Download - Deterministic models: twenty years on
Deterministic models: twenty years on
* Mathematics InstituteUniversity of Warwick
Infectious Disease Dynamics, Cambridge, 19th August 2013
Mick Roberts & Lorenzo Pellis*
II. Spatially inhomogeneous models
Outline
Pair formation models
Metapopulation models
Spatially explicit models
Households models Overview Reproduction numbers Time-related quantities
Network models Key epidemiological quantities (Almost) exact dynamics Approximate dynamics
Comments
PAIR FORMATION MODELS
History
Introduced to better model STD transmission in partnerships
History: First one to model pairs (in demography): Kendall (1949), “Stochastic processes and population growth”
Pair separation added:Yellin & Samuelson (1974), “A dynamical model for human populations”
For STDs:Dietz (1988), “On the transmission dynamics of HIV”
Dietz & Hadeler (1988), “Epidemiological models for sexually transmitted diseases”
Kretzschmar & Dietz (1988), “The effect of pair formation and variable infectivity on the spread of an infection without recovery”
Definition of for models with pairs:Diekmann, Dietz & Heesterbeek (1991), “The basic reproduction ratio for sexually transmitted diseases. Theoretical considerations”
Dietz, Heesterbeek & Tudor (1993), “The basic reproduction ratio for sexually transmitted diseases. II. Effects of variable HIV-infectivity”
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The basic idea
Kretzschmar (2000)
Kretzschmar & Dietz (1998)
Comparison with homogeneous mixing
Very fast pair dynamics lead back to homogeneous mixing
Pairs reduce the spread (and ) thanks to sequential infection: pairs of susceptibles are protected pairs of infectives “waste” infectivity
Comparison at constantKretzschmar & Dietz (1998), “The effect of pair formation and variable infectivity on the spread of an infection without recovery”
Endemic equilibrium is always higher The same can have 2 growth rates and 2 endemic equilibria,
so one needs information about partnership dynamics (for both prediction and inference)
Acute HIV less important and asymptomatic stage more than expected (unless partnerships are very short)
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Extensions
No concurrency: Heterosexual (many) With various infectious stages (many) With heterogeneous sexual activity With maturation period Hadeler (1993)
Two types of pairs (“steady” VS “casual”) Kretzschmar et al (1994)
With concurrency Instantaneous infection from outside a stable partnership Watts & May (1992)
Pairwise approximation on networkFerguson & Garnett (2000); Eames & Keeling (2002)
Full network models (Monte Carlo)
Concurrency
Working definition: the mean degree of the line graphMorris & Kretzschmar (1997), “Concurrent partnerships and the spread of HIV”
Impact: always bad as it amplifies the impact of many partners Faster forward spread (no protective sequencing) Backwards spread Dramatic effect on connectivity and resilience of the sexual network
Quantitative impact:
2
1
( ) ,, size of giant component e tr I t
METAPOPULATION MODELS
History
Born in ecology, but cross-fertilisation with epidemiology
Ideas lurking around before 1970
First formalisation: patches can be occupied or unoccupiedLevins (1969), Some demographic and genetic consequences of environmental heterogeneity for biological control
Extensions: Patches with different sizes Hanski (many papers: 1982, 1985)
Patches with 2 internal states Gyllenberg & Hanski (1991)
Within patch dynamics Hastings & Harrison (1994)
Since the beginning, aim is to study extinction mostly stochastic
Basic model
General formulation:
It is a multitype model, but usually: Spatial interpretation SIS (or SIR with demography), to study oscillations, extinction
and (a)synchrony
Simple case of 2 population (with coupling parameter ):
i i i i i i
i i i i i i i i ij j j
i i i i i
j
S B S d S
I S d I N
R d R
I I
I
1 21 1 1
1 21 1 1 1
(1 )
(1 )
IS bN dS
NI
I
IS
IS I dI
N
2 12 2 2
2 12 2 2 2
(1 )
(1 )
IS bN dS
NI
I
IS
IS I dI
N
A mechanistic approach
Special case of 2 populations Keeling & Rohani (2002)
number of susceptibles with home in but who are now in rate of leaving home rate of coming home
Let fraction of time away from home
Then, assuming fast movements compared to disease history:
Correlation surprisingly well fit by
( estimated from data; 2 populations)
xyS x y
xx yxxx xx xx xy xx
