System Dynamics Modeling:
Overview & Causal Loop Diagrams
Nathaniel Osgood
CMPT 394
What is System Dynamics?
•
A feedback-oriented
perspective
•
A broad, evolving
methodology
to help
Conceptualize
Describe
Analyze
Manage
System Dynamics offers…
•
Qualitative & quantitative components
•
A defined, incremental and iterative modeling
process that delivers value throughout
•
Time-honed techniques for working with diverse
interdisciplinary stakeholders
•
Evolved software permitting focus on the
“what” is being described – not
how
it is run
•
A rigorous mathematical foundation
•
A rich set of analysis tools
•
Techniques for interfacing closely with cognate
areas
(e.g. statistics, decision sciences, evidence-based
Differences in Framing the Issues
•
Modeling methodologies are often
distinguished more fundamentally by the
questions being asked than by the answers
being given
•
Such methodologies will often be most
distinguished by the way in which they
frame problems
•
In comparing, we must be conscious of
these differences
Engagement in the Human Theatre
Uses of SD Models [Hovmand]
[Modeling] Power to the People:
Fostering Participatory Discourse
•
To empower participatory modeling, System
Dynamics tends towards simpler formalisms
•
Capturing qualitative understanding
•
Easy understanding by stakeholders:
Simulation model
• Inspection
• Dialogue
• Manipulation
•
Declarative (“programming free”) specification
•
Intuitive graphical representation
Features support high stakeholder involvement in
model conceptualization, formulation & analysis
Stages of the System Dynamics
Modeling Process
A Key Deliverable!
Model scope/boundary
selection.
Model time horizon
Identification of
key variables
Reference modes for
explanation
Causal loop diagrams
Stock & flow diagrams
Policy structure
diagrams
Group model building
Specification of
•
Parameters
•
Quantitative causal
relations
•
Decision rules
Initial conditions
Reference mode
reproduction
Matching of
intermediate time
series
Matching of
observed data point
Constrain to sensible
bounds
Structural sensitivity
analysis
Specification &
investigation of
intervention scenarios
Investigation of
hypothetical external
conditions
Cross-scenario
comparisons (e.g.
CEA)
Parameter sensitivity
analysis
Cross-validation
Robustness&extreme case
tests
Unit checking
Problem domain tests
Formal analysis
Learning
environm
ents/Micr
oworlds/f
light
simulator
s
Qualitative &
Semi-quantitative insights
Infectives
New Infections
People Presenting
for Treatment
Waiting Times
+
+
Health Care Staff
-Susceptibles
-Contacts of
Susceptibles with
Infectives
+
+
+
+
Normal and Underweight Weight Overweight Pregnant with GDM History of GDM T2DM Developing Obesity Pregnant Normal Weight Mothers with No GDM History Completion of Pregnancy to Non-Overweight State Completion of GDM Pregnancy Women with History of GDM Developing T2DM Overweight Individuals Developing T2DMNormal Weight Individuals Developing
T2DM Pregnant with
T2DM New Pregnancies from Mother with T2DM Completion of Pregnancy for Mother with T2DM
Pregnant Overweight Mothers with No GDM History Pregnancies of Overweight Women Completion of Pregnancy to Overweight State Pregnancies of Non-Overweight Women Pregnancies to Overweight Mother Developing GDM Pregnancies to Non-Overweight Mother Developing GDM Pregnant with Pre-Existing History of GDM Pregnancies for Women with GDM Pregnancies Developing GDM from Mother with GDM History Completion of Non-GDM Pregnancy for Woman with History of GDM Shedding Obesity Pregnant Women
Developing Persistent Overweight/Obesity Oveweight Babies Born
from T2DM Mothers Pregnant Women with GDM
that Continue on to Postpartum T2DM Normal Weight Babies Born from Non-GDM Mother with History of GDM Overweight Babies Born from
Non-GDM Mother with History of GDM
Normal Weight Babies Born from GDM Pregnancy Overweight Babies Born
from GDM Pregnancy Overweight Babies Born to Pregnant Normal Weight Mothers Overweight Babies Born from Pregnant Overweight Mothers
Normal Weight Babies Born to Mothers without GDM
Normal Weight Babies Born from T2DM Pregnancy
Pregnancy Duration
<Birth Rate>
Normal Weight Babies Born to Overweight Mothers without GDM Normal Weight Deaths Overweight Deaths T2DM Deaths
Deaths from Non-T2DM Women with History of GDM
2
2
2
2
(
)
(
)
S
I
h
I
I
h
S
I
h
I
I
h
Baseline
50%
60%
70%
80%
90%
95%
98%
100%
Average Variable Cost per Cubic Meter
0.6
0.45
0.3
0.15
0
0
1457
2914
4371
5828
Time (Day)
Quantitative insights
Value of the Modeling Process
•
Often the
modeling process
itself – rather
than the models created – offers the
greatest value
•
Modeling as theory building: Refinement of
mental models
•
Reflecting on
•
Mental models
•
What is & is not known / data
Benefits of Rich Stakeholder Participation
•
Developing rich, grounded understanding
•
Building community capacity
•
Critiquing model behavior
•
Fostering stakeholder cooperation
•
Implementing policy recommendations
•
Facilitating data collection design
•
Aiding in replanning
•
Keeping model updated
•
Empowering community self-guidance
Model Conceptualization: Feedback Loops
•
Loops in a causal loop diagram indicate
feedback
in the system being represented
•
Qualitatively speaking, this indicates that a
given change kicks off a set of changes that
cascade through other factors so as to either
amplify (“reinforce”) or push back against
(“damp”, “balance”) the original change
•
Loop classification: product of signs in
loop (best to trace through conceptually)
•
Balancing loop: Product of signs negative
Example: Physical Systems
With & Without Feedbacks
•
Balancing
•
Driving / flying
•
Thermic regulation: “Hot blooded”
(homeothermy) vs. “cold blooded”
(ecothermic) animals
Introducing a feedback can
fundamentally alter a system’s
behaviour
Recall: A Common Problem:
Overly Narrow
Mental
Models
•
Most decisions are based on mental models
•
Frequently the failure to anticipate &
account for policy resistance is due to
narrow mental models
•
Deleterious effects are blamed on “side effects”
of anticipated process
•
Failure to consider interactions between diverse
pieces of system (each of which may be well
understood)
•
System dynamics seeks a broader
Examples of Deleterious Feedbacks
•
Cutting cigarette tar levels reduces cessation
•
Cutting cigarette nicotine levels leads to
compensatory smoking
•
Targeted anti-tobacco interventions lead to equally
targeted coupon programs by tobacco industry
•
Charging for supplies for diabetics leads to higher
overall costs by increases costs due to reduced
self-management, faster disease progression
•
ARVs prolong lives of HIV carriers, but lead to
resurgent HIV epidemic due to lower risk perception
•
“Saving money” by understaffing STI clinics, leads to
long treatment wait, greater risk of transmission by
infectives & bigger epidemics
•
Antibiotic overuse worsens pathogen resistance
•
Antilock breaks lead to more risky driving
Causal Loop Diagram
• Focuses on capturing causality
–
and
especially
feedback
effects
• Indicates sign of causal impact (+ vs.
