RD processes on heterogeneous metapopulations: Continuous-time formulation and simulations

wANPE08 – December 15-17, Udine

Joan SaldañaUniversitat de Girona

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Outline of the talk

1. Introduction1. SIS model with homogeneous mixing

2. Epidemic models on contact networks 1. Regular (homogeneous) random networks

2. Complex random networks

3. EM on complex metapopulations 1. Discrete-time diffusion

2. Continuous-time diffusion

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SIS model

The force of infection λ = rate at which susceptible individuals become infected

Proportional to the number of infective contacts

µ = recovery rate

ISdt

dISI

dt

dS ,

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Homogeneous mixing

Any infective is equally likely to transmit the disease to any susceptible

λ = transmission rate across an infective contact x contact rate x proportion of infective contacts

= β · c · I / N

If c ≈ N → λ ≈ β · I (non-saturated) If c ≈ 1 → λ ≈ β · I / N (saturated) (c is the average rate at which new contacts are made and can

take into account other aspects like duration of a contact, etc.)

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Basic reproductive number

= Average number of infections produced by an infective individual in a wholly susceptible

population

= c ·β ·T = c ·β ·1/μ

In a non-homogeneous mixing,

c ~ structure of the contact network → Consider the probability of arriving at an infected

individual across a contact instead of considering the fraction of infected individuals !!

0R

0R

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Contact network epidemiology

What are the implications of network topology for epidemic dynamics?

(May 2001; Newmann 2002; Keeling et al. 1999, 2005; Cross et al. 2005, 2007; Pastor-Satorras & Vespignani 2001, …; Lloyd-Smith et al. 2005; etc. )

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Contact network epidemiology

(Meyers et al. JTB 2005)

Meyers et al.JTB 2005

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Complex contact networks

The contact structure in the population is given by

the degree (or connectivity) distribution P(k)

the conditional probability P(k’|k)

If these two probabilities fully determine the contact structure → Markovian networks

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Degree distributions

Poisson: non-growing random networks

Exponential: growing networks with new nodes randomly attached without preference

Scale free (power law): preferential growing networks → existence of highly connected nodes (= superspreaders)

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Network architectures

Meyers et al.JTB 2005

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A special degree distribution

Distribution of the degrees of nodes reached by following a randomly chosen link:

which has < k² > / < k > as expected value.

This is the value to be considered for c !!

k

kPkkq

)()(

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for contact networks

2

2

0 )(1 CVkTk

kTR

kTR 0

For this value of c, we have

(Anderson & May 1991; Lloyd & May 2001; May & Lloyd 2001)(Pastor-Satorras & Vespignani 2001, Newmann 2002)

For regular random networks, CV = 0 and hence

0R

Absence of epidemic threshold

in SF networks!!

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Epidemics on metapopulations

Schematically (Colizza, Pastor-Satorras & Vespignani, Nature Physics 2007):

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An example

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More modern examples

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A nice picture of the 1st example

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Modern examples

(Colizza et al., PNAS 2006)

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Global invasion threshold

is not sufficient to predict the invasion success at the metapopulation level with small local population sizes (Ball et al. 1997; Cross et al. 2005, 2007)

Disease still needs to spread to different populations

= number of subpopulations that become infected from a single initially infected population Size of the local population (N), Rate of diffusion among populations (D) the length of the infectious period (1/μ).

(Cross et al. 2005, 2007)

0R

*R

*R

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An alternative approach

Consider a complex metapopulation as a structured population of nodes classified by their connectivity (degree)

Include local population dynamics in each node

Forget about the geographical location of nodes and consider only the topological aspects of the network

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Global invasion threshold *R

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Global invasion threshold

In a regular random network with

Similar expressions can be derived for complex metapopulations. For instance, if D = const,

*R

111

11 00

*

R

k

kN

D

RkR

iN

110 R

(Colizza & Vespignani

Phys.Rev.Lett. 2007,

JTB 2008)

102

2

*

Rk

kkN

DR

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A discrete-time model

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Assumptions

The spread of a disease is assumed to be two sequential (alternate) processes:

1) Reaction (to become infected or to recover) Homogeneous mixing at the population level

2) Diffusion: A fixed fraction of individuals migrate at the end of each time interval

(after react !!)

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Transmission rates

In type-I (non-saturated) spreading:

In type-II (saturated) spreading:

kk 0

kBkAkk

k ,,0 with

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The discrete equations

'

1)|'()(

))(1()(

'

1)|'()(

))(1()(

',','

'',',

,,,,,

',','

'',',

,,,,,

kkkPDk

Dt

kkkPDk

Dt

kBkAk

kkBkBB

kBkBkAkkBBkB

kBkAk

kkBkAA

kBkAkkBkAAkA

Susceptible individuals:

Infected individuals:

Diffusion at theend of the

time interval

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The continuous equations

'

1)|'()(

))(1(

'

1)|'()(

))(1(

',','

'',',

,,,,,,

',','

'',',

,,,,,,

kkkPDk

Ddt

d

kkkPDk

Ddt

d

kBkAk

kkBkBB

kBkBkAkkBBkBkB

kBkAk

kkBkAA

kBkAkkBkAAkAkA

Taking the approximation dρ/dt ≈ ρ(t + 1) – ρ(t) it follows:

(Colizza et al., Nature Physics 2007)

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For sequential Type-I processes

1 BA DD

(Colizza et al., Nature Physics 2007)

The number of infectives and susceptibles are linear in the node degree k → Diffusion effect

Constant prevalenceacross the metapopulation

Lack of epidemic threshold in SF networks

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For sequential Type-II processes

1 BA DD

(Colizza et al., Nature Physics 2007)

The number of infectives and susceptibles are linear in the node degree k → Strong diffusion effect

Constant prevalenceacross the metapopulation

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The continuous-time model

The limit of the discrete model as τ → 0 is not defined !!!

