Epidemics on random graphs with a given degree sequence Malwina - - PowerPoint PPT Presentation

Introduction The model Results Results about near-criticality

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

14 April 2015 Workshop on Limit Shapes ICERM

1Joint work with Svante Janson and Peter Windridge 2The work of Luczak and Windridge was supported by EPSRC Leadership

Fellowship, grant reference EP/J004022/2.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

SIR process on a graph

◮ The SIR process is a simple Markovian model for a disease

spreading around a finite population.

◮ Each individual is either susceptible, infective or recovered. ◮ Individuals are represented by vertices in a graph G; edges

correspond to potentially infectious contacts.

◮ Susceptible vertices become infective at rate β times the

number of infective neighbours in G.

◮ Infective vertices become recovered at rate ρ.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Random graphs with a given degree sequence

Take a given degree sequence (di)n

i=1 = (d(n) i

)n

i=1, and let Gn be

uniform over all graphs with this degree sequence. (We assume n

i=1 di is even.)

In fact, we prove our results by first studying a suitable random multigraph G ∗

n (allowing multiple edges and loops) with the given

degree sequence. G ∗

n 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

Introduction The model Results Results about near-criticality

Random graphs with a given degree sequence

If we assume that degree sequences satisfy n

i=1 d2 i = O(n) (i.e.

the second moment of the degree of a random vertex is bounded), then lim inf

n→∞ P(G ∗ n is simple) > 0.

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 G ∗

n is simple.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Initial conditions

◮ At time 0 there are nS = n(n) S

susceptible, nI = n(n)

I

infective, and nR = n(n)

R

recovered vertices (individuals), with nS + nI + nR = n.

◮ Of these, nS,k = n(n) S,k, nI,k = n(n) I,k and nR,k = n(n) R,k,

respectively, have degree k (k = 0, 1, . . . ).

◮ Thus nS = ∞ k=0 nS,k, nI = ∞ k=0 nI,k, nR = ∞ k=0 nR,k. ◮ Also nS,k + nI,k + nR,k = nk, the total number of vertices of

degree k.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Initial conditions

◮ As n → ∞, the fractions of initially susceptible, infective and

recovered individuals converge to constants αS, αI and αR. That is nS n = ∞

k=0 nS,k

n → αS; nI n = ∞

k=0 nI,k

n → αI; nR n = ∞

k=0 nR,k

n → αR. Thus αS + αI + αR = 1.

◮ αS > 0.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Initial susceptible degree distribution

◮ The degree of a random initially susceptible individual has an

asymptotic probability distribution (pk)∞

0 , i.e.,

lim

n→∞

nS,k nS = pk, k ≥ 0.

◮ This distribution (pk) has finite mean λ:

λ =

∞

• k=0

kpk < ∞.

◮ The average degree of the susceptible vertices converges to λ:

∞

k=0 knS,k

nS → λ.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Average degree of vertices

◮ The average degree of all vertices converges to a constant µ,

as n → ∞: 1 n

n

• i=1

di = 1 n

∞

• k=0

knk → µ.

◮ For the infected and recovered vertices, we have

1 n

∞

• k=0

knI,k → µI; 1 n

∞

• k=0

knR,k → µR; for some constants µI and µR.

◮ Thus µ = αSλ + µI + µR. ◮ Also we assume that i d2 i = O(n).

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

◮ Let S∞ be the number of infectives that ultimately escape

infection.

◮ Let vS be the scaled pgf of the asymptotic susceptible degree

distribution: vS(x) = αS

∞

• k=0

pkxk.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Theorem

Suppose µI = 0, let Z = nS − S∞, the number of susceptible vertices that ever get infected, and set R0 =

• β

ρ + β αS µ

k

k(k − 1)pk. (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

• ccurrence of a large epidemic,

S∞ n

p

− → vS(θ∞), where θ∞ is the solution to a fixed-point equation.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

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

Introduction The model Results Results about near-criticality

States of the process

◮ We consider a stochastic process on the given set of vertices,

with each vertex i having di incident half-edges. The key to

• ur 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.’07 and Janson and L.’09.

◮ 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 some given initial sequence of types

• f the n vertices.

◮ 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

Introduction The model Results Results about near-criticality

Events

◮ 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.