–
)
– x →
+
y indicates
– x →
-
y indicates
Hunger
Food Ingested
+
-0
y
x
0
y
x
Causal Loop Diagram
– An arrow with a positive sign (+):
“
all
else remaining equal, an increase
(decrease) in the first variable increases
(decreases) the second variable above
(below) what it would otherwise have
been.
”
– An arrow with a negative sign (-):
“
all
else remaining equal, an increase
(decrease) in the first variable decreases
(increases) the second variable below
(above) what it otherwise would have
been.
”
Reasoning about Link Polarity
•
Easy to get confused regarding link
polarity in the context of a causal chain
•
Tips for reasoning about link polarity
for X→Y
•
Reason about this link in isolation
• Do not be concerned about links preceding X or
following Y
•
Ask
“
if X were to INCREASE, would Y
increase or decrease
”
?
• Increase in Y implies
“
+
”
,decrease in Y implies
“
-
”
• If answer is not clear or depends on value of X, need to
think about representing several paths between X and Y
Tips
•
Variables should be noun phrases
•
Variables should be at least ordinal
•
Links should have unambiguous
polarity
•
Remember factors involving people
•
Avoid mega-diagrams
•
Label loops
•
Distinguish perceived and actual
situation
•
Incorporate targets of balancing loops
Ambiguous Link
•
Ambiguous Link: Sometimes +,
sometimes -
•
Replace this by disaggregating causal
pathways by showing multiple links
Overtime
Fatigue
More Time
Working
Greater Incorporation of
Outside Tasks at Work
Efficiency
+
+
--Work Accomplished
per Day
+
+
+
Overtime
Work Accomplished
per Day
Feedback Loops
•
Loops in a causal loop diagram
indicate
feedback
in the system being
represented
•
Qualitatively speaking, this indicates that a
given change kicks off a set of changes
that cascade through other factors so as
to either amplify (“reinforce”) or damp
(“balance”) original change
•
Loop classification: product of signs in
loop (best to trace through
conceptually)
•
Balancing loop: Product of signs negative
•
Reinforcing loop: Product of signs
positive
Dysphoria &
Stress
Substance Abuse
Costs
Employability
Stigmatization
+
-Nutrition
-
Risk of Injury &
Accidents
+
+
Health
Poverty
-+
+
+
-+
Capacity for
Productive
Work
-+
Impulse towards
Self-Medication
+
+
Reinforcing (positive) feedback
Advantages of Recognizing Feedback
Balancing Feedback
(Stability)
Reinforcing feedback
(Instability)
Desirable
Securing resiliency
Individual
self-regulation
Enabling rapid intervention
success
Viral approaches, peer
messaging, rapid social change
Undesirable
Preventing policy resistance
& adverse “lock in” effects
Low tar & nicotine
cigarettes
Risk perception for
Infectious diseases
Heading off rapidly growing
vicious cycles
Addictions, “cycle of poverty”,
SAVA
Structure interventions (and system) to
achieve this
Gates’ Comments
•
“The biggest advantage we have is that
good developers like to work with good
developers”
•
[Cusumano&Selby,’95]
•
“Most people don’t get millions of
people giving them feedback about their
products…We have this whole group of
[2000] people in the US alone that takes
phone calls about our products and logs
everything that’s done. So we have a
better feedback loop, including the
market”
•
[Cusumano&Selby,’95]
•
“As [NT] got more applications, NT
servers got more popular. As it’s
gotten more popular, we’ve got
more applications.”
•
Computer Reseller News, 9-23-1996
•
“[T] he more users [the internet]
gets, the more content it gets, the
more users it gets.”
•
Red Herring, 10-1995
•
“It’s all about scale economies and
market share. When you’re
shipping a million units of Windows
software a month, you can afford to
spend $300 million a year improving
it and still sell at a low price.”
