Download Models in Ecology

Disease modeling
Maya Groner
University of Prince Edward Island
with slides from Mark Lewis
University of Alberta
Why do we Model?
Provide new understanding of the ecological and evolutionary
processes
Assess the weight of evidence for certain hypotheses
Project/predict outcomes under different management
scenarios
Evaluate new management/control methods
Design experiments/studies or allocate funding that will further
the science
Summarize/synthesize large amount of data
Identify areas of ignorance
Modelling
Process
Formulation
Verification
Calibration
Analysis and
Evaluation
Formulation:
Objectives: What is system? What are questions?
What is the scale and timeframe? How will
output be analyzed, summarized and used?
Hypotheses: verbal statements that connect
objectives and current knowledge.
Mathematical formulation: hypotheses are
translated in specific quantitative relationships.
These form a model that, given inputs, yield an
output (prediction or answer to question).
Verification: Verify that the method used to solve the
model is correct.
Calibration: Measure inputs for the model.
Analysis and Evaluation:
Computation, mathematical analysis etc. is used to
answer questions.
The answer is validated or corroborated against
independent data sets.
Modelling
Process
Formulation
Verification
Calibration
Analysis and
Evaluation
Haefner (2005)
Modelling
Process
Formulation
Verification
Calibration
Analysis and
Evaluation
Haefner (2005)
Modelling
Process
Formulation
Verification
Calibration
Analysis and
Evaluation
Haefner (2005)
Modelling
Process
Formulation
Verification
Calibration
Analysis and
Evaluation
Haefner (2005)
Classical Hypothesis Testing
Key elements are a confrontation between a single
hypothesis and data.
A test data set or “critical experiment” is used to evaluate
hypothesis
Relies on “doctrine of falsification” (Popper, hypotheses
cannot be proved, only disproved).
The hypothesis is given as a “null” statement that could
possibly be falsified using a test of observed data.
Two kinds of errors can arise:
Erroneously rejecting the null hypothesis when it is true
(Type I)
Failure to reject the null hypothesis when it is false (Type
II)
Classical Hypothesis Testing
Advantage: provides a rigorous framework for assessing
the weight of scientific evidence for a particular
hypothesis
Limitations:
If we derive a model we cannot reject, then we may
believe it is the correct model. It may be that we
just need more data to be able to reject.
Sequential passes through the modelling process
should use really new data sets for testing
How do we Fit Parameters to Models?
Real data is best, but not always possible
Sometimes we use data from other models (e.g., coupled
models)
If we don’t know, try several values and test the sensitivity of the
model to these values
Multiple Working Hypotheses
• Evaluates several competing hypotheses and models
simultaneously (in parallel).
• Models are compared to see which is the most likely.
• A common auxiliary is simplicity, which is the basis
for the Principle of Parsimony or Occam’s Razor.
• Advantage: multiple hypotheses can be evaluated
simultaneously with a single data set.
• Limitation: if we start off with a poor set of
hypotheses, the one we choose as “best” may still be
poor.
Multiple Working Hypotheses
• Eg. What are the factors governing movement behaviour
of ants foraging for seeds (Haefner and Crist)?
– H1: ants are moving randomly, H2: ants rely on
memory of places previously foraged, H3: ants rely on
pheromone trails, H4: ants rely on memory and
pheromones, H5: ants are omniscient and thus forage
optimally.
– Models: simulations of ants moving according to rules
that describe the hypotheses.
– Analysis: compare resulting ant movement paths to
those observed.
– Evaluation: assess which which model “best”
describes the movement paths.
Types of Models
Phenomenological (posits a relationship without delving into the
underlying processes driving the relationship) versus
mechanistic (describes relationship in terms of the underlying
processes).
Dynamical (changes with time) versus static.
Spatial (changes with location) versus nonspatial
Stochastic (uncertainty included) versus deterministic.
Some common sorts of models aso include: compartment, stagestructured, transport, particle, finite state.
A model for the plague
Yersinia pestis bacillus
(Science Picture Library)
Susceptible
Infectious
Removed
A classic model (SIR)
Yersinia pestis bacillus
(Science Picture Library)
Susceptible
Infectious
Removed
A model for the plague
Yersinia pestis bacillus
(Science Picture Library)
Susceptible
Infectious
dS
dt
  IS
rate of loss
through
infection
rate of change
of susceptibles
dI
dt
 IS 
dI
rate of gain
through
infection
rate of loss
through
death
rate of change
of infectives
Removed
dR
dt
rate of change
of removed

