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