Use of Mathematical Models to Evaluate

Complex Public Health Interventions

Dr Zaid Chalabi

London School of Hygiene and

Tropical Medicine

Outline of the WorkshopSeminar Interspersed with Discussion

• Motivating Examples

• What are the characteristics of complex public health interventions?

• Why do we need mathematical models to evaluate complex interventions?

• What is complexity?

• What type of mathematical models are required to handle complexity?

• What are agent-based models and can they be of any use?

MOTIVATING EXAMPLES

Transport Intervention

• Ogilvie D, Mitchell R, Mutrie N, Petticrew M,

Platt S (2006) Evaluating health effects of

transport interventions. Methodological case

study. Am J Prev Med 31 (2), 118-126.

• Transport intervention:

– A new 5-mile stretch urban motorway in Glasgow

linking the M74 to the M8

Claims Related to Health and Well-Being Made

For and Against the New Motorway Link

Adapted from Ogilvie et al (2006)

Domain Benefits Harms

Economic o Create new jobs

o Improve business

activity (reduce

journey times to

Glasgow)

o Redistribute economic activity (from

other parts of Scotland)

o Displace some local businesses

Traffic o Reduce risk of traffic

injuries on local roads

o Encourage active travel

in local area

o Increase use of motor vehicles

o Increase risk of traffic injury at

motorway junction

Environmental pollution o Reduce noise and air

pollution on local roads

o Increase air pollution near the junctions

o Increase pollution from contaminated

land (due to construction work)

Social justice o Improve Quality of Life

(QoL) locally

o Funds could be spent elsewhere (e.g.

improving public transport)

o Improve QoL only for people who own

motor vehicles

Built Environment Interventions

• Lorenc et al (2012) Crime, fear of crime,

environment, and mental health and

wellbeing. Health & Place 18, 757-765

• Crime prevention measures

Individual

attitudes

Perceived

individual risk

Perceived

crime rate

Emotional

responses

Violent crime

Environmental

crime e.g.

vandalism

Public space

and transportHousing

Perceived

physical

environment

Perceived

social

environment

Mental

health

Physical

health

Interpersonal

relationships

& networks

Health

behaviours

Social

inequalities

Neighb’hood

& community

factors

Fear of crime

Health and

wellbeing

Social

environ-

ment

Built environment

Crime

Social

representations

Individual

demographics

Individual

crime risk

Avoidance

behaviours

Economic

policy

Social policy Crime and justice

policy

National

policies

National and

international economy

Mass media

C o g n i t i v e h e u r i s t i c s a n d b i a s e s

Perceived

vulnerability

Drug- and

alcohol- related

crime

Lorenc et al (2012)

Built environment, social environment, crime, fear of crime, health and well-being: causal pathway

Individual

attitudes

Perceived

individual risk

Perceived

crime rate

Emotional

responses

Violent crime

Environmental

crime e.g.

vandalism

Public space

and transportHousing

Perceived

physical

environment

Perceived

social

environment

Mental

health

Physical

health

Interpersonal

relationships

& networks

Health

behaviours

Social

inequalities

Neighb’hood

& community

factors

Fear of crime

Health and

wellbeing

Social

environ-

ment

Built environment

Crime

Social

representations

Individual

demographics

Individual

crime risk

Avoidance

behaviours

Economic

policy

Social policy Crime and justice

policy

National

policies

National and

international economy

Mass media

C o g n i t i v e h e u r i s t i c s a n d b i a s e s

Perceived

vulnerability

Drug- and

alcohol- related

crime

Adapted from

Lorenc et al (2012)

Built environment, social environment, crime, fear of crime, health and well-being: causal & intervention pathways

COMPLEX INTERVENTIONS

Some Characteristics of Complex Interventions

• Ill-defined start and end times

• Causal pathways are multi-dimensional with feedback

• System boundaries are blurred (affected population and geographical area are not well defined)

• Behaviour of individuals are influenced by their interaction with other individuals and with their (local and distant) environment

• Multiple health and non-health outcomes

• Outcomes can have widely different response times

• Associations can be non-linear

• .....

Use of Mathematical Models in Evaluating Public

Health Interventions

• It is not always possible to conduct trials due to practical, logistics, cost or ethical reasons.

• In such circumstances, models can be used to evaluate complex interventions ex ante.