xx yx
S bNN
I IdS S S
N
2
2
( )1xy
xx
N
N
C
Important insights
Coupling: no independent subpopulations strong synchrony weak interesting behaviour
Critical community size: Stochastic extinction likely in small subpopulations unlikely in large populations
Rescue effect: Large populations keep “feeding” small populations
Asynchrony: Can increase overall persistence
Application to measles
Simple deterministic SEIR Good fit to data (biennial oscillations) and good CCS estimates
Add age structure and schools Better fit to data
Make the model stochastic Wrong CCS estimates by an order of magnitude
Possible improvements: Metapopulation models Cellular automata or pairwise models More realistic latent and infectious period
Metapopulation models seems to be key to explain the pattern of measles post-vaccination era (much lower prev, still no extinction)
SPATIALLY EXPLICIT MODELS
Reaction-diffusion models
Based on diffusion and PDEs: a lot older than 20 years
Used when space should be treated continuously spatial proximity is key to transmission
Based on the concept of Brownian motion reasonable for dispersing animal populations
No explicit solution, but analytic expression for the asymptotic travelling wave in isotropic environment
Successful applications: Fox rabies: Murray, Stanley & Brown (1986)
Bubonic plague: Nobel (1974)
Dynamics: where:
Travelling waves ( ):
Basic equations
Extensions: kernel-based models
Reaction-diffusion model are inadequate for, e.g. plants (and wind-bourne spore dispersal) farms (and market trade) stationary agents (and long-range dispersal)
Kernel-based models: When probability of transmission decreases with distance Typically stochastic (also deterministic with uniform host density) Kernel usually homogeneous and isotropic:
Analytic results: Kernel decreases exponentially with distance travelling
waves Kernel too fat forward jump to new foci
( , ) ( , ) ( )dx t I y t K x y y
HOUSEHOLDS MODELS
Overview
Reproduction numbers
Time-dependent quantities
History
First ‘attempt’ with many large groups:Bartoszyński (1972), “On a certain model of an epidemic”
Highly infectious disease:Becker & Dietz (1995)
Milestone ( , final size and vaccination ‘equalizing’ strategy):Ball, Mollison & Scalia-Tomba (1997)
Overlapping groups:Ball & Neal (2002)
Real-time growth rate (approximate):Fraser (2007); Pellis, Ferguson & Fraser (2010)
Many reproduction numbers:Goldstein et al (2009); Pellis, Ball & Trapman (2012)
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HOUSEHOLDS MODELS
Overview
Reproduction numbers
Time-dependent quantities
Motivation
Strongest interaction
Lowest level in a hierarchical society
Concept of household is reasonably well defined (some issues)
Data availability
Homogeneous mixing is reasonably justified
Natural target of intervention
Laboratory for detailed parameter estimationCauchemez (2004); Cauchemez et al. (2009); Donnelly et al (2011)
Households model
Considerations
Mostly SIR. Some exceptions:Ball (1999); Neal (2006)
Mostly within-household density dependence. Some exceptions:Cauchemez (2004); Cauchemez et al (2009); Fraser (2007); Pellis, Ferguson & Fraser (2009, 2011)
Mostly stochastic, because of small groups Unless Markovian model using the Master equation House & Keeling (2008)
But many results (reproduction numbers, real-time growth rate) are ‘almost’ deterministic
Early phase (with homogeneous global mixing and large population and lack of prior immunity) is a lot simpler: The household is infected only once It can be treated as a super-individual
HOUSEHOLDS MODELS
Overview
Reproduction numbers
Time-dependent quantities
Household reproduction number R
Consider a within-household epidemic started by one initial case
Define: average household final size,
excluding the initial case average number of global
infections an individual makes
Linearise the epidemic process at the level of households:
: 1G LR
L
G
Individual reproduction number RI
Attribute all further cases in a household to the primary case
is the dominant eigenvalue of :
0G G
IL
M
IR IM
41 1
2G L
IG
R
Individual reproduction number RI
Attribute all further cases in a household to the primary case
is the dominant eigenvalue of :
0G G
IL
M
IR IM
41 1
2G L
IG
R
R0 – naïve construction
Consider a within-household epidemic with a single initial case.
Type of an infective = generation they belong to.
