How mathematics can help explain vaccine scares and associated disease dynamics

How mathematics can help explain vaccine scares and

associated disease dynamics

Chris Bauch Department of Mathematics and Statistics

University of Guelph

Seminar, WICI, University of Waterloo, 5 February 2013

Source: CDC

Mortality rate due to infectious diseases in the United States

sanitation/hygiene, antibiotics, vaccines

Smallpox 1977

Measles 20??

Polio 20??

Measles vaccine distribution

§ Smallpox vaccine scare, 18th century § Fear: turns humans into cow-like hybrids

§ Whole cell pertussis scare, 1970s § Fear: neurological damage

§ MMR vaccine scare, UK, 1998 § Fear: autism, IBS

§ Polio vaccine scare, Nigeria, 2004 § Fear: infertility, AIDS

The interplay between disease incidence and vaccinating behaviour

Disease incidence

Individual vaccination decisions

imitation imitation

vaccination ris

k av

ersi

on

Chapman and Coops (1999) Preventive Medicine 29: 249

Free-riding and vaccine scares l  Behaviour-incidence interplay can result in free-riding

l  Strong herd immunity à less incentive to vaccinate l  Vaccine scares are related to free-riding behaviour.

l  Strong herd immunity à little disease à even a small increase in vaccine risk can cause significant drop in coverage

Access

Choice

eclipses

Psychohistory l  From Isaac Asimov’s Foundation series:

l  A branch of mathematical statistics predicting behaviour of very large groups of people

l  How can human vaccinating behaviour be predicted?

l  More difficult than predicting motion of a billiard ball…

l  …but, the stakes are high so we have to try.

versus

Pertussis vaccine scare, England & Wales, 1970s

Pertussis vaccine scare, England & Wales, 1970s

l  Why did it take 5 years for coverage to bottom out? l  Why is the rebound in vaccine coverage so uniform?

l  Can we capture this with a simple model?

l  Is the population responding to surges in pertussis incidence? l  Can we capture with a game theoretical model?

Behaviour-incidence models l  Mathematical models are needed to formalize this

interplay in a precise, quantitative way. l  One approach: behaviour-incidence models

Disease Transmission

Model

Vaccinating Behaviour

Model

Behaviour-incidence models l  Mathematical models are needed to formalize this

interplay in a precise, quantitative way. l  One approach: behaviour-incidence models

Disease Transmission

Model

Vaccinating Behaviour

Model

SIR model Agent-based model Stochastic model

Game theoretical model Psychological model Phenomenological model

Game theory

The theory of strategic interactions

Introduction to game theory

l  Game theory formalizes strategic interactions between individuals in a group.

l  A game is defined by specifying l  Players: 2 players, n players, population games. l  Strategy Set: Set of actions available to players. l  Currency: The measure with which value of strategy

is quantified (money, morbidity/mortality, evolutionary fitness).

l  Payoffs: The value of strategies, expressed in terms of the currency.

Solution concepts

l  Assume that all individuals are rational (attempt to maximize their payoff). Then, what strategies will players adopt?

l  Solution concepts: l  Strictly dominated strategies, l  Nash equilibrium strategies, l  Evolutionarily stable strategies.

Nash equilibrium

l  Nash equilibrium: a strategy such that, if everyone plays it, no small group of individuals can increase their payoff by switching to another strategy.

l  Populations at Nash equilibria are therefore stable over time. l  Population is supposed to “live” at a Nash equilibrium.

l  Game theoretical analysis seeks to identify Nash equilibria of games

Example: The Prisoner’s Dilemma §  Repeated, 2 player game in a large population.

§  individuals are paired at random to play each round.

§  Strategies: (C)ooperate or (D)efect §  Currency: Money §  Payoffs per round:

$ 1 $ 5 D

$ 0 $ 3 C D C

Payoff to ‘Focal’ Player

Payoff to Opponent

Who gets the highest payoffs: Cooperators or Defectors?

Cooperators: Mr. Spock

“The needs of the many outweigh the needs of the few, or the one.” (Star Trek II: The Wrath Of Khan, 1982)

Defectors: Ayn Rand

“Every man has the right to exist for himself, and not for the benefit of others.” (The Virtue of Selfishness, 1964)

Why ‘Defect’ is a Nash Equilibrium

l  When defectors are the majority, minority cooperators do poorly.

