Download Cours 2.

Spread of epidemics on random graphs
Viet Chi TRAN - Université Lille 1 - France
December 11, 2015
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Course 2 : Convergence of a random individual-based model to Volz’
equations
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SIR on a configuration model graph
FConfiguration model (CM) : Bollobas (80), Molloy Reed (95),
Durett (07), van der Hofstad (in prep.)
F Description of an SIR epidemics spreading on a configuration
model graph :
I
Infinite system of denumberable equations, Ball and Neal
(2008)).
I
5 ODEs, Volz (2008), Miller (2011).
I
Recently: Barbour Reinert (2014), Janson Luczak Winridge
(2014)
F Individuals are separated into 3 classes :
I
Susceptibles St
Infectious It
I
Removed Rt
I
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Stochastic model for a finite graph with N vertices
F Only the edges between the I and R individuals are observed.
The degree of each individual is known.
F To each I individual is associated an exponential random clock with
rate α to determine its removal.
F To each open edge (directed to S), we associate a random
exponential clock with rate β.
F When it rings, the edge of an S is chosen at random. We
determine whether its remaining edges are linked with S, I or R-type
individuals.
(a)
(b)
(c)
I
S
I
I
I
I
R
I
R
R
I
I
I
S
I
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Edge-based quantities
F The idea of Volz is to use network-centric quantities (such as the
number of edges from I to S) rather than node-centric quantities.
F St , It , Rt , St , It , Rt , di , di (St )...
µ finite measure
on N and f bounded or > 0 function:
P
hµ, f i = k ∈N f (k )µ(k ).
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Edge-based quantities
F The idea of Volz is to use network-centric quantities (such as the
number of edges from I to S) rather than node-centric quantities.
F St , It , Rt , St , It , Rt , di , di (St )...
µ finite measure
on N and f bounded or > 0 function:
P
hµ, f i = k ∈N f (k )µ(k ).
F We introduce the following measures:
X
X
µSt (dk ) =
δdu (dk )
µSI
δdu (St ) (dk )
t (dk ) =
u∈St
µSR
(dk )
t
=
X
u∈It
δdu (St ) (dk )
u∈Rt
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Edge-based quantities
F The idea of Volz is to use network-centric quantities (such as the
number of edges from I to S) rather than node-centric quantities.
F St , It , Rt , St , It , Rt , di , di (St )...
µ finite measure
on N and f bounded or > 0 function:
P
hµ, f i = k ∈N f (k )µ(k ).
F We introduce the following measures:
X
X
µSt (dk ) =
δdu (dk )
µSI
δdu (St ) (dk )
t (dk ) =
u∈St
µSR
(dk )
t
=
X
u∈It
δdu (St ) (dk )
u∈Rt
This sums up the evolution of the epidemic (but does not allow the
reconstruction of the complicated graph on which the illness
propagates).
X
It = Card(It ) = hµSI
NtSI = hµSI
du (St )
t , 1i,
t , ki =
u∈It
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Dynamics
F Global force of infection: βNtSI
.
−
F Choice of a given susceptible of degree k : k /NtS− .
So that the rate of infection of a given susceptible of degree k is:
βkptI− .
F The probability that its k − 1 remaining edges are linked to I or R
is:
` m j
p(j, `, m|k − 1, t) =
NtSI −1
−
NtSR
−
k −1
NtS −1
NtSS
−
1j+`+m=k −1 1j<NtSI 1`≤NtSR
−
−
−
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Dynamics
F Global force of infection: βNtSI
.
−
F Choice of a given susceptible of degree k : k /NtS− .
So that the rate of infection of a given susceptible of degree k is:
βkptI− .
F The probability that its k − 1 remaining edges are linked to I or R
is:
` m j
p(j, `, m|k − 1, t) =
NtSR
NtSI −1
−
−
k −1
NtS −1
NtSS
−
1j+`+m=k −1 1j<NtSI 1`≤NtSR
−
−
−
SR
F To modify the degree distributions µSI
t− (idem for µt− ):
We draw a sequence e = (eu )u∈It− of integers.
• eu is the number of edges to the infectious individual u at t− .
• not all sequences are admissible.
The probability of drawing the sequence e is
Q
du
ρU (e | j, µSI
T− ) =
u∈It−
NtSI −
j+1
eu
1{P eu =j+1, e is admissible} .
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Renormalization
F We are interested in increasing the number of vertices N without
rescaling the degree distribution.
µN,S , µN,SI , µN,SR .
F We now consider µ(N),S , µ(N),SI and µ(N),SR where for ex:
(N),S
µt
(dk ) =
(N),SI
(idem for µ0
1 N,S
µ (dk )
N t
with
(N),S
lim µ0
N→+∞
(N),SR
with N̄0SI > ε and µ0
= µ̄S0 in MF (N)
with N̄0SR > ε)
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Renormalization
F We are interested in increasing the number of vertices N without
rescaling the degree distribution.
µN,S , µN,SI , µN,SR .
F We now consider µ(N),S , µ(N),SI and µ(N),SR where for ex:
(N),S
µt
(dk ) =
(N),SI
(idem for µ0
1 N,S
µ (dk )
N t
with
(N),S
lim µ0
N→+∞
(N),SR
with N̄0SI > ε and µ0
= µ̄S0 in MF (N)
with N̄0SR > ε)
F 3 SDE:
(N),SI
(N),SI
(N),SI,f
(N),SI,f
hµt
, f i = hµ0
, f i + At
+ Mt
,
where M (N),SI,f is a square integrable martingale started from 0 and
with previsible quadratic variation in 1/N.
