuations are important

• Stochastic process: Galton-Watson (mean field)
• each infected individual transmits the disease at rate Nβ
• each infected individual recovers at rate γ

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The good candidate: Brownian process with branching and death

In dt, each infected can:

• recovers with probability γ dt
• infects with probability βN dt = γR0dt
• otherwiese, it diffuses (D diffusion const.)

How far in space can an epidemic spread?

Problem 1: How to model the space?

Problem 2: How to quantify the area that needs to be quarantined?

CONVEX HULL

Algorithms: Graham Scan (Nlog(N))

Monitoring the outbreak

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Day 1 Day 2 Day 3

x y x y x y

Real applications

Eric Dumonteuil

How to compute the convex hull of Branching processes?

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y x M(θ)

C

θ

L = Z 2π M(θ) dθ A = 1 2 Z 2π ⇥ M 2(θ) − (M 0(θ))2⇤ dθ

Cauchy formulas

Support Function

M(θ) = max

0≤τ≤t [xτ cos θ + yτ sin θ]

M(0) = xτ=tm = xm(t)

• xm(t) x-maximum up to time t
• tm time location of the maximum

M 0(θ = 0) = −xtm sin θ + ytm cos θ|θ=0 = ytm

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x y x t t y xm y(tm) xm y(tm) tm tm

hL(t)i = 2πhxm(t)i hA(t)i = π ⇥ hx2

m(t)i hy2(tm)i

⇤

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x y x t t y xm y(tm) xm y(tm) tm tm

consider a 1d branching process evolving in (0, t)

• xm is the global maximum
• tm is the location of the maximum
• hy2(tm)i = . . . = 2D tm

Qt+dt(xm) = γdt + R0γdtQ2

t(xm) + [1 γ(R0 + 1)]dthQt(xm ∆x)i

Backward Fokker Planck equation

Qt(xm) = Proba[global max up to t < xm]

• hQt(xm ∆x)i = Qt(xm) h∆xiQ0

t(xm) + h ∆x2 2 iQ00 t (xm) + . . .

• h∆xi = 0
• h∆x2i = 2Ddt

hQt(xm ∆x)i = Qt(xm) + Ddt∂2

xQt(xm) + . . .

hL(t)i = 2π Z ∞ [1 Qt(xm)]dxm. ∂ ∂tQ = D ∂2 ∂x2

m

Q − γ(R0 + 1)Q + γR0Q2 + γ

• initial condition Qt=0(xm) = θ(xm)
• boundary condition Qt(xm < 0) = 0
• boundary condition Qt(xm → ∞) = 1

20 40 60 80 100 101 102 103 104 105 < L(t) > t 10-6 10-5 10-4 10-3 10-2 101 102 Prob(L,t) L

R0=1.15 R0=1 R = 1 . 1 R0=0.99 R0=0.85

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 100 101 102 103 104 Prob(A,t) 500 1000 1500 2000 2500 3000 3500 101 102 103 104 105 < A(t) >

R0=1.15 R0 = 1 R0=1.01 R0=0.99 R0=0.85

Solid lines: our predictions Blue lines: super-critical Red lines: critical Green lines: sub-critical Symbols: Monte Carlo simulations Dashed lines: analytical asymptotic results

500 1000 1500 2000 2500 3000 3500 101 102 103 104 105 < A(t) > t

R0=1.15 R0=1 R0 = 1 . 1 R = . 9 9 R0=0.85

20 40 60 80 100 101 102 103 104 105 < L(t) > t

R0=1.15 R0=1 R = 1 . 1 R0=0.99 R0=0.85

The critical case

hL(t ! 1)i = 2π s 6D γ + O(t−1/2) hA(t ! 1)i = 24πD 5γ ln t + O(1)

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 100 101 102 103 104 105 106 Prob(A,t) A 10-6 10-5 10-4 10-3 10-2 101 102 103 Prob(L,t) L

When t → ∞ the perimeter remains finite, but the area diverges! How it is possible ? ... Fluctuations Prob(L)

t=∞

− − − − →

L→∞ L−3

Prob(A)

t=∞

− − − − →

A→∞

24πD 5γ A−2

500 1000 1500 2000 2500 3000 3500 101 102 103 104 105 < A(t) > t

R0=1.15 R0=1 R0 = 1 . 1 R = . 9 9 R0=0.85

20 40 60 80 100 101 102 103 104 105 < L(t) > t

R0=1.15 R0=1 R = 1 . 1 R0=0.99 R0=0.85

Out of criticality

When R0 6= 1, characteristic time t∗ ⇠ |R0 1|−1. For times t < t∗ the epidemic behaves as in the critical regime. In the subcritical regime, for t > t∗ the epidemic goes to extinction. In the supercritical regime, with probability 1 1/R0 epidemic explodes.

Supercritical

hL(t t∗)i = 4π ✓ 1 1 R0 ◆ p D γ (R0 1) t hA(t t∗)i = 4π ✓ 1 1 R0 ◆ D γ (R0 1) t2 t∗ ⇠ |R0 1|−1

100 101 102 103 104 105 100 101 102 103 < A(t) > t

R = 2 . 5 R = 1 . 5 R = 1 . 2 5 R0 = 1

101 102 100 101 102 103 < L(t) > t

R = 2 . 5 R0=1.5 R0=1.25 R0=1

1 20 40 60 80 100 120 140 x

t=100 t=200 t=400 t =

∞

10 30 40 x

t = 2 t=400 t =∞

1 − 1 R0

Traveling front solution

∂ ∂tW = D ∂2 ∂x2

m

W + γ(R0 − 1)W − γR0W 2

Conclusions:

• Branching Brownian motion with death as a

model for the spatial extent of animal epidemics

• Using Cauchy Formulas we can map the convex

hull problem in the extreme statistic of the 1- dimensional process

• Backward F-P equations for the extreme

distributions

• Critical case has very large fluctuations
• Super Critical case: traveling front solution

How far in space can an epidemic spread?

Problem 1: How to model the space? Brownian motion is the paradigm of animal migration The population is uniformly distributed At time t = 0 an infected individual appears ... and moves in space while human beings take the plane (even when they are sick)

hA(t)i = π Z ∞ dxm [2xm(1 Qt(xm)) Tt(xm)] Similar calculations allows to express the mean area as: Where the evolution of Tt(xm) is governed by: ∂ ∂tTt + ∂xQt(xm) =  D ∂2 ∂x2

m

+ 2γR0Qt − γ (R0 + 1)

• Tt,

Both PDE can be integrated numerically and solved in some asymptotic limit

L = Z 2π M(θ) dθ A = 1 2 Z 2π ⇥ M 2(θ) − (M 0(θ))2⇤ dθ

hL(t)i = 2πhM(0)i hA(t)i = π ⇥ hM 2(0)i hM 0(0)2i ⇤

hL(t)i = 2πhxm(t)i hA(t)i = π ⇥ hx2

m(t)i hy2(tm)i

⇤

This relation is valid ONLY in average

Dimensional reduction

If the process is rotationally invariant any average is independent of θ