Examples: Vicious/Virtuous Cycles
•
Positive (reinforcing) feedback can
lead to extremely rapid changes in
situation
Existing Users
Likelihood of Cross Listing
and Listing on Search
Engines
+
New Users
Discovering Site
+
Number of Connections to
Music Download Server
Length of Time Per
Download
Likelihood of User Starting
Multiple Simultaneous
Downloads
+
+
+
Confusing Code
Ease of Understanding
where to Make a Change
Confusing
Additions
+
-Word of
Mouth Sales
Customers
+
+
+
Simple Causal Loops
Change Requests
Project Duration
Remaining Work
+
+
+
Overbearing PM
M anagement Style
Willingness of Project
Part icip ant s t o share info
with PM
PM Suspicion
-+
Target Budget
Estimated Design Cost s
beyond Target Budget
Design Scope
-+
-Changes to
Schedule
Job Rhy thm
Aggregate
Product ivit y
-
+
-Example “Balancing Loops”
•
Balancing loops tend to be
self-regulating
Aggregate Computer
Responsiveness
Programs Run
Simultaneously
Virtual Memory
Swapping
+
+
-Mistakes
Learning from
Mistakes
+
-Risk Management
Unmanaged Risks
Schedule
Disruptions
+
Time taken for Risk
Assessment and
Management
-Longer Term Cost of Pressure:
Cutting Corners
Vicious Cycles
Turnover Vicious Cycle
Developer Fatigue
Morale
Resignations
-Backlog of Work
+
-Work per Team
Member
+
Extra Hiring
Related Work
+
Team Productivity
-Reinforcing Loop Dynamics:
Exponential Growth
•
Example
•
Dynamic
implications
Word of
Mouth Sales
Customers
+
+
+
Graph for Stock
20,000
15,000
10,000
5,000
0
0
10
20
30
40
50
60
70
80
90
100
Time (M onth)
Stock : Current
Site Popularity
Likelihood of Cross Listing
and Listing on Search
Engines
+
Causal Loop Dynamics: Goal Seeking
(Balancing Loop)
•
Example:
•
Dynamic behavior
Hunger
Food Ingested
+
-Treshold for Policy
Dissatisfaction to Lead to
Action
Threshold Hunger to
Causal Loop Dynamics: Oscillation
(Balancing Loop with
Delay
)
•
Causal Structure
Growth and Plateau
•
Loop structure:
•
Reinforcing Loop
•
Balancing Loop
•
Dynamic Behavior:
Word of
Mouth Sales
Customers
+
+
+
Potential
Customers
-+
-Graph for Customer
1 0 0 ,0 0 0
7 5 ,0 0 0
5 0 ,0 0 0
2 5 ,0 0 0
0
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
Tim e ( Mo n th )
Cu sto mer : Cu r ren t
Existing Users
Likelihood of Cross Listing
and Listing on Search
Engines
+
New Users
Discovering Site
+
Internet Users Yet to
Discover Site
-+
Regulatory Mechanisms
Perverse Incentives Under Stress
Elaborating Causal Loops
Work Remaining
Work Pressure
Product ivity
+
+
-Work Remaining
Work Pressure
Product ivity
+
+
-Fatigue
+
-Elaborating Causal Loops 2
Project
Performance
Managerial Desire
to Blame
Information
Availability
Quality of Management
Decision Making
-
-+
+
Project
Performance
Managerial Desire
to Blame
Information
Availability
Quality of Management
Decision Making
Managerial Trust of
Developers
-+
+
+
Developer's Trust of
Manager
+
+
+
-+
Exercise 1: Link & Loop Polarity
•
Label the polarity of each
link in this diagram
•
What factor is missing?
•
Is the feedback
positive or negative?
Population of
Downloading Users
Time per
Download
Bandwidth
Available to Server
Bandwidth Available
Per Connection
Exercise 2: Unanticipated Side
Effects
•
Seeking modest investment in
Blackberries for productivity
enhancement
•
Muddied by unanticipated side
effects
Exercise 3: Link & Loop Polarity
•
Create one or more causal loops
relating
•
Project Morale
•
Project Turnover
•
Workload
•
Project delay
•
Are these loops positive or negative?
•
Do these loops all have the same
time to ‘kick in’ (time constants)?
One Set of Loops
Developer Fatigue
Morale
Resignations
-Backlog of Work
+
-Work per Team
Member
+
Extra Hiring
Related Work
+
Team Productivity
-Recall: Dealing with Symptoms vs.
Causes
•
Our focus is often on undesirable
symptoms (high cost, schedule
delay, poor quality) rather than on
underlying causes
•
Often a project is in severe trouble by
the time these obvious (and easily
quantifiable) symptoms appear
•
Often choices of interventions fail to
consider broader (and perhaps less
quantifiable) effects of actions
Managerial Pressure
•
Narrow mental model
aims for this goal
•
Unanticipated side effects “push back”
vs. time savings & cause quality
A Bigger Picture
Project Lateness
Thoroughness of
Testing
Reliability of Bug
Fixes
Managerial
Pressure
Overtime
Time for Bug Fix
Task
Fatigue
Multiplexing of
Tasks
Coordination and
Degree of Thought for
Fixes
-+
+
+
-+
+
-+
+
Pressure for "Quick
and Dirty" Fixes
-New Defect
Resolution Rate
-Total Patent or
Latent Defects
Defects
Resolution Rate
+
-
+
Work Accomplished
per Day
+
-Debugging Work to
be Done
+
+
+
Quality of Released
Product
+
-Morale
-
--
+
-Resignations
-New Hiring
Training Needs
Fraction of Staff that is
New to Project
Familiarity of Workers
with Codebase
-Path Dependence/Network Effects
& Lock-In
•
In the presence of
positive feedback,
can get “lock in”
effects
•
Similar early
conditions result
in
divergent
outcomes (instability)
•
Example: Product largely in balance vs.