dI
rate of gain
through
death
Kermack and McKendrick (1927)
A model for the plague
Yersinia pestis bacillus
(Science Picture Library)
Bombay plague of 1905-6
dS
dt
Susceptible
  IS
rate of loss
through
infection
rate of change
of susceptibles
dI
dt
Infectious
 IS 
dI
rate of gain
through
infection
rate of loss
through
death
rate of change
of infectives
dR
dt
Removed
rate of change
of removed
Time (Weeks)

dI
rate of gain
through
death
A key prediction of epidemiological modeling is the disease
basic reproduction number, R0.
Number of
infectious
R0 > 1
R0 < 1
Time
Time
R0=2
R0 is defined as the number of secondary infections arising from the
introduction of a single infective into an otherwise susceptible population.
In the model for the plague
Initial density
Average
R0  Contact rate 

of susceptibles
infectious period


N
1/d
Use of R0 to understand global
warming and disease
Harvell et al. 2002
Many variations on the SIR model
Susceptible-Exposed-Infectious-Recovered
(SEIR)
Susceptible-Infected (SI)
S-I-I-I-I-I-I-R (Waning Infection)
Multi-host SIR model (e.g, coupled SIR-SIR
model)
Etc.
Sea lice as a ‘model’ system for
modeling
What are we modeling?
Ricker Model for Stock-recruitment
ni(t)=ni(t-2)exp[r - bni(t-2)]
Population growth rate
Density dependent mortality
• Nonlinear model
• Carrying capacity
• Compensation
• Overcompensation
• Common in Fisheries
• Common in Ecology
Model for Pink Salmon: Stochastic Ricker
Is a given pink salmon population (unexposed to sea lice but exposed to fishing) growing
or declining?
Stochastic Ricker equation:
ni(t)=ni(t-2)exp[r-bni(t-2)+Z(0,2)]
r is growth rate
r>0 means population will grow
r<0 means population will decline
Parameter estimate: r=0.62
95% Confidence Interval: (0.55, 0.69)
Conclusion: r>0 for this population
Log transformed Ricker:
log[ni(t)/ni(t-2)] = r – bni(t-2) +Z(0,2)
Can fit to data using least squares
Quantifying the role of temperature and
mate limitation in sea lice epidemiology
Population matrix model
P1
P2
Egg
Larvae
(freeG1 swimming)
P3
G2
P5
P4
Chalimu
s
G3
Preadult
G4
P6
Gravid
I
G5
Adult
Betwee
nClutch
P7
G
6
G7
F5
F7
Gravid
II
Quantifying the role of temperature and
mate limitation in sea lice epidemiology
Temperature causes
dramatic increases in
population growth as
a result of increases
in net reproductive
rate and decreases in
generation time
Groner et al. 2014 PLoS One
Quantifying the role of temperature and
mate limitation in sea lice epidemiology
Quantifying the role of temperature and
mate limitation in sea lice epidemiology
Identification of sensitive stages
1) What stages should be targeted for control?
2) When should controls be administered?
Initial questions
Can we use modeling techniques to investigate
the optimal ratio of wrasse: salmon?
To what extent does using wrasse reduce the
need for chemical treatments?
Modeling approaches
Factors to
incorporate
Sea Lice Growth
and Survival
Effects of Wrasse
(Dynamic process)
Bath Treatments
(Instantaneous
event)
Behaviour and
individual variation
Stochastic
processes
Solve for
equilibriums
Differential
Equation
Delay-differential Individual-Based
Equation
Model
Individual-based model of sea lice in
Anylogic
Individual-based model of sea lice in
Anylogic
Temperature-dependent
development
Temperature ⁰C
after meta-analysis
data from Stien et al.
Additional causes of mortality
Treatments
Fish surveyed weekly.
If Mobiles > treatment
threshhold,
we ‘treat’
Wrasse
Predation
Feed at a constant rate
Wrasse: salmon ratios
0, 1:200, 1:100, 1:50, 1:25, 1:10
What is driving infestations?
External
What is driving infestations?
Reinfection
What is driving infestations?
Run the model
http://www.runthemodel.com/models/koketzEcHJltLMX4KZ5Zs/
Low reinfection, High external
infection
No control of sea lice
Wrasse: Salmon is
1:50
Treated if mobiles >
4
17