= expected number of cases in each generation
= average number of global infections from each case
The next generation matrix is:
Then: Pellis, Ball & Trapman (2012)
0 1 2 11, , ,...,Hn
1
2 1
1
0
0H H
G G G G G
n n
K
G
0 ( )R K
Recall that, without households,
Assume
Assume a perfect vaccine
Define as the fraction of the population that needs to be vaccinated (at random) to reduce below 1
Then
such that
Vaccine-associated reproduction number RV
1R
CpR
1:1
V
C
Rp
0
11
: Cp R
11 C
V
pR
Conclusions
Comparison between reproduction numbers:Goldstein et al (2009); Pellis, Ball & Trapman (2012)
At the threshold:
In a growing epidemic:
In a declining epidemic:
, so vaccinating is not enough
But bracketed between two analytically tractable approximations
0VR R0
11 p
R
1R 1IR 1VR 0 1R
R IR VR 0R
R IR 0R
VR
Extensions
Overlapping groups model: Ball & Neal (2002)
Clump-to-clump reproduction number
Households-workplaces model: Pellis, Ferguson & Fraser (2009)
Household reproduction number Workplace reproduction number
Household-network model: Ball, Sirl & Trapman (2009)
Household reproduction number
Basic reproduction number calculated for all these extensionsPellis, Ball & Trapman (2012)
R
HR
WR
R
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HOUSEHOLDS MODELS
Overview
Reproduction numbers
Time-dependent quantities
Time-related quantities
Real-time growth rate Linearise at the level of households:
where is the infectivity profile of a household Markovian model: can be found exactly using CTMC Non-Markovian model: only approximate resultsFraser (2007); Pellis, Ferguson & Fraser (2010)
Full dynamics: Markovian model and Master equation House & Keeling (2008)
( )H 0
( ) 1e drH
r
( )H
, , , , , 1, 1
, , 1, 1,
, , 1, 1,
( ) ( 1) ( )
( ) ( 1
( )
)( 1) ( )
( ) ( ) ( 1) ( )
x y z x y z x y z
x y z x y z
x y z x y z
H t yH
xyH
x
t y H t
t x y H t
I t t x H tH
NETWORKS
Key epidemiological quantities
(Almost) exact dynamics
Approximate dynamics
History
A few milestones Roots in graph theory: Euler (1736) on the 7 bridges of Köningsberg
Random graph theory: Erdős & Rényi (1959); Gilbert (1959) Small world network: Watts & Strogatz (1998)
Scale-free network: Barabási & Albert (1999); Bollobás et al (2001)
Many different branches: Static VS dynamic Small VS large Clustered VS unclustered Correlated VS uncorrelated Weighted VS unweighted Markovian VS non-Markovian SIS VS SIR
NETWORKS
Key epidemiological quantities
(Almost) exact dynamics
Approximate dynamics
R0 for SIR – basics
Simple case: large, static, unclustered, unweighted, - regular
is bounded
Repeated contacts: Markovian model: prob of transmission
instead of
First infective is special: All others have 1 link less to use Formally:
20 1| 1R X X E
0R
p
p
n
R0 for SIR – basics
Simple case: large, static, unclustered, unweighted, - regular
is bounded
Repeated contacts: Markovian model: prob of transmission
instead of
First infective is special: All others have 1 link less to use Formally:
20 1| 1R X X E
0R
p
p
n
R0 for SIR – extensions
Degree-biased (or excess degree) distribution: A node of degree is times more likely to be reached than a
node of degree 1 A node of degree is times more susceptible and times
more infectious So ( mean and variance):
Next-generation matrix (NGM) approach: Degree correlation Weighted networks Bipartite networks
Dynamic network: Slow dynamics Fast dynamics
Markovian example:
d d
d
0 1dd
d
R p
0[
[ 1)
]
]( )(lm lmK p R
m m
lm mK
d 1d
d d
0 1dd
d
R
0
dd
d
R
R0 for SIR – clustering
Clustering by trianglesNewman (2009), “Random graphs with clustering”
Clustering by asymptotic expansion on transmissibility (and weighted network)
Miller (2009), “Spread of infectious diseases in clustered networks”
Clustering by households: Use tools from households models to get and Ball, Sirl & Trapman (2010), “Analysis of a stochastic SIR epidemic on a random network incorporating household structure”
Pellis, Ball & Trapman (2012), “Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R0”