Unstable, Critical, and Subject to
Lock-in
•
Software Quality
•
Trust
•
Respect
•
Morale
Elaborating Causal Loop Diagrams:
Most Basic Feedbacks
With Risk Perception-Driven
Behavioral Change
Public Health System
Blowback:
Perverse Evolutionary
& Behavioural Feedbacks
Structure as Shaping Behaviour
•
System structure is defined by
•
Stocks
•
Flows
•
Connections between them (yielding feedbacks)
•
Nonlinearity: The behaviour of the whole is
more than the sum of the behaviour of the
parts
•
“Emergent” behaviour would not be anticipated
from simple behaviour of each piece in turn
•
Stock and flow structure (including feedbacks)
of a system determines the qualitative
behaviour modes that the system can take on
(parameters determine particulars)
•
Changes to the feedback structure can change
Causal Loop Structure: Dynamic Implications
•
Each loop in a causal loop diagram is
associated with qualitative dynamic
behavior
•
Most Common Single-Loop Modes of
Dynamic Behavior
•
Exponential growth
•
Goal Seeking Adjustment
•
Oscillation
•
When composed, get novel behaviors due to
shifting loop dominance
•
Behaviour of system more than sum of parts
Causal Loop Dynamics: Exponential Growth
(First Order Reinforcing Loop)
•
Examples
•
Dynamic implications
# of Infectives
# New Infections
+
+
# of Susceptibles
-+
Weight Perceived
as Normal
Individual Target
Weight
Mean Weight in
Population
+
+
+
Causal Loop Dynamics: Goal Seeking
(Balancing Loop)
•
Example:
•
Dynamic behavior
Hunger
Food Ingested
+
-Treshold for Policy
Dissatisfaction to Lead to
Action
Threshold Hunger to
Causal Loop Dynamics: Oscillation
(Balancing Loop with
Delay
)
•
Causal Structure
Growth and Plateau
•
Loop structure:
•
Reinforcing Loop
•
Balancing Loop
•
Dynamic Behavior:
Word of
Mouth Sales
Customers
+
+
+
Potential
Customers
-+
-Graph for Customer
1 0 0 ,0 0 0
7 5 ,0 0 0
5 0 ,0 0 0
2 5 ,0 0 0
0
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
Tim e ( Mo n th )
Cu sto mer : Cu r ren t
Existing Users
Likelihood of Cross Listing
and Listing on Search
Engines
+
New Users
Discovering Site
+
Internet Users Yet to
Discover Site
-+
State of the System: Stocks
(“Levels”, “State Variables”,
“Compartments”)
•
Stocks (Levels) represent accumulations
•
These capture the “state of the system”
•
These can be measured at
one instant in time
•
Stocks start with some initial value & are
thereafter changed only by
flows
into & out of
them
•
There are no inputs that immediately change stocks
•
Stocks are the source of delay in a system
The Critical Role of Stocks in Dynamics
•
Stocks determine current state of
system
•
Stocks often provide the basis for
making choices
•
Stocks central to most disequilibria
phenomena (buildup, decay)
•
Lead to inertia
State Changes: Flows
(“Fluxes”, “Rates”, “Derivatives”)
•
All changes to stocks occur via
flows
•
Always expressed per some unit time: If
these flow into/out of a stock that keeps
track of things of type
X (
e.g. persons), the
rates are measured in
X
/(Time Unit) (e.g.
persons/year, $/month, gallons/second)
•
Typically measure over certain period of
time (by considering accumulated quantity
over a period of time)
•
e.g. Incidence Rates is calculated by
accumulating people over a year, revenue is
$/Time, water flow is litres/minute
Key Component: Stock & Flow
Stock
Flow
+
Stock
Flow
Stocks
Determine
Flows
Flows
Dictate
Change in
Stocks
Auxiliary Variables
•
Auxiliary variables are convenience names we
give to concepts that can be defined in terms
of expressions involving stocks/flows at
current time
•
Adding or eliminating an auxiliary variable does not
change the mathematical structure of the system
•
Critical for model transparency
•
Can be reused at many places
•
References to auxiliary variables prevents need for
modeler to think about all of details of definition
•
Enhanced modifiability: Single place to define
•
Convenient for reporting (graphing, tables) &
Example Model: Auxiliary
Variables
Constants & Time Series Parameters
•
For similar reasons to auxiliary
variables, we give names to
•
Model constants
Handling Heterogeneity
•
Step 1: Test (via scenarios) if heterogeneity is
likely to have substantive impact on results
•
Step 2: Where necessary, disaggregate
model according to heterogeneity
•
Small model: Duplicate stocks
•
Larger model: Subscripting
•
Step 3: Express inter-group contact patterns
via mixing & preference matrices
Susceptible (X-16fb) Infected (Y-16fb) Undetected CIN1 (CIN1-16fb) Detected CIN1 (DCIN1-16fb) Treated and Infected CIN1 (ICIN1-16fb) Undetected CIN2 (CIN2-16fb)
Undetected CIN3 (CIN3-16fb)
Detected CIN2 (DCIN2-16fb) Detected CIN3 (DCIN3-16fb)
Treated and Infected CIN2 (ICIN2-16fb) Immune (Z-16fb)
Treated and Infected CIN3 (ICIN3-16fb) Treated and Cured TCINs (TCINs-16fb) Undetected Carcinoma s1 (CIS1-16fb)
Detected Carcinoma s1 (DCIS1-16fb) Treated and Infected Carcinoma s1 (ICIS1-16fb) Detected Carcinoma s2 (DCIS2-16fb) Treated and Infected Carcinoma s2 (ICIS2-16fb) Undetected Carcinoma s2 (CIS2-16fb)
Treated and Cured TCISs (TCISs-16fb) Undetected Local (CCL-16fb)
Incidence Per Contact Riskof Infection(beta-f)
Total Infectives (f) by Age SmokingStatus Sexual Activity Group and Screening Category Immune (Z-16m)
Mean duration of acute HPVinfection females
Progression from Infected to Undetected CIN1 Undetected CIN1 Rate (theta-16fbs1)
Progression from UndetectedCIN1 to Detected CIN1 Detected CIN1 Rate(kappa-16fbis1)