$ 1 $ 5 D

$ 0 $ 3 C

D C

Why ‘Cooperate’ is not a Nash equilibrium

l  When cooperators are the majority, minority defectors thrive.

$ 1 $ 5 D

$ 0 $ 3 C

D C

An interesting prediction

l  In a population consisting of defectors: average payoff = $ 2

l  In a population consisting of cooperators: average payoff = $ 5

Paradoxically, defection is the predicted behavior … … even though the population as a whole would do best if everyone cooperated.

$ 1 $ 5 D

$ 0 $ 3 C

D C

The Prisoner’s Dilemma in other contexts (Tragedy of the Commons) l  Forests l  Cocktail Parties

Voluntary vaccination policy as a Prisoner’s Dilemma l  High coverage levels are not Nash equilibria:

l  Non-vaccinators (defectors) gain benefits of herd immunity without paying costs of vaccination;

l  Vaccinators (cooperators) accept the costs of vaccination.

l  Therefore, non-vaccinators (free riders) may make it difficult to sustain high coverage levels.

Previous models of free-rider effects in vaccination

l  The free-rider problem in vaccination has received attention primarily from l  Economists l  Theoretical population biologists

l  Theoretical population biologists have highly suitable tools for studying this problem l  Transmission modelling literature l  Evolutionary game theory literature

l  Number of publications on behaviour-incidence modelling for vaccine-preventable infections has grown considerably in recent years l  Reviewed in Funk et al (2010) Interface 7: 1247.

Previous models of free-rider effects in vaccination

l  Models generally predict that free-riding should occur for a range of diseases and vaccine types (Fine & Clarkson 1986, Geoffardson & Philipson 1997, Bauch et al 2003, Galvani et al 2007)

l  They show that the individually optimal equilibrium coverage (e.g. Nash equilibrium) is unique and lower than the socially optimal coverage.

l  However, some studies have explored conditions under which exceptions occur l  Multiple individually optimal equilibria are possible in some systems

l  Sufficiently imperfect vaccine for SIS-type infections (Chen 2006) l  Age-dependent virulence (Reluga et al 2006)

l  Free-riding can be averted through taxes and subsidies (Brito et al 1991).

Vaccination games for childhood diseases (Bauch and Earn, PNAS 2004)

l  Players: population game (payoff to an individuals depends upon average behavior of population)

l  Strategies: an individual vaccinates their child with probability P, where 0 ≤ P ≤ 1. l  Pure nonvaccinator strategy: P = 0 l  Pure vaccinator strategy: P = 1

l  If everyone plays P, then P = vaccine uptake (the current rate of vaccination in the population).

l  Currency: perceived probability of morbidity (from vaccine or disease).


payoff to an individual who vaccinates with probability P = (probability of vaccinating) x (payoff to vaccinate) + (probability of not vaccinating) x (payoff not to vaccinate)

n  Payoffs:

n  Let rv = perceived probability of morbidity upon becoming vaccinated.

n  Hence, payoff to vaccinate, ev is

= P x (payoff to vaccinate) + 1-P x (payoff not to vaccinate)

eV = −rv

Vaccination games for childhood diseases (Bauch and Earn, PNAS 2004) n  Let ri = perceived probability of morbidity upon becoming

infected. n  Let π(p) = probability that an unvaccinated individual

eventually becomes infected, if vaccine coverage is p. n  Payoff not to vaccinate, eN = - (probability of infection) x

(probability of morbidity upon becoming infected)

n  Perceived payoff E(P) to an individual playing P in population with vaccine coverage p is

eN = −riπ (p)

EP (p) = PeV + (1−P)eN= P(−rv )+ (1−P)(−riπ (p))

0 1 p

π(p)

1

0

pcrit

n  π(p) is a decreasing function of p. n  à can prove a unique,

socially suboptimal Nash equilibrium vaccine coverage (i.e. free-riding occurs)


0 1 p

π(p)

1

0

pcrit

n  π(p) is a decreasing function of p. n  à can prove a unique,

socially suboptimal Nash equilibrium vaccine coverage (i.e. free-riding occurs)

n  For quantitative predictions, a functional form for π(p) can be obtained from a dynamic model of disease spread.