→ The previous equations are not the continuous time limit of the discrete equations !!

Assuming uniform diffusion during each time interval (with probability τ ·Di), the limit as τ → 0 becomes well-defined and one obtains …

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The discrete equations

'

1)|'()(

))(1()(

'

1)|'()(

))(1()(

',','

'',',

,,,,,

',','

'',',

,,,,,

kkkPDk

Dt

kkkPDk

Dt

kBkAk

kkBkBB

kBkBkAkkBBkB

kBkAk

kkBkAA

kBkAkkBkAAkA

Susceptible individuals:

Infected individuals:

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The limit equations

','

,,,,

','

,,,,

'

1)|'()()(

'

1)|'()()(

kBk

BkBBkAkkBkBt

kAk

AkAAkAkkBkAt

kkkPDkDt

kkkPDkDt

Susceptible individuals:

Infected individuals:

(Saldaña, Phys. Rev. E 78 (2008))

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Conserv. of number of particles

Consistency relation between P(k) and P(k’|k):

Mean number of particles:

Conservation of the number of particles:

)'|()'(')|'()( kkPkPkkkPkPk

0)]()([

ttt BA

)()()( , tkPt kik

i

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Equilibrium equations

*,

'

*,

*,

**,

*',

'

*,

*,

**,

'

1)|'()(

'

1)|'()(

kBk

kBBkAkkB

kAk

kAAkAkkB

kkkPkD

kkkPkD

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Uncorrelated networks

In uncorrelated networks:

= Degree distribution of nodes that we arrive at by following a randomly chosen link

)()'('

)|'( kqk

kPkkkP

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Equilibrium equations in U.N.

**,

*,

**,

**,

*,

**,

)(

)(

BkBBkAkkB

AkAAkAkkB

k

kD

k

kD

*

,**

,* )(,)( kB

kBkA

kA kPkP

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Disease-free equilibrium

In this case, and

→ the number of individuals is linear in the

node degree k → Diffusion effect

0*, kB

0*,

k

kkA

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Endemic equilibrium in type-II

Saturation in the transmission of the infection → all the local populations have the equal

prevalence of the disease:

0*

*

0

*,

*

0

*, 1,

k

kk

kkBkkA

All are linear in thedegree k

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Endemic equilibrium in type-II

Therefore, the condition for its existence at the metapopulation level is the same as the one for each subpopulation:

There is no implication of the network topology for the spread and prevalence of the disease

100

R

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Endemic equilibrium in type-I

Increase of the prevalence with node degree (being almost linear for large k)

Absence of epidemic threshold in networks with unbounded maximum degree

There is an implication of the network topology for the spread and prevalence of the disease

When DA = DB, the size a each population is linear with k, as in type-II

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A sufficient condition in type I

The disease-free equilibrium will be unstable whenever the following condition holds:

This condition follows from the localization of the roots of the Jacobian matrix J of the linearized system around the disease-free equilibrium

0max

0

BD

k

k

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A sufficient condition in type I

Precisely, with

The roots of being simple and satisfying

B

CAJ

0)()()( BAJ pDp

)(

1

00

max

00

00

B

BB

Dk

k

Dk

kD

k

k

maxk

k

λ

λ

)(Bp

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A remark on the suff. condition

For regular random networks, k = <k> and the condition reads as

which is more restrictive than the n. & s. condition that follows directly from the model, namely,

100

BD

100

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Simulations under type-I trans.

(Saldaña, Phys.Rev. E 2008)

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Monte Carlo simulations

(Baronchelli et al., Phys .Rev. E 78 (2008))Not when

D and R occur simultaneously !!

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Monte Carlo simulations - 2

The length τ of the time interval must be small enough to guarantee that events are disjointThe diffusing prob. of susceptibles and infectives

are τ·DA and τ·DB , respectivelyThe prob. of becoming infected after all the

infectious contacts is σ = σ(τ,k) τμ is the recovering probability

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Monte Carlo simulations - 3

For infective individuals

For susceptible individuals

with

BB D

D

11

1)( kAD

kkkkB /1)1(1)( , and

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Monte Carlo simulations - 4

This last inequality can be rewritten as

If we consider the minimum of these τ’s over the network: , the value of τ we take for each time step is

)(* t

)()1( *)(, tD kt

kAkB

)(,

1,

1min)( * t

Dt

kB

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MC simulations for type-II trans.

nodes5000

1

5.1

30

BA DD

(Juher, Ripoll,Saldaña, in preparation)

The same output as withdiscrete-time diffusion

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MC simulations for type-I trans.

nodes5000

1

3

5.10

BA DD

(Juher, Ripoll,Saldaña, in preparation)

Prevalence is NOTconstant with k !!

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Future work

Analytical study of the properties of the equilibrium in type-I transmission

More general diffusion rates (for instance, depending on the population degree)

Impact of degree-degree correlations

Introduction of local contact patterns