Treatments
10
Treatments
Treated if mobiles > 4
is 1:50
Wrasse: Salmon
Time (days)
Using wrasse can reduce the number of
chemical treatments for all infection
scenarios
18
Number of Treatments
16
14
12
10
8
6
4
2
0
0
0.02
0.04
0.06
Wrasse: Salmon
0.08
0.1
0.12
Individual-based model of evolution of
pesticide resistance in sea lice
How do treatment regimens affect the rate the resistance
evolves?
How does the genetic mechanism of resistance effect
the rate of evolution?
Can integrated pest management slow the rate that
resistance evolves?
How do chemical refugia (e.g., wild salmon) influence
the rate that pesticide resistance evolves in sea lice?
Model Structure
Upper level variables
Preliminary data
Temperature influences the rate that resistance
evolves and the number of treatments used
*these results vary with each iteration, so this will
need to be scaled up to look at average effects.
Transmission processes
Most disease models
assume that transmission
is density- or frequencydependent (e.g., from a
host perspective),
however, transmission in
aquatic systems is
dynamic and
environmentally
dependent
Transmission processes
Hydrodynamic approaches, 3D particle tracking model
Erin Rees
Transmission processes
Calculate force of infection among farm network
and analyze with Who Acquires Infection from
Whom (WAIFW) matrix
Paull et al. 2012
Dobson & Fofopolous 2002
Various levels of complexity
Deterministic SIR model
Population matrix model w/ temperaturedependency and demographic stochasticity
Agent Based Model w/ demographic stochasticity
Hydrodynamic transmission model coupled with
ODE model of farm transmission processes
Additional Considerations
Sensitivity analysis: How sensitive are model
predictions to small changes in model inputs
(parameters or initial/boundary conditions)?
Model validation: does the model accurately predict
outcomes from a new experiment/field study?
Bayesian versus frequentist approach: How do we deal
with prior information? What philosophical approach
do we use to assess the weight of scientific
evidence?
Hierarchical modeling: Is there a hierarchy of structure
in the model? Are parameters fixed or do they they
actually have a probability distribution?
Additional Considerations
Consider your audience!
Who tunes out during a modeling talk?
Can you add visualizations?
When to simplify or not?
Consider your strengths
If you aren’t a modeler, can you find one?
How can you collect your data to be model-friendly?
New modeling frontiers with ‘big
data’
Googling ‘influenza’
New modeling frontiers with ‘big
data’
Syndromic surveillance
real-time monitoring
early detection
use of technology (apps, cellphones)
Law of Unintended Consequences: An intervention in a
complex system invariably creates unanticipated and often
undesirable outcomes
In Borneo in the 1950s many Dayak villagers had malaria and the World
Health Organization had a solution that was simple and direct. Spraying
DDT seemed to work: mosquitoes died and malaria declined. But then an
expanding web of side effects…started to appear.
• The roofs of peoples houses began to collapse because DDT had killed tiny
parasitic wasps that had previously controlled thatch-eating caterpillars.
• The colonial government issued sheet-metal replacement roofs, but people
could not sleep when tropical rains turned the tin roofs into drums
• The DDT-poisoned bugs were eaten by geckoes which were eaten by cats.
The DDT invisibly built up in the food chain and began to kill cats.
• Without the cats the rats multiplied.
• The World Health Organization , threatened by potential outbreaks of
typhus and sylvatic plague, which it had itself created, was obliged to
parachute fourteen thousand live cats into Borneo
• Thus, occurred Operation Cat Drop, one of the odder missions of the
British Royal Air Force.
Hawken (1999)