Ball, Britton & Sirl (2013), “A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon”
There is a lot more than just this type of clustering:House & Del Genio (in preparation)
R 0R
Real-time growth rate
Assume: Infection rate on each link at time after infection The network is static and independent from the infectivity profile No clustering
Then the real-time growth rate is the solution of:
where
The story is not very different from
( )
r
0
( 1)e dr
0( )
1( ) ( )d
es s
dd
d
0R
NETWORKS
Key epidemiological quantities
(Almost) exact dynamics
Approximate dynamics
Small populations
Requires a Markovian model (constant rates)
Master equation:
is the distribution over all possible system states Problem: curse of dimensionality
Automorphism-driven lumping: Simon, Taylor & Kiss (2011)
Allows reducing the number of equations by exploiting symmetry Still curse of dimensionality No reduction at all if there are no symmetries
d
d
p
tQp
p
Large populations
Requirements: Markovian model and configuration network
Basic deterministic models: Effective degree (ED) model: Ball & Neal (2008), “Network epidemic models with two levels of mixing”
‘Probability generating function’ (PGF) model:Volz (2008), “SIR dynamics in random networks with heterogeneous connectivity”
Miller (2011), “A note on a paper by Erik Volz: SIR dynamics in random networks”
Effective degree (ED2) modelLindquist et al (2011), “Effective degree network disease models”
Some extensions: Clustered networks: Volz (2010)
Dynamic networks: Volz & Meyers (2007)
Effective degree (ED) model
Proved to be exact (mean behaviour, conditional on non-extinction)
Assumptions: number of S and I nodes of effective degree ED at : number of links still available for transmission A node decreases its ED by one when infected by or transmitting
to a neighbour, or when a neighbour recovers
,k kS I kt
1
1 1 1 1
1 :
11
kkk k k
l ll
k k k k k k k kk
kIS kS k S
l S I
I kI I k I I k II S
Effective degree (ED) model
Proved to be exact (mean behaviour, conditional on non-extinction)
Assumptions: number of S and I nodes of effective degree ED at : number of links still available for transmission A node decreases its ED by one when infected by or transmitting
to a neighbour, or when a neighbour recovers
Infecting Being infected Being contacted Neighbour recovering
,k kS I kt
1
1 1 1 1
1 :
11
kkk k k
l ll
k k k k k k k kk
kIS kS k S
l S I
I kI I k I I k II S
Effective degree (ED) model
Proved to be exact (mean behaviour, conditional on non-extinction)
Assumptions: number of S and I nodes of effective degree ED at : number of links still available for transmission A node decreases its ED by one when infected by or transmitting
to a neighbour, or when a neighbour recovers
Infecting Being infected Being contacted Neighbour recovering
,k kS I kt
1
1 1 1 1
1 :
11
kkk k k
l ll
k k k k k k k kk
kIS kS k S
l S I
I kI I k I I k II S
Effective degree (ED) model
Proved to be exact (mean behaviour, conditional on non-extinction)
Assumptions: number of S and I nodes of effective degree ED at : number of links still available for transmission A node decreases its ED by one when infected by or transmitting
to a neighbour, or when a neighbour recovers
Infecting Being infected Being contacted Neighbour recovering
,k kS I kt
1
1 1 1 1
1 :
11
kkk k k
l ll
k k k k k k k kk
kIS kS k S
l S I
I kI I k I I k II S
Effective degree (ED) model
Proved to be exact (mean behaviour, conditional on non-extinction)
Assumptions: number of S and I nodes of effective degree ED at : number of links still available for transmission A node decreases its ED by one when infected by or transmitting
to a neighbour, or when a neighbour recovers
Infecting Being infected Being contacted Neighbour recovering
,k kS I kt
1
1 1 1 1
1 :
11
kkk k k
l ll
k k k k k k k kk
kIS kS k S
l S I
I kI I k I I k II S
Probability generating function (PGF) model
Volz: probability that a random edge has not transmitted yet
fraction of degree-1 nodes still susceptible at time probability that a node of degree is still susceptible probability that a susceptible node is connected to a
