dissecting the dynamics bryan grenfell recurrent …satellite’ dynamics: next step : produced a...

Dissecting the dynamics of acute immunizing

infections

Bryan Grenfell Recurrent epidemics, herd immunity and the dynamic clockwork of measles

External forcing and nonlinear dynamics

The dynamics of immune escape

Dynamics and immune escape• Nita Bharti

• Ottar Bjornstad

• Ben Bolker

• Andrew Conlan

• Janet Daly

• David Earn

• Steve Ellner

• Bärbel Finkenstädt

• Katie Glass

• Bryan Grenfell

• Eddie Holmes

• Matt Keeling

• Natalia Mantilla

• Jenny Mumford

• Ollie Pybus

• Pej Rohani

• Colin Russell

• James Wood

• Darren Shaw

• Yingcun Xia

Wellcome Trust, CDC, WHO, NIH Fogarty Center

Measles in England and Wales

Year

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Start of vaccination

Anatomy of a measles epidemicBirths

Susceptible

Infected

Recovered

Vaccination

Simple epidemic: no births

Time

‘simplest picture’: measles dynamics, where epidemics are ended by strong herd immunity, then start when susceptibles have built up via births

R0

Or is it so simple?

( )( )

( )

1= − − −

= − +

= + −

dSN p S IS

dtdI

IS IdtdR

I NP Rdt

ββββ

ββββ

µ µµ µµ µµ µ

µ γµ γµ γµ γ

γ µ µγ µ µγ µ µγ µ µ

Lessons from Measles EpidemicsLessons from Measles EpidemicsBefore the onset of mass-vaccination in England & Wales, strong spatial synchrony due to:

human mobilityvery infectious virus seasonal forcing

After vaccination, case reports in cities very much out-of-phase:

dynamic complexity - multiple attractorsfewer infections ⇒ weaker epidemiological coupling

•Other key features of dynamics: Stochastic fadeout in small places (persistence above a critical community sizeof about 250K); seasonal forcing of infection

•What drives the cycle period? First use time series analysis to examine the cyclicity

Rather than conventional ‘global’ Fourier analysis, use wavelets–locates periodicities in time

0 20 40 60 80

(a) Constant oscillationData wl 15wl 10wl 5

0 0.05 0.1 0.15 0.20

10

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Frequency per unit time

Pow

er

(b)..standard global power spectrum

20 40 60 70 80 90Time

(c) Variable period

Data scale 10scale 5

(d)

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20 40 60 80 100

5

10

Time

Per

iod

log(Power) (e)

peak at wavelength 10

(b)

Vaccine era

Use wavelet spectra to dissect changes in cycle period….

Modelling local dynamicsBirths

Susceptible

Infected

Recovered

Vaccination

Simple epidemic: no births

Tim e

Note that we know the inflow and outflow from the susceptible class…

Time

RAS model susceptibles

Reconstructed susceptibles (Z)

0

Estimate susceptibles from births and cases

Finkenstädt and Grenfell JRSS C, (2000); Grenfell et al, Nature (2001); Ferrari et al, Nature (2008)

We can use this trick to help estimate transmission rates…

Expected casesObserved cases: crudely use log least squares to estimate parameters..

In practice, I is NB; also allow for measurement process..

(a) TSIR model fit, R squared=0.98

Predicted

Obs

erve

d

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(b) Observed and fitted time series

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000 o Obs Fit

(c) 20 year simulation of the TSIR model

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Modeling local dynamics: The TSIR model:

(a) TSIR model fit, R squared=0.98

Predicted

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erve

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(b) Observed and fitted time series

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000 o Obs Fit

(c) 20 year simulation of the TSIR model

Year

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000 o Obs Simulation

Birth rate

Modelling local dynamicsBirths

Susceptible

Infected

Recovered

Vaccination

Simple epidemic: no births

Time

Baby booms drive more annual dynamics (low birth rates act like vaccination) –‘slow’ forcing

Aside: these dynamics are at moderate levels of seasonality: if seasonal forcing is very high, dynamics could be chaotic

…but implies unrealistically deep troughs, so need to allow for stochastic dynamics/fadeouts

So these ‘exotic’ dynamics are irrelevant for real measles….?

Returning to temporal dynamics and demography: high birth rates tend to lead to regular, annual dynamics:(Earn et al Science, 2000)

Underlined by stochastic simulations at ‘moderate (UK) seasonality:

Birth rate:

low (/vaccination) Developed Developing

Measles still a major killer in many high birth rate developing countries….Overall, do see more annual epidemics there

But seasonality complicates this (Matt Ferrari, Andrew Conlan, Rebecca Grais, Ottar Bjornstad, Lisa Cairns, Lara Wolfson)…(Ferrari et al Nature (2008))

Seasonal Dynamics: Niger

• Overall, Annual outbreaks in most of Niger

• but local outbreaks are erratic...

• Transmission much more strongly seasonal than in UK (8-fold variation)

• Seasonality (due to annual agricultural migrations?) forces local extinction??

tran

smis

sion

rat

e

Seasonal Dynamics: Niamey (pop 700,000)

Birthrate and Seasonality

Stronger seasonality leads to more erratic dynamics at all birthrates – in the chaotic region , but this really implies irrgular epidemics with local extinctions

Strength of seasonality

Birt

h ra

te

London Niamey

+

+

Seasonality and Persistence• Strong seasonality in

Niamey leads to local extinction

CCS Much higher than the 250K expected from developed country analyses

Regional Persistence varies• Niamey: more fadeouts than

expected• North : more fadeouts than

expected• Maradi & Zinder : fewer than

expected– Perhaps more movement between

towns

Maradi

ZinderCurrently exploring implications for Routine/reactive vaccination – at high seasonality epidemics may be irregular even at high vaccination rates as we approach the measles endgame

Need regional metapopulationmodels to do this (ferrari et al 08)

To get at space-time dynamics, need rich data sets, illustrate with the

historical UK notifications

Strong waves of infection

Spatial correlation patterns confirm this picture

•A significant drop in correlation with distance (suggests a regional scale of influence of large cities?)