The SIR model with births/deaths

€

dSdt

= µ(1− p) −βSI −µS

dIdt

= βSI − γI −µI

dRdt

= µp+ γI −µR€

S = proportion susceptibleI = proportion infectedR = proportion recovered/immune

€

µ = birth rate/death ratep = proportion vaccinated

S I R γ βΙ

µp

µ(1-p)

µR µI µS

€

β = transmission rateγ = recovery rate

µ

Nash equilibrium P* versus relative risk r = rv / ri

pure non-vaccinator state

mixed state

relative risk rv/ri

pcrit N

ash

equi

libriu

m

cove

rage

leve

l P*

R0 is the average number of secondary infections produced by an infected individual, in an otherwise susceptible population

R0 > 1: epidemic may occur R0 < 1: epidemic dies out Bauch and Earn, PNAS 2004

Nash equilibrium P* versus relative risk r = rv / ri

pure non-vaccinator state

mixed state

relative risk rv/ri

pcrit N

ash

equi

libriu

m

cove

rage

leve

l P*

R0 is the average number of secondary infections produced by an infected individual, in an otherwise susceptible population

R0 > 1: epidemic may occur R0 < 1: epidemic dies out Bauch and Earn, PNAS 2004

Free-riding always occurs!!

Challenges for modeling

1)  How was smallpox eradicated despite voluntary vaccination?

Homogeneous mixing behaviour-incidence models

l  Previous models assume homogeneous mixing to a greater or lesser extent.

l  However, in close contact infections, disease is more likely to be acquired from close associates.

l  Smallpox is spread primarily through close contact, despite rare but extensively reported cases of long-distance dispersal (CDC 2008, Gelfand & Posch 1971).

Close contact infection models l  Close contact infections can be

modelled using networks. l  Infection dynamics on networks and

lattices are qualitatively different from dynamics in homogeneous mixing populations. l  Lack of threshold behaviour (May & Lloyd

2001). l  More realistic time series and critical

community sizes (Keeling & Grenfell 1997). l  Slower pathogen invasion (Rand et al 1995).

l  Hence, it is reasonable to investigate whether the incentive to vaccinate changes for transmission through a social contact network.

Network model (Perisic & Bauch PLoS Comp. Biology, 2009)

l  Simulated a random, static network. l  A vaccine-preventable SEIR infection spreads through

edges. l  τ = probability per day that an infectious node transmits

to a neighbouring susceptible node l  rv = perceived probability of death due to vaccination l  ri = perceived probability of death due to infection l  ε = vaccine efficacy

Payoffs and decision-making

l  On any given day, a node vaccinates if

€

eV > eN

n  Currency is expected life years accrued n  Nonvaccinator payoff

n  Vaccinator payoff

€

eN = (1− λ)α + λ 1− ri( )L[ ]

€

eV = 1−ε( ) 1− rv( )eN{ } + ε 1− rv( )L{ }

Condition for vaccination

l  It can be shown that: l  A node with 0 infectious neighbours does not vaccinate. l  A node with ≥1 infectious neighbour vaccinates if:

l  This condition is satisfied for (1) Sufficiently safe vaccine (low rv) (2) Sufficiently dangerous disease (high ri) (3) Sufficiently high node-to-node transmission probability τ

τ >rv

ri ε + 1−ε( )rv"# $%

Rapid control through voluntary ring vaccination

τ > r⇒ for large ε

Measles versus smallpox

Measles Smallpox

Large perceived relative risk r =rv/ri Small perceived relative risk r =rv/ri

Relatively large number of potential infectious contacts,

relatively small τ for any given contact.

Relatively small number of potential infectious contacts, relatively large

τ for any given contact

τ > r

Simulation Design

l  To isolate effect of contact structure, explore network dynamics for a wide range of neighbourhood sizes Q.

l  For each value of Q, set τ such that the average force of infection λ does not change relative to baseline.

l  For Q sufficiently large (τ sufficiently low) l  A homogeneously mixing population is approximate. l  A free rider effect is expected.

l  For Q sufficiently small (τ sufficiently high) l  A population on a localized contact structure is obtained. l  Infection control through voluntary vaccination is possible.

l  Since λ is constant, differences between low and high Q limits are attributable to changes in contact localization.

