Epidemics on random graphs with a given degree
sequence
Malwina Luczak1
2
School of Mathematical Sciences
Queen Mary, University of London
e-mail: m.luczak@qmul.ac.uk
31 March 2016
Complex networks and emerging applications
ICMS
1
Joint work with Svante Janson, and Peter Windridge
The work of Luczak and Windridge was supported by EPSRC Leadership
Fellowship, grant reference EP/J004022/2.
2
Malwina Luczak
Epidemics on random graphs with a given degree sequence
SIR process on a graph
I
We study a simple Markovian model for a disease spreading
around a finite population.
I
Each individual is either susceptible, infective or recovered.
I
Individuals are represented by vertices in a graph G ; edges
correspond to potentially infectious contacts.
I
Each susceptible vertex becomes infective at rate β > 0 times
the number of its infective neighbours in G .
I
Infective vertices become recovered at rate ρ ≥ 0.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Recovered vertices are immune to the disease and cannot be
infected again.
In contrast, for an SIS epidemic, the population consists of
susceptible and infective individuals. When infective individuals
recover, they become susceptible again.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Random graphs with a given degree sequence
(n)
Take a given degree sequence (di )ni=1 = (di )ni=1 , and let Gn be
uniform
over all graphs with this degree sequence. (We assume
Pn
d
i=1 i is even.)
In fact, we prove our results by first studying a random multigraph
Gn∗ (allowing multiple edges and loops) with the given degree
sequence.
Gn∗ is constructed by the configuration model: we equip each
vertex i with di half-edges, and then take a uniform random
matching of all half-edges into edges.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Random graphs with a given degree sequence
P
If we assume that degree sequences satisfy ni=1 di2 = O(n) (i.e.
the second moment of the degree of a random vertex is bounded),
then
lim inf P(Gn∗ is simple) > 0;
n→∞
see Janson (2009).
Any result for the random multigraph that concerns convergence in
probability will also hold for the random simple graph, by
conditioning on the event that Gn∗ is simple.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Initial susceptible degree distribution
(n)
I
We start
P with nS,k = nS,k susceptible vertices, for each k. So
nS = ∞
k=0 nS,k is the total number of initially susceptible
vertices.
I
The degree of a random initially susceptible individual has an
asymptotic probability distribution (pk )∞
0 , i.e.,
lim
n→∞
I
nS,k
= pk ,
nS
k ≥ 0.
This distribution (pk ) has finite mean λ:
λ=
∞
X
kpk < ∞.
k=0
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
The average degree of the susceptible vertices converges to λ:
P∞
k=0 knS,k
→ λ.
nS
I
The fraction of susceptible vertices converges:
P∞
nS,k
nS
= k=0
→ αS ,
n
n
where αS > 0.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Other types of vertices
(n)
(n)
I
At time 0 there are nI = nI infective and nR = nR
recovered vertices (individuals).
I
Of these, nI ,k = nI ,k and nR,k = nR,k , respectively, have
degree k (k = 0, 1, . . . ).
I
As n → ∞, the fractions of initially infective and recovered
individuals converge to constants αI and αR :
P∞
nI
k=0 nI ,k
=
→ αI ;
n
P∞n
nR
k=0 nR,k
=
→ αR .
n
n
(n)
Malwina Luczak
(n)
Epidemics on random graphs with a given degree sequence
I
For the infected and recovered vertices, we have
∞
∞
1X
knI ,k → µI ;
n
1X
knR,k → µR ;
n
k=0
k=0
for some constants µI and µR .
I
The average degree of all vertices converges to a constant µ:
n
∞
i=1
k=0
1X
1X
di =
knk → µ.
n
n
I
I
Thus µ = αS λ + µI + µR .
P
Also we assume that i di2 = O(n).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
Let S∞ be the number of susceptibles that ultimately escape
infection.