• Models can also be used to help in the design of trials of complex interventions

• If mathematical models are to be used to evaluate complex interventions, what type of models are appropriate?

Discussion

• Can you think of other examples of complex

interventions that you have encountered or are

working with?

• Have you used mathematical models to help

inform the evaluation of complex interventions?

• Do you think that the standard epidemiological

methods are adequate to evaluate complex

interventions? How would they cope with

feedback? nonlinearity?.....

COMPLEX DYNAMICS

Example of a Complex Time Series

Time

Observation

Can we unwrap its complex dynamics?

Characteristics of Complex Dynamics

(“Chaos”)

• Determinism

• Sensitivity to initial conditions

• Nonlinearity

• Presence of an attractor

http://demonstrations.wolfram.com/LogisticMap

OnsetOfChaos/

Phase Space Construction

kk-2 k-1

Time series

Phase space

xk-1

xk

*

Phase Space Representation of the

Time Series

x1

x2 Map system dynamics to

geometrical space

SimulationObservation x1

Observation x2

x1 and x2 are

coupled in a

complex way

Can we unwrap

their complex

interaction ?

Phase Space Representation

-1 -0.5 0.5 1

x1

-0.4

-0.2

0.2

0.4

0.6

x2

Map complex dynamics interactions to

geometrical space

Discussion

• Can you think of epidemiological or

physiological time series which exhibit

complex dynamics?

– Hints

• Atrial fibrillation ?

• Infectious disease processes?

• Panic behaviour?

• .....

AGENT-BASED MODELS

“The economy needs agent-based modelling.....The

leaders of the world are flying the economy by the

seat of their pants, say J. Doyne Farmer and Duncan

Foley. There is, however, a better way to help guide

financial policies”

Nature 460, 685-686 (6 August 2009)

“Modelling to contain pandemics.... Agent-based

computational models can capture irrational

behaviour, complex social networks and global scale

— all essential in confronting H1N1, says Joshua M.

Epstein”

Nature 460, 687 (6 August 2009)

Applications of Agent-Based Models in

Public Health and Social Science (1)

• Gorman et al. Agent-based modelling of drinking behaviour: a preliminary model and applications to theory and practice. Am J Public Health 2006; 96: 2055-2060.

• Auchincloss & Diez Roux. A new tool for epidemiology: the usefulness of dynamic-agent models in understanding place effects on health. American Journal of Epidemiology 2008; 168(1), 1-8.

• Diez Roux & Auchincloss. Understanding the social determinants of behaviours: can new methods help? International Journal of Drug Policy 2009; 20, 227-229.

• Galea et al. Social epidemiology and complex system dynamic modelling as applied to health behaviour and drug use research. International Journal of Drug Policy 2009; 20, 209-126.

Applications of Agent-Based Models in

Public Health and Social Science (2)

• Auchincloss et al. An agent-based model of income inequalities in diet in the context of residential segregation. Am J Prev Med 2011; 40(3), 303-311.

• Yang et al. A spatial agent-based model for the simulation of adults’ daily walking within a city. Am J Prev Med 2011; 40(3), 353-361.

• Epstein JM. Modelling civil violence: an agent-based computational approach. Proceedings of the National Academy of Sciences 2002; 99 (suppl. 3), 7243-7250.

• Maglio et al. Agent-based models and systems science approaches to public health. Am J Prev Med 2011; 40(3):392-394

Main Characteristics of Agent-based Models (1)

• Simulates “agents” (individuals) who – are heterogeneous in their characteristics

– make decisions autonomously (independently )

– interact with other individuals and with their environment using individually-tailored “behaviour rules”

• “Behaviour rules” are defined as those which govern the behaviour of individuals (can be derived from qualitative studies)

– respond dynamically to interventions (disturbances)

– can “adapt and learn” but are not necessary “rational” • “Rational” behaviour is defined in a strict narrow sense as that

which maximises explicitly the individual’s payoff (utility)

• “Adaptive and learning” behaviour is defined (also narrowly) as that which uses the individual’s experience of the consequences (positive or negative) of their past decisions on their payoff(utility)

Main Characteristics of Agent-based Models (2)

• Uses a “bottom-up” approach rather than a “top-down” approach to modelling behaviour– Deduces macro-level (community or population) behaviour by

allowing it to evolve (emerge from) “micro-level” (individual) interactions

– Can simulate complex macro-level behaviour which cannot be generated except by modelling interactions at the micro-level

• Can simulate macro-level behaviour which is not necessarily at “equilibrium”– e.g. In a “Nash equilibrium” there is no incentive for any individual to

change unilaterally their chosen behaviour (strategy) because no individual can choose an alternative behaviour which is more rewarding given the behaviour of all other individuals.