Progression from Detected CIN1 to Treated & Infected CIN1 Cure Rate of CIN1 (cap-gamma-s1) % CIN1 infectedafter treament(psi-s1)
Progression from Detected CIN1 to Treated & Cured TCINs Female
Mortality Rate
<Cure Rate of CIN1 (cap-gamma-s1)> <% CIN1 infected after treament (psi-s1)> Progression from Infected to UndetectedCIN3 Undetected CIN3 Rate (theta-16fbs3) Undetected CIN2 Rate (theta-16fbs2) Progression from Undetected CIN1 toUndetected CIN2 Undetected CIN1 Rate (pi-16fbs1) Progression from Undetected CIN2 to Undetected CIN3 UndetectedCIN2 Rate(pi-16fbs2) Undetected CIN3 Rate (pi-16fbs3) Progressionfrom UndetectedCIN3 to Undetected Carcinoma s1 Progression from Detected CIN1 to Undetected CIN2 Progression from Undetected CIN2 toDetected CIN2 Detected CIN2 Rate (kappa-16fbis2) Detected CIN3 Rate (kappa-16fbis3)
Progression from Detected CIN2 toTreated & Infected CIN2 Cure Rate of CIN2(cap-gamma s2)
% CIN2 infected after treatment(psi-s2)
Recurrence of CIN2(theta- rs2) Recurrence of CIN1(theta-rs1)
<Cure Rate of CIN2(cap-gamma s2)> <% CIN2 infected aftertreatment (psi-s2)>
Cure Rate of CIN3 (cap-gamma s3) % CIN3 infected after treatment(psi-s3) Regression from
Treated & Infected CIN3 to UndetectedCIN3
Reoccurence of CIN3 (theta-rs3)
Detected CIS1 Rate(kappa-16fbis4)
% CIS s1 infected after treatment(psi-s4) Cure Rate of CIS s1(cap-gamma s4) Undetected CIS1 Rate (pi-16fs4)
Reoccurenceof CIS1 (theta-rs4)
Reoccurenceof CIS2(theta-rs5)
Detected CIS2 Rate (kappa-16fbis5)
% CIS s2 infected aftertreatment (psi-s5) Cure Rate of CIS s2 (cap-gamma s5) <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate>
Male Population
Female Population
Female TotalDeaths
Detected Local (DCCL-16fb)
Regression CIN1 to InfectedRate (tau-16fbs1) Regression CIN2 to Infected Rate(tau-16fbs2) Regression CIN3 to Infected Rate (tau-16fbs3)
Regression CIN2to CIN1 Rate (tau-16fbs21) Regression CIN3 to CIN1 Rate (tau-16fbs31)
Regression CIN3to CIN2 Rate(tau-16fbs32) Regressionfrom UndetectedCIN2 to UndetectedCIN1 Regression from Undetected CIN3 to Undetected CIN2 Regression from Undetected CIN2 toInfected Regression from Undetected CIN3 to Undetected CIN1
Regression from Treated & Infected CIN1 to Undetected CIN1 Recovered from Detected CIN1
Recovered from Treated & Infected CIN1 Recovered from Undetected CIN1
Regression from Treated & Infected CIN2 to Undetected CIN2 Progression from Detected CIN2 to Treated & Cured TCINs Recovered from Undetected CIN2
Recovered from Detected CIN2
Recovered from Treated & Infected CIN2 Recovered fromUndetected
CIN3
Progression from Undetected CIN3 to Detected CIN3
Progression from Detected CIN3 to Treated & InfectedCIN3
Progression from Detected CIN3 toTreated & Cured TCINs Progression from
Undetected Carcinoma s1 to Detected Carcinoma s1 Progression from UndetectedCarcinoma s1 to UndetectedCarcinoma s2
Progression from Detected Carcinoma s1 to Treated &Infected Carcinoma s1 Recovered from Detected CIN3
Recovered from Treated & InfectedCIN3
Regression from Treated & Infected Carcinoma s1 to Undetected Carcinoma s1 Progression from
Detected Carcinoma s1 to Treated & Cured TCISs Progression from Undetected Carcinoma s2 to Detected Carcinoma s2
Progression from Detected Carcinoma s2 to Treated & Infected Carcinoma s2 Progression from Detected Carcinoma s2 toTreated & Cured TCISs Progression from Detected Carcinoma s1 to UndetectedCarcinoma s2
Regression from Treated & Infected Carcinoma s2 to Undetected Carcinoma s2 Recovered from Treated
& Infected Carcinoma s1 Recovered from Treated & Infected Carcinoma s2
Undetected Regional (CCR-16fb) Cancer Death
Undetected Distant (CCD-16fb) Detected Regional (DCCR-16fb) Cancer Survivors (SCC) Detected Distant (DCCD-16fb) Progression from Undetected Local to Undetected Regional Progression from Undetected Regional to Undetected Distant Progression from
Undetected Local toDetected Local Progession from Detected Local toCancer Death Progression from Detected Local to Cancer
Survivors Detected Distant toProgression fromCancer Survivors Progression from Detected Distant toCancer Death Progression from Detected Regional toCancer Death Progression from Undetected Regional toDetected Regional Progression from
Undetected Local toCancer Death
Progression from Undetected Distant to Cancer Death Progression from Undetected Regional toCancer Death
Progression from Detected Regional toCancer Survivors Progression from
Undetected Carcinoma s2 to Undetected Local
Progression from Detected Carcinoma s2 to Undetected Local
Regression from UndetectedCIN1 to Infected Progression from Infected to Undetected CIN2 Regression from Undetected CIN3 toInfected
Progression from DetectedCIN3 to UndetectedCarcinoma s1
<Cure Rate of CIS s2 (cap-gamma s5)> <% CIS s2 infectedafter treatment(psi-s5)>
Undetected Local to Undetected RegionalRate (pi-L) Undetected Local to Detected Local Rate(upsilon-L)
DCC death rate (chi-D) LCC death rate
(chi-L)
Undetected Regional to Undetected Distant Rate(pi-R) RCC death rate(chi-R)
Detected Regional Cancer Survivors Rate(omega-R) Detected Local CancerSurvivor Rate(omega-L)
Detected Distant Cancer Survivor Rate (omega-D) Undetected Regional to Detected Regional Rate(upsilon-R)
Undetected Distant to Detected Distant Rate(upsilon-D) Progression from Undetected Distant toDetected Distant muX-fb death muY-fb death mu-CIN1 death mu-DCIN1death mu-ICIN1 death mu-ICIN2death mu-DCIN2 death mu-CIN2 death mu-CIN3 death mu-TCINsdeath mu-DCIN3death <Female Mortality Rate> mu-CIS1death <Female Mortality Rate> mu-ICIN3death <Female Mortality Rate> mu-ICIS1 death mu-DCIS1death