I
Let vS be the scaled pgf of the asymptotic susceptible degree
∞
X
distribution: vS (x) = αS
pk x k .
k=0
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Theorem (Janson, L., Windridge (2014))
Suppose µI = 0, let Z = nS − S∞ , the number of susceptible
vertices that ever get infected, and set
X
αS
β
k(k − 1)pk .
R0 =
ρ+β
µ
k
(i) If R0 ≤ 1 then P(Z /n > ε) → 0 for any ε > 0.
(ii) If R0 > 1, then there exists c > 0 such that
lim inf P(Z /n > c) > 0. Moreover, conditional on the
occurrence of a large epidemic,
S∞ p
−→ vS (θ∞ ),
n
where θ∞ is the solution to a fixed-point equation.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
In the case µR = 0, the threshold for the emergence of a large
epidemic was identified heuristically by Andersson’99, Newman’02
and Volz’07, and rigorously proved by Bohman/Picollelli’12 for
bounded degree sequences.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Analysis of the process
I
We think of each vertex i having di incident half-edges
initially. The key to our analysis is to reveal the edges (i.e.,
pairings of the half-edges) only as they are needed, just as in
a similar analysis in Janson and L. (2007) and Janson and L.
(2009).
I
At each time t, each vertex is either susceptible, infective or
recovered, and a half-edge has the same type as its vertex.
We start at time 0 with the given initial sequence of types of
the n vertices.
I
A half-edge not yet paired with another is free – initially all
half-edges are free.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Events
I
Infection Each infective half-edge has an exponential clock
with intensity β. When the clock rings, the partner of the
half-edge is revealed, chosen uniformly from among the other
free half-edges. The two half-edges are connected by an edge,
and are no longer free. If the partner is a susceptible half-edge,
then its vertex becomes infective, along with all its half-edges.
I
Recovery Each infective vertex has an exponential clock with
intensity ρ. When the clock rings, the vertex and all its
half-edges become recovered.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
The process ends when there are no further infective vertices.
There may still be free half-edges: in order to find the graph, these
should be paired uniformly at random. This will not affect the
course of the epidemic.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
More notation
I
Let St , It and Rt denote the number of susceptible, infective
and recovered individuals at time t.
I
Let Xt be the number of free half-edges at time t, and let
XS,t , XI ,t and XR,t denote the number of free susceptible,
infective and recovered half-edges.
I
So St + It + Rt = n and XS,t + XI ,t + XR,t = Xt for every t.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Parameterisation θt
I
Under suitable conditions, the processes St /n, . . . , XR,t /n
converge to deterministic functions. Their limits are functions
of a parameterisation θt of time solving an ODE.
I
θt can be interpreted as the asymptotic probability that a
half-edge has not been paired with a (necessarily infective)
half-edge by time t, i.e. that an infectious contact has not
happened.
I
For a given vertex of degree k that is initially susceptible, the
probability that the vertex is still susceptible at time t is
asymptotically close to θtk .
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
Recall that
vS (θ) = αS
∞
X
pk θ k ,
k=0
so at time t the limiting fraction of susceptibles is vS (θt ).
I
Similarly,
hS (θ) = αS
∞
X
kθk pk = θvS0 (θ),
k=0
so that at time t the limiting fraction of free susceptible
half-edges is hS (θt ).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
The limiting proportion of all free half-edges is
hX (θt ) = µθt2 .
I
For free half-edges of the other types, we define
µρ
θt (1 − θt ),
β
hI (θt ) = µθt2 − hS (θt ) − hR (θt ).
hR (θt ) = µR θt +
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
The limit functions for infective and recovered vertices are
defined as follows.
I
Let bIt be the unique solution to
βhI (θt )hS (θt )
db
It =
− ρbIt , t ≥ 0, bI0 = αI ,
dt
hX (θt )
I
bt = 1 − vS (θt ) − bIt .
Also set R
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
Let
pI (θ) =
hI (θ)
hX (θ)
be the ‘infective pressure’, the proportion of infective free
half-edges.