◮ 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

Introduction The model Results Results about near-criticality

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

Introduction The model Results Results about near-criticality

More notation

◮ Let St, It and Rt denote the number of susceptible, infective

and recovered individuals at time t.

◮ 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.

◮ 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

Introduction The model Results Results about near-criticality

Parameterisation θt

◮ Under suitable conditions, the processes St/n, . . . , XR,t/n

converge to deterministic functions. Their limits are functions

• f a parameterisation θt of time solving an ODE.

◮ 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.

◮ 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 θk

t .

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

◮ Recall that

vS(θ) = αS

∞

• k=0

pkθk, so at time t the limiting fraction of susceptibles is vS(θt).

◮ Similarly,

hS(θ) = αS

∞

• k=0

kθkpk = θv′

S(θ),

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

Introduction The model Results Results about near-criticality

◮ The limiting proportion of all free half-edges is

hX(θt) = µθ2

t . ◮ For free half-edges of the other types, we define

hR(θt) = µRθt + µρ β θt(1 − θt), hI(θt) = µθ2

t − hS(θt) − hR(θt).

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

◮ The limit functions for infective and recovered vertices are

defined as follows.

◮ Let

It be the unique solution to d dt

• It = βhI(θt)hS(θt)

hX(θt) − ρ It, t ≥ 0, I0 = αI,

◮ Also set

Rt = 1 − vS(θt) − It.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

◮ Let

pI(θ) = hI(θ) hX(θ) be the ‘infective pressure’, the proportion of infective free half-edges.

◮ We prove that there is a unique continuously differentiable

function θ : [0, ∞) → [0, 1] such that d dt θt = −βθtpI(θt), θ0 = 1.

◮ We also prove that 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∞).

◮ Then limt→∞ θt = θ∞.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Epidemics starting with many infectives

Theorem

Suppose µI > 0.

◮ Then, uniformly on [0, ∞),

XS,t/n

p

− → hS(θt), XI,t/n

p

− → hI(θt), XR,t/n

p

− → hR(θt), St/n

p

− → vS(θt), It/n

p

− → It, Rt/n

p

− → Rt.

◮ Consequently, S∞ = limt→∞ St satisfies S∞/n p

− → vS(θ∞).

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Epidemics starting with few infectives: R0 ≤ 1

◮ Suppose now that µI = 0 but there is initially at least one

infective vertex with non-zero degree.

◮ Recall that

R0 = β ρ + β αS µ

∞

• k=0

(k − 1)kpk.

◮ 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

Introduction The model Results Results about near-criticality

Epidemics starting with few infectives: R0 > 1

◮ We choose a reference value s0 < αS; if R0 > 1, then the

probability that the number of susceptibles falls to s0n is bounded away from zero. The time until this occurs is a random variable T0.

◮ The epidemic (in real time) starting from few infectives runs

“slowly” until T0, but after that it is almost deterministic, up to the time-shift.

◮ The exact choice of s0 is not important, but we have to make

• ne in order to state the result precisely.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

◮ Assume R0 > 1. ◮ 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∞).

◮ Fix any s0 ∈ (vS(θ∞), vS(1)). There is a unique continuously

differentiable function θ : R → (θ∞, 1) such that d dt θt = −βθtpI(θt), θ0 = v−1

S (s0). ◮ Let T0 = inf{t ≥ 0 : St ≤ ns0}. Then

lim infn→∞ P(T0 < ∞) > 0. Furthermore, if ∞

k=1 knI,k → ∞, then P(T0 < ∞) → 1.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Theorem

◮ Suppose µI = 0, but the initial number of infectives is at

least 1. Suppose R0 > 1, and let T0 be as above.

◮ Conditional on T0 < ∞ we have, uniformly on R,

XS,T0+t/n

p

− → hS(θt), XI,T0+t/n

p

− → hI(θt), XR,T0+t/n

p

− → hR(θt), ST0+t/n

p

− → vS(θt), IT0+t/n

p

− → It, RT0+t/n

p

− → Rt.

◮ Consequently, conditional on T0 < ∞, the number of

susceptibles that escape infection satisfies S∞/n

p

− → vS(θ∞).