mu-TCISsdeath Mortality Rate><Female
<Female Mortality Rate> mu-CIS2 death mu-DCIS2 death mu-ICIS2death <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> mu-CCL death mu-CCRdeath mu-DCCLdeath mu-DCCRdeath mu-SCC
death mu-DCCDdeath
<Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <Female Mortality Rate> <DCC death rate(chi-D)>
Undetected CIS2 Rate (pi-16fbs5)
<RCC death rate(chi-R)> <LCC death rate
(chi-L)> <Infected (Y-16fb)>
Progression from Immune to Susceptible(f) Rate of waning natural immunity (sigma-zf)
<mu-CCL death>
<mu-CCRdeath> <mu-CIN1death>
<mu-CIN2death> <mu-CIN3death> <mu-CIS1 death> <mu-CIS2death> <mu-DCCDdeath> <mu-DCCLdeath> <mu-DCCRdeath> <mu-DCIN1death> <mu-DCIN2death> <mu-DCIN3death> <mu-DCIS1death> <mu-DCIS2death> <mu-ICIN1 death> <mu-ICIN2death> <mu-ICIN3death> <mu-ICIS1 death> <mu-ICIS2death> <mu-SCC death> <mu-TCINsdeath> <mu-TCISs death> <muX-fbdeath> <muY-fb death>
<Female TotalDeaths>
Regression from Detected CIN3 toInfected
Regression from Detected CIN2 toInfected
<Regression CIN2 toInfected Rate(tau-16fbs2)>
Regression from Detected CIN1 toInfected
<Regression CIN1 to Infected Rate (tau-16fbs1)>
Regression from Detected CIN3 to Detected CIN2<Regression
CIN3 to CIN2Rate (tau-16fbs32)>
Regression from Detected CIN3 to Detected CIN1
<Regression CIN3 toCIN1 Rate (tau-16fbs31)> Regression from Detected CIN2 to Detected CIN1 <Regression CIN2 to CIN1 Rate (tau-16fbs21)> <Cancer Survivors (SCC)> <Detected Carcinomas1 (DCIS1-16fb)> <Detected Carcinomas2 (DCIS2-16fb)> <Detected CIN1(DCIN1-16fb)> <Detected CIN2(DCIN2-16fb)> <Detected CIN3 (DCIN3-16fb)> <Detected Distant (DCCD-16fb)> <Detected Local(DCCL-16fb)> <Detected Regional(DCCR-16fb)>
<Treated and Cured TCINs (TCINs-16fb)><Treated and Cured
TCISs (TCISs-16fb)> <Treated and InfectedCarcinoma s1(ICIS1-16fb)> <Treated and InfectedCarcinoma s2 (ICIS2-16fb)> <Treated and Infected CIN1 (ICIN1-16fb)> <Treated and InfectedCIN2 (ICIN2-16fb)> <Treated and Infected CIN3 (ICIN3-16fb)> <Undetected Carcinoma s1 (CIS1-16fb)> <Undetected Carcinoma s2 (CIS2-16fb)> <Undetected CIN1(CIN1-16fb)>
<Undetected CIN2(CIN2-16fb)> <Undetected CIN3 (CIN3-16fb)> <Undetected Distant (CCD-16fb)> <Undetected Local(CCL-16fb)> <Undetected Regional (CCR-16fb)> <Total Infectives (f) by Age
SmokingStatus Sexual ActivityGroup and ScreeningCategory>
Fractional PrevalenceFemales Susceptible (X-16m)
Infected (Y-16m) Incidence(m)
Per Contact Risk of Infection (beta-m) Recovery(m) Male Mortality Rate <Male Mortality Rate> muX-m death muY-m death
Fractional Prevalence Males muZ-m death <muX-m death> <muY-m death> <muZ-m death>
Male Total Deaths Mean duration of acute
HPV infection in males
Rate of Recovery frominfection wtih HPV (gamma-16f) Progression from Immune to Susceptible (m)
Rate of waning natural immunity(sigma-zm)
muZ-fbdeath
<Female Mortality Rate>
Fraction of CINs regressions clearing CIN that also clear infection(gamma bar-16f)
<Regression CIN1 toInfected Rate(tau-16fbs1)>
<Fraction of CINs regressionsclearing CIN that also clearinfection (gamma bar-16f)>
<Regression CIN2 toInfected Rate (tau-16fbs2)>
<Rate of Recovery frominfection wtih HPV(gamma-16f)>
<Fraction of CINs regressionsclearing CIN that also clearinfection (gamma bar-16f)>
<Regression CIN3 toInfected Rate(tau-16fbs3)> <Rate of Recovery frominfection wtih HPV(gamma-16f)> <Rate of Recovery frominfection wtih HPV(gamma-16f)>
<Female Mortality Rate> mu-CCD death <mu-CCD death> Mean Duration of Sexual Activity <Mean Duration of Sexual Activity> <Fractional Prevalence Males> <Infected (Y-16fb)> Recovery (f) Cumulative number of
female with infections Cumulative femaleinfective years
Vaccinationfemales Annual likelihood ofvaccination forfemales Fraction of female population vaccinated Initial fraction of females vaccinated new female infections <Susceptible (X-16fb)> <Force of Infection (lambda-16fb)>
Total population of females byCervicalScreeningCategorySexualActivityGroupSmokingStatusAgeCategory
<Cancer Survivors(SCC)> <Detected Carcinoma s1(DCIS1-16fb)>
<Detected Carcinoma s2(DCIS2-16fb)> <Detected CIN1(DCIN1-16fb)> <Detected CIN2 (DCIN2-16fb)> <Detected CIN3 (DCIN3-16fb)> <Detected Distant (DCCD-16fb)> <Detected Local (DCCL-16fb)> <Detected Regional(DCCR-16fb)> <Infected (Y-16fb)> <Susceptible (X-16fb)> <Treated and Cured TCINs
(TCINs-16fb)> <Treated and Cured TCISs (TCISs-16fb)>
<Treated and InfectedCarcinoma s1(ICIS1-16fb)> <Treated and InfectedCarcinoma s2(ICIS2-16fb)>
<Immune (Z-16fb)> <Treated and Infected CIN1 (ICIN1-16fb)> <Treated and InfectedCIN2 (ICIN2-16fb)>
<Treated and Infected CIN3 (ICIN3-16fb)>
<Undetected Carcinoma s1(CIS1-16fb)> <Undetected Carcinoma s2
(CIS2-16fb)> <Undetected CIN1 (CIN1-16fb)> <Undetected CIN2(CIN2-16fb)> <Undetected CIN3(CIN3-16fb)> <Undetected Distant(CCD-16fb)> <Undetected Local (CCL-16fb)> <Undetected Regional(CCR-16fb)> <Total population of females byCervicalScreeningCategory>SexualActivityGroupSmokingStatusAgeCategory
<Total population of females bySexualActivityGroupSmokingStatusAgeCategory CervicalScreeningCategory> Vaccinated (V-16fb) muV-fb death <muV-fb death> <Female Mortality Rate> <Vaccinated(V-16fb)> Vaccination males Annual likelihood ofvaccination formales Fraction of malepopulation vaccinated Initial fraction of males vaccinated
Vaccinated (V-16m) muV-m death
