Use of Mathematical Models to Evaluate Complex Public Health ...
Transcript of Use of Mathematical Models to Evaluate Complex Public Health ...
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?
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
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?.....
Characteristics of Complex Dynamics
(“Chaos”)
• Determinism
• Sensitivity to initial conditions
• Nonlinearity
• Presence of an attractor
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?
• .....
“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
– .....