τ > r

Simulation Results

Imperfect voluntary ring vaccination: infection escapes through susceptible gaps in ring and percolates through network.

Successful voluntary ring vaccination: infection is quickly controlled.

Simulation Results

Imperfect voluntary ring vaccination: infection escapes through susceptible gaps in ring and percolates through network.

Successful voluntary ring vaccination: infection is quickly controlled.

smallpox measles

Three Dynamical Regimes

Regime Homogeneous mixing model Network model

1 No vaccination

very large final size No vaccination

very large final size

2 Partial vaccination, moderate final size

Partial vaccination, moderate final size

(imperfect ring vaccination)

3 n/a Little vaccination small final size

(efficacious ring vaccination)

l  Univariate sensitivity analysis reveals three dynamic regimes in the network model, versus two in the homogeneous mixing model.

Summary of findings

l  Spatial/contact structure changes the incentive to vaccinate. l  Reduces probability of a free-rider problem l  Provides one way to reconcile smallpox

eradication under a voluntary vaccination program with theory of behaviour-incidence modelling.



2)  Do individuals really act according to assumptions of classical game theory?

l  Can introduce evolutionary game dynamics such as the “imitation dynamic”.

l  Evolutionary game dynamics describe how a population evolutions toward a Nash equilibrium

l  Imitation dynamics: individuals learn successful strategies by sampling others

The interplay between disease prevalence and vaccinating behaviour

Disease prevalence

Individual vaccination decisions

imitation imitation

vaccination ris

k av

ersi

on

dSdt

= µ(1− p)−βSI −µS

dIdt= βSI −γ I −µI

Vaccine coverage fixed

Classic SIR Model with births and deaths

The SIR model with births/deaths

!

dSdt

= µ(1" p) "#SI "µS

dIdt

= #SI " $I "µI

dRdt

= µp+ $I "µR!

S = proportion susceptibleI = proportion infectedR = proportion recovered/immune

!

µ = birth rate/death ratep = proportion vaccinated

S I R !"#$"

µp"

µ(1�p)"

µR"µI"µS"

!

" = transmission rate# = recovery rate

µ"

Social learning vaccination model l  Combines SIR model with social learning model:

l  Individuals sample others at a constant rate and l  Switch to their strategy with a probability proportion to expected

gain in payoff

dSdt

= µ(1− p)−βSI −µS

dIdt= βSI −γ I −µI

€

dSdt

= µ(1− x) −βSI −µS

dIdt


dxdt

=κx(1− x)(−1+ωI)

κ = imitation rate ω = relative risk of disease/vaccine

Vaccine coverage fixed Vaccine coverage is determined by behaviour

Bauch, Proceedings of the Royal Society of London B, 2005

Parameter plane

Imitation rate

Sen

sitiv

ity to

dis

ease

pre

vale

nce

Oscillations in frequency of vaccinators

Constant, nonzero frequency of vaccinators

Pure nonvaccinator state




3)  Aren’t coverage dynamics during a scare simply due to individual risk perception?




3)  Aren’t coverage dynamics during a scare simply due to individual risk perception?

4)  Lots of models being developed, but how do we know they are “right”?

l  Can they fit empirical data? l  Can they predict actual behaviour-disease dynamics?

Simulation Design (Bauch and Bhattacharyya, PLoS Comp. Biol. 2012)

l  Parsimony analysis l  Predictive analysis

€

dSdt

= µ(1− x) −βSI −µS

dIdt


dxdt

=κx(1− x)(−1+ωI)

Simulation Design: Parsimony Analysis l  Compared the Akaike Information Criterion of this model

to AIC of reduced models with l  Social learning but no feedback l  Feedback but no social learning l  Neither feedback nor social learning

l  Under five possible descriptions of how perceived vaccine risk evolved over time l  1: l  2: l  3: l  4: l  5:

!

!

!

!