Total population of males by AgeCategory SmokingStatusSexualActivityGroup
<Immune (Z-16m)> <Susceptible (X-16m)>
<Infected (Y-16m)> <Male MortalityRate>
<Vaccinated (V-16m)> <Total population of males by
AgeCategory SmokingStatusSexualActivityGroup>
<Total population of males by AgeCategory SmokingStatusSexualActivityGroup>
infective yearsfemales
<Infected (Y-16fb)>
Cumulative number of males with infections Cumulative male infective years new male infections infective years males
<Susceptible(X-16m)> <Force of Infection(lambda-16m)> <Infected (Y-16m)>
<muV-m death>
waning vaccination (f)
<muZ-fb death>
Rate of waning fromvaccine protection(sigma-vf) waning vaccination (m) Rate of waning fromvaccine protection(sigma-vm)
Initial Infected Aging of vaccinated
males ()
Aging of suceptiblemales () Aging of infected males () Aging of immunemales()
Aging of vaccinatedfemales ()
Aging of susceptiblefemales () Aging of infectedfemales () Aging of undetectedCIN1 ()
Aging of detected CIN1 ()
Aging of treated and infected CIN1 Aging of immunefemales () undetected CIN3 ()Aging of
Aging of undetected CIN2 ()
Aging of detected CIN2 ()
Aging of treated and infected CIN2 ()
Aging of undetectedcarcinoma s1 ()
Aging of detected CIN3 ()
Aging of treated and cured TCINs ()
<Undetected CIS1Rate (pi-16fs4)>
Aging of detected carinoma s1 ()
<Fraction of CINs regressionsclearing CIN that also clearinfection (gamma bar-16f)>
Aging of treated and infected carcinoma s1() Aging of treated and cured TCISs () Aging of
undetected carinoma s2()
Aging of treated andinfected CIN3 () Aging of detected
carcinoma s2 () Aging of treated and infected carcinoma s2 () Aging of
undetected local ()
Aging of undetectedregional () undetected distantAging of() Aging of
detected local ()
Aging of detected regional ()
Aging of detecteddistant () Aging of cancer
survivors() Mean Time Until Age
Progression for Females
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females> <Mean Time Until Age
Progression for Females>
<Mean Time Until Age Progression for Females> <Mean Time Until Age
Progression for Females> <Mean Time Until Age
Progression for Females>
<Mean Time Until Age Progression for Females> <Mean Time Until Age Progression for Females> <Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females> <Mean Time Until Age
Progression for Females> <Mean Time Until Age Progression for Females> <Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females> <Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females> <Mean Time Until Age Progression for Females> <Mean Time Until Age
Progression for Females> <Mean Time Until Age Progression for Females>
<Mean Time Until Age Progression for Females>
Sexual Partner Change Rate c subkeli Mixing Matrix Elements rho Age Category MixingParameter Epsilon 1 Age Category IdentityMatrix small delta ij
Fraction of Total Initial Partnership Changes forActivity Group Total Initial Partnership Changesfor Sex
Total Initial Partnership Changes for Sex andActivity Group
Total Initial Partnership Changes for Sex and ActivityGroup and Age Category Fraction of Total Initial
Partnership Changes for Sex Occuring with People in a GivenAge Category
Capital B Adjusted Sexual Partner Change Rateckeflmij
<Sexual Partner Change Rate c subkeli>
Theta Fraction of Contacts by someone of Age Category and Sex to assume that take place with people of each AgeCategory Fraction of Contacts by someone of a specific Sexual Activity Group and Sex toassume that take place with people of each Sexual Activity Group Age Category Mixing Parameter Epsilon 2 Activity Category Identity Matrix smalldelta lm
<Total Initial Partnership Changes for Sex and AgeCategory> <Fraction of Total Initial PartnershipChanges for Sex Occuring withPeople in a Given Sexual ActivityGroup>
<Force of Infection(lambda-16fb)>
Force of Infection(lambda-16fb)
<Force of Infection(lambda-16m)>
Force of Infection(lambda-16m)
<Per Contact Risk of Infection (beta-f)> <Per Contact Risk ofInfection (beta-m)>
Transmission Probability Beta
SmokingStatus Mixing Parameter Epsilon 3 (Level of Non-assortivity in mixing) SmokingStatus Identity Matrix small delta ef
Fraction of Contacts by someone of a specific SmokingStatus and Sex to assume that take place with people ofeach SmokingStatus <Fraction of Total Initial PartnershipChanges for Sex Occuring withPeople in a Given Sexual Activity
Group> <Total population of females bySexualActivityGroupSmokingStatusAgeCategory CervicalScreeningCategory>
<Total population of males by AgeCategory SmokingStatus SexualActivityGroup>
Total Population by Sex SmokingStatus Age andSexual Activity Group
Total population of females by SmokingStatus Age andSexual Activity Group
Total Initial Partnership Changes for Sex and SmokingStatus and Activity Group and Age Category Total Initial Partnership Changes for Sex and SmokingStatus and ActivityGroup Total Initial Partnership Changes for Sex andSmokingStatus Fraction of Total Initial PartnershipChanges for Sex Occuring with People in a Given Sexual ActivityGroup Fraction of Total Initial PartnershipChanges for Sex Occuring with People in a Given SmokingStatusGroup Contacts with Infectives betweenPeople of Potentially DifferentSmokingStatus Sexual Activity
Groups and Ages Fractional Prevalence by SexSmokingStatus SexualActivity Group and Age Fractional PrevalenceFemales Across All Screening Groups
Total Infectives (f) by SmokingStatus SexualActivity Group Age