I
If µI > 0, then there is a unique continuously differentiable
function θ : [0, ∞) → [0, 1] such that
d
θt = −βθt pI (θt ), θ0 = 1.
dt
I
There is a unique θ∞ ∈ (0, 1) with hI (θ∞ ) = 0. Further, hI is
strictly positive on (θ∞ , 1] and strictly negative on (0, θ∞ ).
I
Also, limt→∞ θt = θ∞ .
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Epidemics starting with many infectives
Theorem (Janson, L., Windridge (2014))
Suppose µI > 0.
I
Then, uniformly on [0, ∞),
p
p
XS,t /n −→ hS (θt ), XI ,t /n −→ hI (θt ),
p
p
XR,t /n −→ hR (θt ), St /n −→ vS (θt ),
p
p
bt .
It /n −→ bIt , Rt /n −→ R
I
p
Consequently, S∞ = limt→∞ St satisfies S∞ /n −→ vS (θ∞ ).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Epidemics starting with few infectives: R0 ≤ 1
I
Suppose now that µI = 0 but there is initially at least one
infective vertex with non-zero degree.
I
Recall that
R0 =
∞
β αS X
(k − 1)kpk .
ρ+β µ
k=0
I
If R0 ≤ 1, then the number of susceptible vertices that are
ever infected is op (n).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Epidemics starting with few infectives: R0 > 1
I
We choose a reference value s0 < αS ; if R0 > 1, then the
probability that the number of susceptibles falls to s0 n is
bounded away from zero. The time until this occurs is a
random variable T0 .
I
The epidemic starting from few infectives runs “slowly” until
T0 , but after that it is almost deterministic, up to the
time-shift.
I
The exact choice of s0 is not important, but we have to make
one in order to state the result precisely.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
I
Assume R0 > 1 and µI = 0.
I
There is a unique θ∞ ∈ (0, 1) with hI (θ∞ ) = 0. Further, hI is
strictly positive on (θ∞ , 1) and strictly negative on (0, θ∞ ).
I
Fix any s0 ∈ (vS (θ∞ ), vS (1)). There is a unique continuously
differentiable function θ : R → (θ∞ , 1) such that
d
θt = −βθt pI (θt ), θ0 = vS−1 (s0 ).
dt
I
Let T0 = inf{t ≥ 0 : St ≤ ns0 }. Then
lim inf n→∞ P(T0 < ∞) > 0. Furthermore, if
P
∞
k=1 knI ,k → ∞, then P(T0 < ∞) → 1.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Theorem (Janson, L., Windridge (2014))
I
Suppose µI = 0, but the initial number of infectives is at
least 1. Suppose R0 > 1, and let T0 be as above.
I
Conditional on T0 < ∞ we have, uniformly on R,
p
p
XS,T0 +t /n −→ hS (θt ), XI ,T0 +t /n −→ hI (θt ),
p
p
XR,T0 +t /n −→ hR (θt ), ST0 +t /n −→ vS (θt ),
p
p
bt .
IT0 +t /n −→ bIt , RT0 +t /n −→ R
I
Consequently, conditional on T0 < ∞, the number of
susceptibles that escape infection satisfies
p
S∞ /n −→ vS (θ∞ ).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Conditional on T0 = ∞, S0 − S∞ = op (n) in the sense that, for all
> 0, P(T0 = ∞, S0 − S∞ > n) = o(1) as n → ∞.
Similarly, XS,0 − XS,∞ = op (n), supt≥0 XI ,t = op (n),
supt≥0 (X0 − Xt ) = op (n) conditional on T0 = ∞.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
A special case: the giant component
In the special case ρ = 0, µI = 0, and µR = 0 (so almost all
individuals are susceptible at time 0 and there are no recoveries),
the above equation for θ∞ becomes
θ∞ vS0 (θ∞ )
−
2
µθ∞
=
∞
X
k
2
kpk θ∞
− λθ∞
= 0,
k=1
which is the well known equation for the size of the giant
component in the configuration model.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Volz’s equations
The differential equation for θ first appeared in biological literature
in Miller’11. It was found as a simplification of the following system
of three equations used in Volz’08 to describe the SIR process.