!

€

dSdt

= µ(1− x) −βSI −µS

dIdt


dxdt

=κx(1− x)(−1+ωI)

(Hence, risk evolution not modelled mechanistically)

Fitted the model using a hill-climbing algorithm with random re-start of Initial conditions

Pertussis

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!

! ! ! !Supplementary,Figure,3:,,Parsimony,analysis,of,behaviour9incidence,model,,pertussis,vaccine,scare.!!Best!fitting!model!(red)!versus!data!(black)!on!whole!cell!pertussis!vaccine!uptake,!for!5!risk!evolution!curves!and!4!cases,!using!the!behaviourDincidence!model.!!The!numerical!value!in!the!inset!of!each!subpanel!is!the!corresponding!AICc!value!for!the!fit.!!See!Methods!in!main!text!for!definition!of!risk!evolution!curves.!!

!

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MMR

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! ! ! !Supplementary,Figure,3:,,Parsimony,analysis,of,behaviour9incidence,model,,pertussis,vaccine,scare.!!Best!fitting!model!(red)!versus!data!(black)!on!whole!cell!pertussis!vaccine!uptake,!for!5!risk!evolution!curves!and!4!cases,!using!the!behaviourDincidence!model.!!The!numerical!value!in!the!inset!of!each!subpanel!is!the!corresponding!AICc!value!for!the!fit.!!See!Methods!in!main!text!for!definition!of!risk!evolution!curves.!!

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!

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Supplementary,Figure,4:,,Parsimony,analysis,of,behaviour9incidence,model,,MMR,vaccine,scare.!!Best!fitting!model!(red)!versus!data!(black)!on!MMR!vaccine!uptake,!for!5!risk!evolution!curves!and!4!cases,!using!the!behaviourEincidence!model.!!The!numerical!value!in!the!inset!of!each!subpanel!is!the!corresponding!AICc!value!for!the!fit.!!See!Methods!in!main!text!for!definition!of!risk!evolution!curves.!!!

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Summary of findings

l  We repeated the analysis by fitting behavioural model to empirical incidence dataà same results

l  Adding l  Social learning l  Disease incidence feedback on behaviour

(strategic interactions/game theory) Improves model fit, sometimes dramatically, with little or no parsimony penalty

Caveats/Questions

l  Does parsimony analysis “stack the cards” against the behaviour-incidence model? l  Risk evolution curve #5:

l  Would findings hold up if a mechanistic model of risk perception were used instead?

!

l  Fitted the model to data points for times t ≤ tfit l  Vaccine coverage l  Disease incidence

l  And then checked how well it predicted data for t > tfit l  Vaccine coverage l  Disease incidence

l  Picked epidemiological parameters from previous publications and fitted: l  Imitation rate, risk evolution curve parameters

l  Used risk evolution curve #1 l  To assess parameter uncertainty, conducted:

l  Probabilistic sensitivity analysis

l  Bootstrapping analysis

!

Simulation Design: Predictive Analysis (Bauch and Bhattacharyya, PLoS Comp. Biol. 2012)

Pertussis Vaccine Scare

Vaccine Coverage Disease Incidence

black: data

1975


black: data blue: best-fit model red: PSA samples

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! 4&)+=+789:+

+++

++

! 4&)+=+789>+

+++

++

! 4&)+=+7899+

+ +

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+ +

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1976



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! 4&)+=+789:+

+++

++

! 4&)+=+789>+

+++

++

! 4&)+=+7899+

+ +

! 4&)+=+789;+

+ +


1977



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! 4&)+=+789:+

+++

++

! 4&)+=+789>+

+++

++

! 4&)+=+7899+

+ +

! 4&)+=+789;+

+ +


1978



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! 4&)+=+789:+

+++

++

! 4&)+=+789>+

+++

++

! 4&)+=+7899+

+ +

! 4&)+=+789;+

+ +


1979



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! 4&)+=+7898+

+ +

! 4&)+=+78;L+

+ +

! 4&)+=+78;7+

+ +

! 4&)+=+78;P+

+ +


1980



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! 4&)+=+7898+

+ +

! 4&)+=+78;L+

+ +

! 4&)+=+78;7+

+ +

! 4&)+=+78;P+

+ +


1988



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! 4&)+=+78;9+

+ +

! 4&)+=+78;;+

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MMR: 2000



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+ +

! 4&)+>+899?+

+ +

! 4&)+>+8999+

+ +

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MMR: 2004



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MMR: 2005



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MMR: 2009



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Summary of findings

l  For pertussis vaccine scare, model appears to have some predictive power for both future vaccine coverage and disease incidence.