•Lower correlations in the vaccination era

Pre-vaccination

Vaccine era

Waves appear to bedue to spatial coupling:

Lag

….. in a stochastic spatially hierarchical world, infection needs to be restarted in small centres after epidemics: ‘core-satellite’ dynamics:

Next step: produced a full metapopulation patch model for urban measles dynamics, use ‘gravity models’ for coupling (Xia et al)

τ τ

ρθ=1 2i j

ijij

N NTransmission

d

0.5

1

Lag

No detailed information on movements -Fit local dynamics with TSIR, coupling from duration of fadeouts

Phase ( °°°° ) relativeto London:

-90

0

90

Phase ( °°°° ) relativeto London:

-90

0

90

ModelObserved

Use a gravity model to capture spatiotemporal waves in the UK (Xia et al, Am Nat 2004). Waves because infection ‘invades’ small places from large centers - very like forest fire dynamics. Historical movement of children key (but also hard to estimate independently…). For seasonal flu in the US, work flows of adults more important (Viboud et al Science 2006) –why the difference?

In fact, measles is unusual in the perfection of the immunity it

causes…Comparative dynamics of herd immunity

How do these dynamics change if pathogen evolution makes herd immunity imperfect?

Explore:�Immune escape and the impact of host demography�Impact of immune escape on herd immunity

Measles: strong herd immunity leads to multi annual cycles, but little phylogeneticstructure

Annual influenza:repeated immune escape, leads to episodic evolution. Fouchier, Smith et al; Koelle, Cobey et al (Science 2006). In-host dynamic history matters!

Before considering implications for cross scale dyn amics, focus first on a more general analysis of immune escape and demograp hy at the population level..

Grenfell et al Science (2004)

Viral phylodynamics

•We know that dynamics of strongly immunizing acute /SIR infections very strongly and promptly track demographic changes Eg measles)

•What of SIRS infections (much more common), where immune escape effectively acts as a massive ‘birth rate’ of susceptibles?

Susceptible

Infected

Immune

Birth rateEpidemics track birth pulse

Epidemics ‘ignore’baby boom!

Possible example of this ‘demographic buffering by immune escape’: parainfluenza 1 ‘tracks’ birth rate decline in US; RSV, also flu, rotavirus independent of birth rate (more ‘SIRS’). Currently exploring impact and use of this phenomenon…

0

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1 3 5 7 9 11 13 15 17 19 21days

new

cas

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Viral shedding

Time

Immunity, pathogenicity

Mutation

transmission

Epidemic dynamics

Host heterogeneity

Spatial dynamics

Kentucky/1/94Alvdalen/96Soderala/94

Newmarket/93Florida/1/93

LaPlata/1/93Ibadan/9/91

Arundel/12369/91Kentucky/1/91

Alaska/1/91Arundel/91

Kentucky/1/92Kentucky/1/92/2

Kentucky/1/90Berlin/3/89

Berlin/2/91Avesta/93

Lambourn/92Lambourn/22778/92

HongKong/1/92HongKong/J92Rome/5/91

Sweden/TBY/91Sussex/89Ella/89

Newmarket/BC89Lichfield/89Sussex/93753/89

Ibadan/6/91Yvelines/2136/89Suffolk/89Visingso/90

Skara/88Kentucky/1/87

Tennessee/5/85Kentucky/2/86

Santiago/1/85Kentucky/1/81

Newmark/D/64/9Aby/84

Romania/80Fontainebleau/76

France/1/76Solvalla/79

NewMarket/76Tokyo/71Algiers/72

Uruguay/1/63Miami/63

Miami/1/63

Exploring seasonal influenza evolution across scales

Really need data across these scales – look at equine influenza (EIV)

Experimental infections in a ‘natural’ host..

Use use these data to quantify the dependence of epidemics (R0>1) on antigenic distance of pre-existing immunity?

Components of R0…

0 5 10 15 ctrl0.0

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prob

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amino acid changes, a

(a)

(b)

(c)

Infe

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per

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Epidemic threshold calibrated against immune escape(?) –extend for heterogeneously immune populations…

Illustration of implications for epidemics: assume all hosts have a given level of heterologousimmunity (OK for thoroughbred horses), and Ro=2(ish?)

Summary; future work• Epidemic clockwork, stochastic and seasonal forcing and

spatiotemporal dynamics of recurrent epidremics• Seasonality and birth rate in measles dynamics and

eradication - general point: ‘local’ insights don’t necessarily work in nonlinear systems; need more data/models for developing countries: social structure/demographic and environmental drivers important: what’s the end game for measles? - could be unpredictable up to the point of eradication, role of coinfection?…

• Imperfect herd immunity often because of evolution: potentially less overcompensatory dynamics affect pattern of spatiotemporal spread -- gravity models a reasonable first approximation…? (compare: pandemic flu and FMD…)

• Dynamic implications of immune escape: ‘demographic buffering for spatiotemporal dynamics. Need data (from vet systems?) to understand cross scale evolutionary dynamics - ARGUABLY THE BIG PROBLEM OVER THE NEXT DECADE