d
θt
dt
d
pI ,t
dt
d
pS,t
dt
= −βpI ,t θt ,
h
i
v 00 (θt )
= pI ,t βpS,t θt S0
− (β + ρ) + βpI ,t ,
vS (θt )
h
v 00 (θt ) = βpS,t pI ,t 1 − θt S0
.
vS (θt )
Malwina Luczak
Epidemics on random graphs with a given degree sequence
In Volz’s equations, pI ,t is the probability that an edge is
connected to an infected node given that it has not transmitted
infection to the target node (for us, pI ,t ≈ XI ,t /Xt ), and pS,t is the
probability that an edge is connected to a susceptible node given
that it has not transmitted infection to the target node (for us,
pS,t ≈ XS,t /Xt ).
Volz’s and Miller’s equations assume that initially there are no
recovered individuals.
Both Volz and Miller assume deterministic spread of disease, and
they do not give a formal proof of their equations.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
This model was also studied by Decreusefond, Dhersin, Moyal and
Tran’12. They analyse a more complicated, measure-valued
process corresponding to the number of edges between different
types of vertices.
Decreusefond, Dhersin, Moyal and Tran’12 prove a law of large
numbers for their measure-valued process. As a corollary, their
result yields convergence of the relevant quantities to Volz’s
equations, under the assumption that the fifth moment of the
distribution of susceptible vertices is uniformly bounded.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Near-criticality
Here we take infection rate βn and recovery rate ρn and look at
what happens just above the ‘large epidemic’ threshold, that is,
where the basic reproductive number is just above 1.
(n)
R0
βn
=
ρn + βn
P∞
(k − 1)knS,k
k=0
P∞
.
k=0 knk
We assume that R0 − 1 → 0 and n(R0 − 1)3 → ∞.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Probability of a large epidemic
From the theory of branching processes, at the start of an
epidemic, each infective individual leads to a large outbreak with
probability of the order R0 − 1.
If the size n of the population is very large, with the initial total
infectious degree XI ,0 much larger than (R0 − 1)−1 , then a large
epidemic will occur with high probability.
If the initial total infectious degree is much smaller than
(R0 − 1)−1 , then the outbreak will be contained with high
probability.
In the intermediate case, a large epidemic can occur with positive
probability, of the order exp(−cXI ,0 (R0 − 1)), for some positive
constant c.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Size of a large epidemic
Roughly, if XI ,0 is much larger than n(R0 − 1)2 , then the total
number of people infected will be proportional to (nXI ,0 )1/2 .
If XI ,0 is much smaller than n(R0 − 1)2 then, in the event that
there is a large epidemic, the total number of people infected will
be proportional to n(R0 − 1).
The intermediate case where XI ,0 and n(R0 − 1)2 are of the same
order ‘connects’ the two extremal cases.
Note that, if XI ,0 is of the same or larger order of magnitude than
n(R0 − 1)2 , then XI ,0 (R0 − 1) is very large, so a large epidemic
does occur with high probability.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Our results will be stated in terms of the related quantity
P
ρn + βn ∞
(n)
k=0 knk
αn = (R0 − 1)
.
βn
nS
Under our assumptions, both
ρn +βn
βn
and
P∞
k=0 knk
nS
are bounded, and
(n)
bounded away from zero, so αn is equivalent to R0 − 1 as a
measure of distance from criticality.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Assumptions, near-critical regime
(D1) The degree of a randomly chosen initially susceptible vertex
converges weakly to a probability distribution (pk )∞
k=0 with a finite
and
positive
mean
λ,
i.e.
n
/n
→
p
(k
≥
0),
where
S,k
S
k
P∞
kp
=
λ.
k
k=0
(D2) The third moment of the degree of a randomly chosen
susceptible vertex is uniformly integrable as n → ∞. That is, given
ε > 0, there exists M > 0 such that, for any n,
X nS,k
< ε.
k3
nS
k>M
(D3) The second moment of the P
degree of a randomly chosen
∞
2
vertex is uniformly bounded, i.e.
k=0 k nk = O(n).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Assumptions, near-critical regime
(D4) As n → ∞, αn → 0 and nS αn3 → ∞.