l  For MMR vaccine scare, model does not have much predictive power.

l  Possible reason for difference: pertussis outbreaks were in deterministic regime, whereas measles outbreaks were in stochastic regime

Caveats/Questions

l  Approach involves a counter-factual since behaviour model is fitted against modelled incidence, not actual historical incidence. l  Is there a better way?

l  Would more sophisticated models work better for predictive analysis? l  e.g. stochasticity, age-structure, etc

Challenges for modeling 1)  How was smallpox eradicated despite voluntary

vaccination? 2)  Do individuals really act according to assumptions

of classical game theory? 3)  Aren’t coverage dynamics during a scare simply

due to individual risk perception? 4)  Lots of models being developed, but how do we

know they are “right”? 5)  How were measles, polio, and whooping cough

eliminated in the UK despite voluntary vaccination?

Future work l  Ongoing work with T. Oraby (postdoc) explores modifying models to

take into account: l  Social norms l  Prospect theory

l  Framing

l  Planned work with M. Garvie and M. Althubyani (PhD student) explores applications of optimal control theory: l  What does public health actually control?

l  Data, data, data!

Actual P

erce

ived

Take-home messages l  Classical models based on assuming

l  Homogeneous mixing l  Humans as purely rational optimizers

generally predict free-riding will be pervasive and do not account for slowly evolving vaccine uptake during a vaccine scare.



l  Introducing contact structure changes the incentive to vaccinate l  Circumvents free-rider problem for close contact infections l  Provides one explanation for smallpox eradication under voluntary vacc.



l  Introducing contact structure changes the incentive to vaccinate l  Circumvents free-rider problem for close contact infections l  Provides one explanation for smallpox eradication under voluntary vacc.

l  Introducing social learning and disease incidence feedback (i.e. strategic interactions/game theory): l  Explains vaccine scare data more parsimoniously than a broad range of

other candidates lacking social learning and/or feedback. l  Confers predictive power for disease dynamics in deterministic regime.

l  Social learning and disease feedback can be major factors in determining how vaccine scares unfold

Implications for public health l  Behavior-incidence models can change perspectives on

vaccinating behaviour l  Vaccine scares are not just historical accidents… they are

exacerbated by inherent instabilities in voluntary vaccination. l  Nonlinear dynamics, as well as static considerations such as

vaccine and disease risk, may contribute to observed patterns. l  It may be necessary to consider vaccine coverage in all groups,

and transmission between groups, to understand vaccine coverage in a given group.

l  à We need a dynamic, meta-population perspective l  Parsimonious, empirically validated behaviour-incidence

models may be useful to public health in some situations l  For example, predicting how vaccine scares will unfold and how

best to mitigate them.

Thank you!

Relevant publications l  C.T. Bauch, S. Bhattacharyya (2012). ‘Evolutionary game theory and

social learning can determine how vaccine scares unfold’. PLoS Computational Biology 8(4): e1002452.

l  A. Perisic, C.T. Bauch (2009). Social contact networks and disease eradicability under voluntary vaccination. PLoS Computational Biology 5(2):e1000280.

l  C.T. Bauch (2005). ‘Imitation dynamics predict vaccinating behaviour’ Proceedings of the Royal Society of London B 272: 1669-75.

l  C.T. Bauch and D.J.D. Earn (2004). ‘Vaccination and the theory of games’. Proceedings of the National Academy of Sciences of the USA 101: 13391-4.

l  C.T. Bauch, A.P. Galvani and D.J.D. Earn (2003).‘Group interest versus self-interest in smallpox vaccination policy’. Proceedings of the National Academy of Sciences of the USA 100: 10564-7.

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How mathematics can help explain vaccine scares and associated disease dynamics

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