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”

0R

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)

0R

0R

0R

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)

R

HOUSEHOLDS MODELS

Overview



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



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

0R

HOUSEHOLDS MODELS

Overview



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




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




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





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






,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






,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






,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




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)

0R r

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!

Documents

Deterministic models: twenty years on