• Can simulate complex organized, segregated or chaotic spatio-temporal social systems– e.g. Panic behaviour during fire escape in confined spaces.

Behaviour Surface (Sudden Transition)

Explanatory variable 2Explanatory variable 1

Outcome

variable

Transition (step change)

Simulated in

Mathematica

Stability of Behaviour

Graphics: http://demonstrations.wolfram.com/StableEquilbria/

Stable equilibrium

Unstable equilibrium

Disturbance

Disturbance

Would behaviour bounce back to its natural state

after a disturbance?

Discussion

• Can you think of examples from public health in which interactions at the individual level lead to step changes at community level?

– Hints

• Alcohol and drug use behaviour?

• Civil violence?

• .....

• Can you think of examples in health behaviour which demonstrate instability?

How Can Simple Rules Governing

Micro-Level Behaviour (i.e. at the

individual level) Generate Complex

Macro-Level Behaviour (i.e. at the

community/population level)?

Example of a Simple Micro-Level Deterministic Rule

Generating Complex Macro-Level Behaviour (1)

Consider a one dimensional grid of cells. Each cell can be in one of two states:

white (0) or black (1).

Initially assume that all cells are white (i.e. in state 0) except for one cell

Now define a set of rules such that at the next (discrete) time step,

each cell either changes its state or stays in the same state, depending

on its current state and the state of its neighbours on the left and the

right)

Apply the same set of rules for subsequent time steps

Example of a Simple Micro-Level Deterministic Rule

Generating Complex Marco-Level Behaviour (2)

1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1

1 1 1 1000 0

An example of a rule: “Rule 30” ≡ (00011110)2

Description of the rule

� If both the cell and its neighbour on the right are of state 0 (white), change the

state of the cell at the next time step to be the same as that of its neighbour on

the left, otherwise change the state of the cell to be opposite to its neighbour on

the left.

(Stephen Wolfram 2002)

Example of a Simple Micro-Level Deterministic Rule

Generating Complex Macro-Level Behaviour (3)

t=0

t=1

t=2

time

As time progresses, a geometrical pattern formed of white and black cells evolve.

Apply “Rule 30”

Apply “Rule 30”

t=...

Example of a Simple Micro-Level Deterministic Rule

Generating Complex Macro-level Behaviour (4)

Pattern after 10 time steps* Pattern after 50 time steps*

* Simulated in Mathematica

Pattern after 100 time steps*

Chaotic pattern with some pockets of embedded

regularity (fractal structures)

Application of “Rule 30”

Why should this happen?

Example of a Simple Micro-Level Deterministic Rule

Generating Complex Marco-Level Behaviour (5)

1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1

1 1 1 1000 0

If both the cell and its neighbour on the right are of state 0, change the state of the cell at the next time step

to be the same as that of its neighbour on the left, otherwise change the state of the cell to be opposite to its

neighbour on the left.

(Stephen Wolfram 2002)

α β γ

x=(α+β+γ+β×γ) mod 2

Definition of “Rule 30” in numerical form

Example of a Simple Micro-Level Deterministic Rule

Generating Complex Macro-Level Behaviour (6)

Pattern after 50 time steps*

* Simulated in Mathematica

“Rule 30”

Pattern after 50 time steps*

“Rule 26”

Simulation of two different rules starting from the same initial conditions

Example of Individual Behaviour Rules

Generating Complex Crowd Behaviour (1)

Simulating random walkers (Nishidate et al 1996; Gaylord & D’Andria 1998)

X

� Space is divided into a number

of grid cells

� Each cell is identified by its

state: an integer number (0,1,2,3,4)

indicating whether the cell is

empty, occupied by an individual

facing N, E, S and W

� No two individuals can occupy

the same cell

� An individual X can move to an

empty cell on its N,E,S or W if

he/she is facing it and if no other

individual in the local

neighbourhood is facing it.