Contacts with Infectives between People of PotentiallyDifferent SmokingStatus andAges Contacts with Infectives between People of PotentiallyDifferent SmokingStatus
Contacts with Infectives forPerson of Given Sex SmokingStatus Sexual ActivityGroup and Age Force of Infection for Person ofGiven Sex SmokingStatus Sexual Activity Group and Age
Mean partner acquisition rate for age category (c jbar) Relative Partner Acquisition Rate for AgeCategory (pa i) Relative Partner Acquisition Rate for Sexual ActivityCategory (pc i) Relative Partner Acquisition Rate forSmokingStatus
Initial Total Population by Sex SmokingStatus Age andSexual Activity Group
Initial Total Population by Sex SmokingStatusand Age Initial Total Population Weighted by pc and pa by Sex SmokingStatus andAge Sexual Partner Change Ratec sub keli for Ages 18 through 59
Sexual Partner Change Rate csub keli for Single-AgeGroup Categories Is Age Categoryamong Ages 18through 59
<Male Mortality
Rate> <Male MortalityRate>
<Regression CIN3 to Infected Rate (tau-16fbs3)> <Fraction of CINs regressionsclearing CIN that also clearinfection (gamma bar-16f)>
Total population of malesSexualActivityGroupby SmokingStatus Total population ofSmokingStatusmales by
Total populationof males Total population of femalesby SmokingStatusSexualActivityGroup Total population ofSmokingStatusfemales by Total population
of females Total population of femalesby SmokingStatusSexualActivityGroupAgeCategory Routine Screening coverage byAge and Cervical ScreeningCategory (cover i)
Liquid-based cytology sensitivity (papsn s) forCIN1 Liquid-based cytology sensitivity (papsn s) forCIN2 CIN3 CIS1 CIS2
<Detected CIN1 Rate (kappa-16fbis1)> <Detected CIN2 Rate (kappa-16fbis2)> <Detected CIN3 Rate(kappa-16fbis3)>
<Detected CIS1 Rate(kappa-16fbis4)> <Detected CIS1 Rate(kappa-16fbis4)>
<Detected CIS2 Rate (kappa-16fbis5)>
Fraction of Females entering cervical screening category(tadpole b) Fraction of Females never undergoing cervical screening category (tadpole b) Initial population females (N-f)SexualActivityGroupby AgeCategorySmokingStatus
Initial Population by Sex SmokingStatus AgeCategory
Fraction of Population in Sexual Activity Category
Initial Population by SexSmokingStatus SexualActivityCategory andAgeCategory Initial population females (N-f) byAgeCategory SmokingStatusSexualActivityGroup CervicalScreeningCategory Initial population of
males (N-m) Male Children 10 to 11
Female Children 10 to 11 Male new Entrants into Sexually Active Population(cap beta-m) Years in Years 10 to 11 Female new Entrants flow Male Children 5 to 9 Male Children 0 to 4 Male Births
Male Aging to Age 5
Male Aging to Age10 Years in Years 0 to 4 Years in Years 5 to 9 Initial Children 0to 4 by Sex Initial Children 5to 9 by Sex
Initial Children 10 to 11 by Sex
Female Children 5 to 9 Female Children 0 to 4 Female Births
Female Aging toAge 5 Female Aging toAge 10
Total Population ofSmokingStatusAgeCategoryfemales by Fertility Rate per 1000 for SmokingStatusAge Category Fertility Rate per Capita for SmokingStatus AgeCategory Babies Born to Mother inSmokingStatus andAgeCategory Babies Born Babies Born by Sex
<Babies Born by Sex>
<Babies Born by Sex>
Total populationby SmokingStatus
<Total population offemales by SmokingStatus>
Population Growth Rate q YearsInAgeCategory(band i)
Mean Time Until Age Progression by SexSmokingStatusAgeCategory
<Female Mortality Rate> <Male MortalityRate>
Mortality Rate bySmokingStatusSex Age
<Population Growth Rate q>
Mean Time Until Age Progression for Males Deaths of Male Children 0 to 4
Deaths of Male Children 5 to 9 Deaths of MaleChildren 10 to 11 Mortality Rates for Children 0 to 4 by Sex Mortality Rates for
Children 5 to 9 by Sex Mortality Rates for Children 10 to 11 bySex
<Mortality Rates for Children 0 to 4 by Sex>
<Mortality Rates for Children 5 to 9 by Sex>
<Mortality Rates forChildren 10 to 11 bySex>
Deaths of Female Children 0 to 4
Deaths of FemaleChildren 5 to 9 Deaths of FemaleChildren 10 to 11
Total Infectives (f) by SmokingStatus SexualActivity Group Total Infectives (f) by SmokingStatus
Total Infectives (f) Fractional PrevalenceAmong Females Fractional Prevalence Among Females bySmokingStatus
<Years in Years0 to 4> <Years in Years5 to 9>
<Years in Years 10 to 11> <Fraction of Females entering cervical screeningcategory (tadpole b)> <Fraction of Populationin Sexual ActivityCategory> <Fraction of Population in Sexual ActivityCategory> <Mean Time Until Age Progression for Males>
<Mean Time Until Age Progression for Males>
<Mean Time Until Age Progression for Males>
Fraction of Initial Population that StartsInfective
Cancer Death by SmokingStatus and CervicalScreeningCategory Cancer Death by CervicalScreeningCategory
Cancer Death Total over Population
Total Population of females by AgeCategory Total Infectives (f)
by SmokingStatusAge Total Infectives(f) by Age
<Total Infectives(f) by Age> <Total Infectives
(f) by Age>
Fractional Prevalence of Infection in Females byAge
Cancer Death bySmokingStatus
<YearsInAgeCategory(band i)>
<Fraction of Babies whoare of a given sex>
<Cancer Death by AgeCategory SmokingStatusand CervicalScreeningCategory>
<Total populationof females> <Total populationof males>
Total population
Fraction of children that initiate smoking by 12 yearsold by Sex Total Female new Entrants per Year
<Fraction of children that initiate smoking by 12 yearsold by Sex>
Total Male new entrants per year