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Conditional on T0 = ∞, S0 − S∞ = op(n) in the sense that, for all ǫ > 0, P(T0 = ∞, S0 − S∞ > ǫn) = 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

Introduction The model Results Results about near-criticality

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∞v′

S(θ∞) − µθ2 ∞ = ∞

• k=1

kpkθk

∞ − λθ2 ∞ = 0,

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

Introduction The model Results Results about near-criticality

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

• f three equations used in Volz’08 to describe the SIR process.

d dt θt = −βpI,tθt, d dt pI,t = pI,t

• βpS,tθt

v′′

S(θt)

v′

S(θt) − (β + ρ) + βpI,t

• ,

d dt pS,t = βpS,tpI,t

• 1 − θt

v′′

S(θt)

v′

S(θt)

• .

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

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

Introduction The model Results Results about near-criticality

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

Introduction The model Results Results about near-criticality

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. R(n) = βn ρn + βn ∞

k=0(k − 1)knS,k

∞

k=0 knk

. We assume that n(R0 − 1)3 → ∞.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

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

Introduction The model Results Results about near-criticality

Size of a large epidemic

Broadly speaking, 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

• rder ‘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

Introduction The model Results Results about near-criticality

Our results will be stated in terms of the related quantity αn = (R(n) − 1) ρn + βn βn ∞

k=0 knk

nS . Under our assumptions, both ρn+βn

βn

and

∞

k=0 knk

nS

are bounded, and bounded away from zero, so αn is equivalent to R(n) − 1 as a measure of distance from criticality.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

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. nS,k/nS → pk (k ≥ 0), where ∞

k=0 kpk = λ.

(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,

• k>M

k3 nS,k nS < ε. (D3) The second moment of the degree of a randomly chosen vertex is uniformly bounded, i.e. ∞

k=0 k2nk = O(n).

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Assumptions, near-critical regime

(D4) As n → ∞, αn → 0 and nSα3

n → ∞.

(D5) The total degree of infective vertices knI,k = o(n), and the limit ν = lim

n→∞ ∞

• k=0

knI,k

• nSα2

n

exists (but may be 0 or ∞). Furthermore, either ν = 0 or dI,∗ = max{k : nI,k ≥ 1} = o(

∞

• k=0

nI,k). (D6) p0 + p1 + p2 < 1. (D7) lim infn→∞ nS/n > 0.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Result for ν = 0

Let λ3 = ∞

k=0 k(k − 1)(k − 2)pk; then λ3 ∈ (0, ∞).

Theorem

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)

Z nSαn < εn (the epidemic is small and ends prematurely), or

(ii) |

Z nSαn − 2λ λ3 | < εn (the epidemic is large and its size is well

concentrated).

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

Result for ν > 0

In the case ν > 0, the epidemic is always large, and its size is again well concentrated.

Theorem (continued)

• 2. If 0 < ν < ∞, then

Z nSαn p

− → λ(1+√1+2νλ3)

λ3

.

• 3. If ν = ∞, then

Z (nS ∞

k=0 knI,k)1/2

p

− →

√ 2λ √λ3 .

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.,

∞

• k=0
• Zk

Z − kpk λ

• < ε.

Malwina Luczak Epidemics on random graphs with a given degree sequence

Introduction The model Results Results about near-criticality

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

Suppose assumptions (D1)–(D7) are satisfied, and that ν = 0.

• 1. If αn

∞

k=0 knI,k → 0, then case 1(a) in the previous theorem

• ccurs w.h.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

Introduction The model Results Results about near-criticality

Theorem (continued)

• 3. If αn

∞

k=0 knI,k is bounded above and below, then both

cases 1(a) and 1(b) occur with probabilities bounded away from 0 and 1. Furthermore, if dI,∗ = o(∞

k=0 knI,k), then the

probability that case 1(a) occurs is exp

• −λ2 + λ + ∞

k=0 knR,k/nS

λ2λ3 αn

∞

• k=0

knI,k

• + o(1).

Moreover, in the case where the epidemic is small, Z = Op(α−2

n ).

In the simple graph case, for part 3. we need the additional assumptions:

k≥1 k2nI,k = o(n) and k≥α−1

n k2nR,k = o(n). Malwina Luczak Epidemics on random graphs with a given degree sequence