� The grid cells indicated by the

letter Ω define the local

neighbourhood of X (known as

Gaylord-Nishidate neighbourhood)

N

S

EW

Ω

Ω

Ω Ω

Ω

Ω

Ω

Ω

Ω

Ω

Ω

Ω

Ω

Example of Individual Behaviour Rules

Generating Complex Crowd Behaviour (2)

� Dimension of the grid cell matrix (10×10)

� Percentage occupancy of cells by individuals (65%)

� Number of discrete time steps (10,000)

� Different shading levels indicate state of cell (white means unoccupied)

� At each time step, the individual selects randomly the direction he/she is facing and then either

stays or moves.

Initial configuration (t=0) Final configuration (t=10,000)

Model simulated in Mathematica

(Gaylord and D’Andria 1998)

Simulating random walkers

Extension of the Random Walkers Model

Simulating social grouping and segregation

�Each individual X is endowed

with a set of beliefs (opinions)

�An individual can (i) either

move away from his/her local

neighbourhood if the majority

(in his/her neighbourhood) do

not share his/her beliefs, (ii)

conform to the beliefs of the

majority by changing his/her

beliefs to fit the majority, or

(iii) be indifferent to the beliefs

of his/her neighbourhood

� Shaded area is known as the

VonNeumann local

neighbourhood

Ω

X

Schelling’s Segregation Model (1)

Y

� A community is divided into a

number of grid cells each

representing a location of a home

� Each cell is identified by an

integer number (0, 1,2) indicating

whether it is empty (0), occupied by

individual of characteristics X or by

an individual of characteristic Y.

� X is only interested in his/her

local neighbourhood.

� X is “happy“ to stay in a cell if

the proportion of Y individuals in

X’s local neighbourhood does not

exceed a threshold; if X is unhappy,

they move.

� “Unhappy“ X individuals move

to their nearest local neighbouring

empty space

� The same above rules apply for Y

individuals

� X and Y individuals keep moving

until they are all “happy”.

X

X

X

X

Y

Y

Schelling’s Segregation Model* (2)

* Simulated in Mathematica. Lu PS. Schelling’s model of residential segregation” From the Wolfram Demonstrations

Project. http://demonstrations.wolfram.com/SchellingsModelOfResidentialSegregation

Individual

attitudes

Perceived

individual risk

Perceived

crime rate

Emotional

responses

Violent crime

Environmental

crime e.g.

vandalism

Public space

and transportHousing

Perceived

physical

environment

Perceived

social

environment

Mental

health

Physical

health

Interpersonal

relationships

& networks

Health

behaviours

Social

inequalities

Neighb’hood

& community

factors

Fear of crime

Health and

wellbeing

Social

environ-

ment

Built environment

Crime

Social

representations

Individual

demographics

Individual

crime risk

Avoidance

behaviours

Economic

policy

Social policy Crime and justice

policy

National

policies

National and

international economy

Mass media

C o g n i t i v e h e u r i s t i c s a n d b i a s e s

Perceived

vulnerability

Drug- and

alcohol- related

crime

Adapted from

Lorenc et al (2012)

Built environment, social environment, crime, fear of crime, health and well-being: causal & intervention pathways

Individual

attitudes

Perceived

individual riskEmotional

responses

Violent crime

Environmental

crime e.g.

vandalism

Public space

and transport

Housing

Mental

health

Physical

health

Interpersonal

relationships

& networks

Health

behaviours

Social

inequalities

Neighb’hood

& community

factors

Fear of crime

Health and

wellbeing

Social environ-

mentBuilt environment

Crime

Perceived

vulnerability

Drug- and

alcohol- related

crime

Adapted from Lorenc et al (2012)

Schematic of an ABM

Intervention

Built environment

Housing

Social environment

Individual in a housing complex interacts

with other individuals and the surrounding

environment (built, social and crime).

Individual behaviour is defined by a set of rules.

Crime

Generated community behaviour

Intervention

Issues with Agent-Based Models

• Opaque/black-box models

• Outcomes sensitive/very sensitive to behaviour rules

• Defining individual behaviour rules– Role for qualitative studies

• Validation of ABMs– Data requirements (qualitative & quantitative studies)

• Is the interest in ABMs driven by “complexity science”, “computer science”, or “social and health” sciences?

• Software for implementation– Mathematica

– REPAST

– .....

Discussion

• Are agent-based models appropriate to evaluate complex interventions such as

– Transport and health

– Built environment and health

– Climate change and health

– Drugs and health behaviour

– Gender violence and health

– Sexual health

– .... ?