susceptible/infected node PGF of the degree distribution Then:
Miller:
( )kt k
0( ) k
kkd xx
( )(1 )
(1)
( )t
,S Ip p
( ))
( )
(
(
)
)1
(
I
I I S I
S IS
p
p p pp
p pp
t
SIS effective degree model (ED2)
number of susceptible/infected nodes with susceptible and infectious neighbours Lindquist et al (2011)
,si siS I si
SIR effective degree model (ED2)
Lindquist et al (2011)
Conclusions: Definition of as
the dominant eigenvalue of a certain matrix
SIR final size identical
to ED’s (Ball & Neal (2008)), and a bit higher than Volz’s (Volz (2008))
0 0SIS SIRR R
0SISR
NETWORKS
Key epidemiological quantities
(Almost) exact dynamics
Approximate dynamics
History
Moment closure: Originated in probability theory to estimate moments of
stochastic processesGoodman (1953); Whittle (1957)
Extended to physics, in particular statistical mechanics
Pair approximations: pioneered by the Japanese schoolMatsuda et al (1992), Sato (1994), Harada & Isawa (1994)
Extensively studied by Keeling and Morris PhD thesis in Warwick
Keeling (1995); Morris (1997)
Moment closure - basics
Approximate average behaviour of a stochastic network model with a few deterministic equations: Large network Assuming non-extinction Markovian model
Mean-field approximation (SIR case as an example): Original system:
Closure:
Closed system:
[ ]
[ ] [ ] [ ]
[ ] SI
I
S
SI I
[ ] [ ][ ]AB A B
[ ][ ]
[ ] [ ][ ] [
[ ]
]
S I
I I I
S
S
Pairwise approximation (SIR)
Original system:
(extensions to multitype
– e.g. degree distribution
are possible)
Basic closures:
Open triplet: Closed triangle: Kirkwood & Boggs (1942)
Overall (clustering coeffficient ): Keeling (1999)
2
2 2
S IS
I IS I
SS ISS
IS ISS ISI IS IS
II ISI IS II
( 1) [ ][ ][ ]
[ ]
n AB BCABC
n B
2
( 1) [ ][ ][ ][ ]
[ ][ ][ ]
n N AB BC ACABC
n A B C
1[ ][ ] [ ][ ] 1
[ ] [ ][ ]
n AB BC N ACABC
n B n B C
Extensions:
Dynamics network (SIS): Eames & Keeling (2004)
Directed network (SIR): Sharkey (2006)
‘Invasory’ pair approximation (SIS): Bauch (2005)
Triple approximation (SIS): Bauch (2005)
Motif-based triple approximation: House et al (2009)
Improved pairwise approximation (SIS): House & Keeling (2010)
Maximum Entropy (SIR): Rogers (2011)
Clustered PGF (SIR): House & Keeling (2011)
1 [ ][ ] [ ][ ][ ])
[ ] [ ] [ ][[ ] (
]1)
[(1
]a
AB BC AB BC AC
n B A aB CAB
a aC n
Applications and results
Theoretical: Simple approximate results for and Full approximate transient dynamics
Practical: Sexually transmitted infections: Ferguson & Garnett (2000); Eames & Keeling (2002)
Foot-and-mouth disease: Ferguson et al (2001)
Contingency planning against smallpox: House et al (2010)
Results: SIR pairwise is exact for unclustered networks: Sharkey (2013)
SIR pairwise model (with some simplifying assumptions) is equivalent to Volz’s PGF model: House & Keeling (2010)
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COMMENTS
Deterministic models
Summary More stochastic, but deterministic still very useful Deterministic-stochastic distinction is blurred Most models are formulated stochastically but results are
deterministic
Many models More effort is needed to compare them in the same context Many ways of comparing them: which one is fair? Suggestion: find relevant quantities to keep fixed that have the
same biological interpretation in both of them (e.g. )0R
Future challenges – beyond R0
Lack of an exponentially growing phase
Small populations
Hierarchical society (lack of group-scale separation)
Long infectious periods (lack of time-scale separation)
Superinfection
Prior immunity
…
Organisers
Funding MRC EPSRC
Collaborators Imperial College: Christophe Fraser, Neil
Ferguson, Simon Cauchemez, Katrina Lythgoe Warwick: Matt Keeling, Thomas House,
Déirdre Hollingsworth, … Others: Frank Ball, Pieter Trapman
Acknowledgments
Organisers
Funding MRC EPSRC
Collaborators Imperial College: Christophe Fraser, Neil
Ferguson, Simon Cauchemez, Katrina Lythgoe Warwick: Matt Keeling, Thomas House,
Déirdre Hollingsworth, … Others: Frank Ball, Pieter Trapman
Acknowledgments
Thank you all!