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(N),IS,f
F At
(N),IS,f
At
:
Z
=−
t
(N),SI
αhµs
, f ids
0
+
Z tX
(N),I (N),S
µs
(k )
0 k ∈N
×
X
e∈U
X
βkps
ρU (e|j +
psN (j, `, m|k − 1, t)
j+`+1≤k
1, µn,SI
)
s
f (m) +
X
f (du − eu ) − f (du ) ds,
u∈IsN
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(N),IS,f
F At
:
t
Z
(N),IS,f
=−
At
(N),SI
αhµs
, f ids
0
+
Z tX
(N),I (N),S
µs
(k )
0 k ∈N
X
×
X
βkps
psN (j, `, m|k − 1, t)
j+`+1≤k
ρU (e|j +
1, µn,SI
)
s
e∈U
f (m) +
X
f (du − eu ) − f (du ) ds,
u∈IsN
Th: Under appropriate moment conditions,
(N),S
(N),SI
(N),SR
(µt
, µt
, µt
SR
(µ̄St , µ̄SI
,
µ̄
)
t∈R+
t
t
hµ̄SI
t ,fi
=
hµ̄SI
0 ,fi
)t∈R+ converge to a deterministic limit
Z
t
−
αhµ̄SI
s , f ids
0
Z
+
t
X
0 k ∈N∗
X
βk p̄sI
j+`+m=k −1
× f (m) + (j + 1)
X
k 0 ∈N∗
j, `, m
(p̄sI )j (p̄sR )` (p̄sS )m
k −1
0 k 0 µ̄SI
s (k )
f (k 0 − 1) − f (k 0 )
µ̄Ss (k )ds
SI
hµ̄s , k i
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Deterministic limit
F Limit equations:
µ̄St (k ) = µ̄S0 (k )θtk ,
θt = e−β
Rt
0
p̄sI ds
hµ̄SI
t , f i = ...
Z t
hµ̄SR
,
f
i
=
αhµ̄SI
t
s , f ids
0
+
Z tX
0 k ∈N
βk p̄sI (k − 1)p̄sR
X
k 0 ∈N
0
k 0 µSR
s (k ) S
µ̄s (k )ds
f (k 0 − 1) − f (k 0 )
SR
N̄s
F This allows us to recover Volz’equations:
• Choosing f ≡ 1 gives S̄t , Īt ,
• Choosing f (k ) = k gives N̄ S , N̄ SI , N̄ SR ,
from which we can deduce p̄I = N̄ SI /N̄ S ...
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Volz’equations P
Prop: let g(z) =
k ∈N
µ̄S0 (k )z k be the generating function of µ̄S0 .
Z t
θt = exp − β
psI ds
0
S̄t =g(θt ),
Īt = Ī0 +
Z t
β p̄sI θs g 0 (θs ) − αĪs ds
0
N̄ SI
p̄tI = t S = p̄0I +
N̄t
p̄tS =
N̄tSS
N̄tS
Z t
g 00 (θs )
I
I
I
−
β
p̄
(1
−
p̄
)
−
α
p̄
ds.
s
s
s
g 0 (θs )
0
Z t
g 00 (θs ) = p̄0S +
β p̄sI p̄sS 1 − θs 0
ds.
g (θs )
0
β p̄sI p̄sS θs
2
Recall the limit for mixing models:
d S̄t
= − β S̄t Īt ,
dt
d Īt
= β S̄t Īt − αĪt .
dt
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Volz’equations P
Prop: let g(z) =
k ∈N
µ̄S0 (k )z k be the generating function of µ̄S0 .
Z t
θt = exp − β
psI ds
0
S̄t =g(θt ),
Īt = Ī0 +
Z t
β × N̄sSI − αĪs ds
0
N̄ SI
p̄tI = t S = p̄0I +
N̄t
p̄tS =
N̄tSS
N̄tS
Z t
g 00 (θs )
I
I
I
−
β
p̄
(1
−
p̄
)
−
α
p̄
ds.
s
s
s
g 0 (θs )
0
Z t
g 00 (θs ) = p̄0S +
β p̄sI p̄sS 1 − θs 0
ds.
g (θs )
0
β p̄sI p̄sS θs
2
Recall the limit for mixing models:
d S̄t
= − β S̄t Īt ,
dt
d Īt
= β S̄t Īt − αĪt .
dt
Here:
d S̄t
=g 0 (θt )θ̇t = −βg 0 (θt )θt p̄tI = −β N̄tS p̄tI = −β N̄tSI .
dt
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Sketch of the proof
(N),SI
(N),S
, 1 + k 5 i < +∞,
Assumption: supN∈N∗ hµ0
, 1 + k 5 i + hµ0
F Tightness: topology on MF (N). Roelly’s criterion.
Aldous-Rebolledo criterion.
|>ε ≤ε
− A(N),SI,f
P |Aτ(N),SI,f
σN
N
P |hM (N),SI,f iτN − hM (N),SI,f iσN | > ε ≤ ε.
F Convergence of the generators.
• The identification of the limit is OK on [0, T ] IF T < τεN where
(N),SI
τεN = inf{t ≥ 0, Nt
< ε}.
F Uniqueness:
• Gronwall’s lemma gives that solutions of the limiting equation
have same mass and same moments of order 1 and 2.
• Uniqueness of the generating function of µ̄IS which solves a
transport equation.
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600
400
0
200
Sizes of classes
800
1000
Degree distribution of the “initial condition”
0
1
2
3
4
5
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Time
Prop: For ε > 0, when N → +∞, the degree distribution when after
[εN] infections converges to:
1 X
pk (1 − z ε )k δk
1−ε
k ≥0
where z ε is the solution of 1 − ε = f (1 − z), f being the generating
function of the original degree distribution.
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