(D5) The total degree of infective vertices knI ,k = o(n), and the
limit
∞
.
X
ν = lim
knI ,k nS αn2
n→∞
k=0
exists (but may be 0 or ∞). Furthermore, either ν = 0 or
dI ,∗ = max{k : nI ,k ≥ 1} = o(
∞
X
nI ,k ).
k=0
(D6) p0 + p1 + p2 < 1.
(D7) lim inf n→∞ nS /n > 0.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Result for ν = 0
Let λ3 =
P∞
k=0 k(k
− 1)(k − 2)pk ; then λ3 ∈ (0, ∞).
Theorem (Janson, L., Windridge (2015+))
Suppose that (D1)–(D7) hold. Let Z be the total number of
susceptible vertices that ever get infected.
1. If ν = 0, then there exists a sequence εn → 0 such that, for
each n, w.h.p. one of the following holds.
(i)
(ii)
Z
nS αn < εn (the epidemic is small and ends prematurely), or
| nSZαn − 2λ
λ3 | < εn (the epidemic is large and its size is well
concentrated).
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Result for ν > 0
In the case ν > 0, the epidemic is always large, and its size is again
well concentrated.
Theorem (continued)
√
λ(1+ 1+2νλ3 )
.
λ3
√
p
√2λ .
P Z
1/2 −→
(nS ∞
λ3
k=0 knI ,k )
2. If 0 < ν < ∞, then
3. If ν = ∞, then
Z
nS αn
p
−→
Moreover, in cases 1(b), 2 and 3, the numbers Zk of susceptible
vertices of each degree k > 0 that get infected satisfy, w.h.p.,
∞ X
Zk
kpk Z − λ < ε.
k=0
Malwina Luczak
Epidemics on random graphs with a given degree sequence
In the case ν = 0, the epidemic may be ‘large’ or ‘small’. The next
result yields an asymptotic formula for the probability of a small
epidemic in this case.
Theorem (Janson, L., Windridge (2015+))
Suppose assumptions (D1)–(D7) are satisfied, and that ν = 0.
P
1. If αn ∞
k=0 knI ,k → 0, then case 1(a) in the previous theorem
occurs w.h.p.
P
2. If αn ∞
k=0 knI ,k → ∞, then case 1(b) in the previous
theorem occurs w.h.p.
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Theorem (continued)
P
3. If αn ∞
k=0 knI ,k is bounded above and below, then both
cases 1(a) and 1(b) occur with probabilities
P∞ bounded away
from 0 and 1. Furthermore, if dI ,∗ = o( k=0 knI ,k ), then the
probability that case 1(a) occurs is
!
P
∞
X
λ2 + λ + ∞
k=0 knR,k /nS
αn
knI ,k + o(1).
exp −
λ2 λ3
k=0
Moreover, in the case where the epidemic is small,
Z = Op (αn−2 ).
In the simple graph
case, for part 3. weP
need the additional
P
2n
k
k 2 nR,k = o(n).
assumptions:
=
o(n)
and
I ,k
k≥1
k≥α−1
n
Malwina Luczak
Epidemics on random graphs with a given degree sequence
Time change
I
If we are in a state with a total of x free half-edges, including
xI infective half-edges, then we multiply all transition rates by
(x − 1)/βxI > 0. This time change accelerates slow parts of
the epidemic, when there are few infectives.
I
Thus each infective vertex recovers at rate ρ(x − 1)/βxI , and
each infective half-edge pairs off at rate (x − 1)/xI .
I
Equivalently, each susceptible half-edge gets infected at unit
rate.
Malwina Luczak
Epidemics on random graphs with a given degree sequence