Transcript ppt - Bama.ua.edu

Lecture 11
BSC 417
Outline
• More on sensitivity analysis
– Spreadsheet on website
– Examples and in-class exercise
• Case analysis
• Discussion of Eisenberg et al. (2002)
paper
• Eisenberg presentation on model
• Translating the model to Stella
Assessing Risk from Environmental
Exposure to Waterborne Pathogens:
Use of Dynamic, Population-Based
Analytical Methods and Models
26 February 2008
The following is based on lecture material prepared by Prof.
Joe Eisenberg, formerly of the University of CaliforniaBerkeley and now at the University of Michigan
Used with his permission
Overview
• Role of water in disease burden
– Water as a route of disease transmission
• Methods of risk estimation
– Direct: intervention trials
– Indirect: risk assessment
• Population-level risks
– Example: the Milwaukee outbreak
Importance of Waterborne Pathogens
Domestic: U.S. interest in water quality
– 1993 Cryptosporidium outbreak
– Increasing number of disease outbreaks
associated with water
– Congressional mandates for water quality
– (Safe Drinking Water Act)
– Emphasis on risk assessment and
regulation
Importance of Waterborne Pathogens
Worldwide: WHO interest in water quality
– Estimating GBD associated with water,
sanitation, and hygiene
– Diarrheal diseases are a major cause of
childhood death in developing countries.
– 3 million of the 12.9 million deaths in
children under the age of 5 attributable
to diarrheal disease
– Emphasis on intervention and control
Pathways of Transmission
• Person-person
– Mediated through fomites (e.g., phone, sink,
etc.)
– Often associated with hygiene practices
• Person-environment-person
– Mediated through water, food, or soil
– Contamination can occur through improper
sanitation (example: sewage inflow into
drinking water source or lack of latrines)
– Animals are often sources (Zoonotic
pathogens)
– Exposure can occur through improper
treatment of food or water
The Disease Transmission
Process
Risk estimation depends on transmission
dynamics and exposure pathways
Transport to other water sources
Agricultural
Runoff
Drinking
Water
Animals
2°
Trans.
Food
Recreational Waters
or
Wastewater reuse
Approaches to Risk
Estimation
• Direct approach: The intervention trial
– Can be used to assess risk from drinking
water and recreational water exposures
– Problems with sensitivity (sample size issue)
– Trials are expensive.
• Indirect approach: Mathematical
models
– Must account for properties of infectious
disease processes
– Pathogen specific models
– Uncertainty and variability may make
interpretation difficult.
Approaches to Risk
Estimation
• Combining direct and indirect
approaches
– Models can define the issues
and help design studies.
– Epidemiology can confirm
current model structure and
provide insight into how to
improve the model.
Approaches for Risk Estimation:
Direct estimates of waterborne infectious
•
illnesses
Surveillance: count waterborne infectious
illnesses
– How can a waterborne disease outbreak be
distinguished from other outbreak causes (food,
fomites, etc.)?
– What about endemic disease?
•
Observational
– Ecologic studies (e.g., serosurvey comparing
communities with and without filtration).
– Time series (e.g., correlation between turbidity and
hospitalization data)
Approaches for Risk Estimation:
Distinguishing waterborne GI disease from other GI
diseases
• Methods for addressing the question
– In a single community: a randomized, blinded,
placebo-controlled trial
– design provides an estimate of the effectiveness of
a drinking water intervention.
• Basic study design: two groups
– “Exposed” group = normal tap water.
– “Treated” group = use a water treatment device to
provide water as pathogen-free as technically
possible
Approaches for Risk
Estimation:
A Tap Water Intervention Trial
• Enroll 1000 subjects
• 500 receive an active home water treatment
device (and carry drinking water to work,
etc. when practical)
• 500 receive a “placebo” home water
drinking device (does nothing to change the
water)
• Follow the subjects for one year with daily
logs of GI illness
• Alternative design: Each household changes
device type after 6 months.
Approaches for Risk
Estimation:
A Tap Water Intervention Trial
• Placebo group (tap water):
– 90 illnesses over course of the study
– “Rate” = 90 / 500
Rate in placebo group = 0.18 per person per
year
• Treated group (active device):
– 60 illnesses in the treated group (active
device)
– “Rate” = 60 / 500
Rate in treated group = 0.12 per person per
Approaches for Risk Estimation:
Epidemiologic Measures
•Relative Risk (RR)
Incidence in exposed group
Incidence in unexposed group
Interpretation: the risk of disease in the tap water
group is 1.5 times higher than that of the treated
group
Approaches for Risk Estimation:
Epidemiologic Measures
Attributable Risk (AR)
Incidence in exposed – Incidence in unexposed
Incidencetapwater  Incidenceactive
 0.18  0.12  0.06
Interpretation: There are 6 excess cases of disease
per 100 subjects receiving tap water
Approaches for Risk Estimation:
Epidemiologic Measures
Attributable Risk Percent (AR%)
Excess cases in exposed
Incidence in exposed
Excess Casestapwater 0.06

 0.33
Incidencetapwater
0.18
Interpretation: 33% of the cases of disease
in the tap water group are due to water
Approaches for Risk Estimation:
Epidemiologic Measures
• To generalize beyond the cohort, need
an estimate of the community incidence.
• PAR: population attributable risk
• PAR%: population attributable risk %
• AR compares completely protected
group with completely unprotected
group.
• PAR incorporates intermediate exposure
Approaches for Risk Estimation:
Epidemiologic Measures
• Population attributable risk
• Incidence in the community–incidence in
the unexposed
IncidenceComm  Incidenceactive
 0.14  0.12  0.02
Interpretation: In the community, 2 excess cases
of disease per every 100 subjects in the
community
Approaches for Risk Estimation:
Epidemiologic Measures
• Population attributable risk percentage
Excess cases in the community
Incidence in the exposed
Excess CasesComm 0.02

 0.14
Incidencetapwater
0.14
Interpretation: 14% of the cases of disease in
the community are due to tap water
Approaches for Risk Estimation:
Tap Water Intervention Trials
Trials in immunocompetent populations
 Canada (Payment)--challenged surface water
– AR = 0.35 (Study 1), 0.14-0.4 (Study 2)
 Australia (Fairley)--pristine surface water
– No effect
 Walnut Creek (UCB) – pilot trial
– AR = 0.24 (non-significant effect)
 Iowa (UCB)--challenged surface water
– No effect
Trials in sensitive populations
 HIV+ in San Francisco (UCB)--mixed sources
 Elderly in Sonoma (UCB)--intermediate quality surface
Approaches for Risk Estimation:
Tap Water Intervention Trials
• Davenport, Iowa study
– Comparing sham vs. active groups
– AR = - 365 cases/10,000/year (CI: -2555,
1825)
– Interpretation: No evidence of a
significantly elevated drinking water
risk
– Is the drinking water safe?
Approaches for Risk Estimation:
Risk Assessment vs. Intervention Trial
Comparing estimates from a risk assessment to
randomized trial results (Eienberg et al. AJE, submitted)
Data collected during the intervention trial
– Self-report illnesses from participants: Weekly
diaries
– Source water quality: Cryptosporidium, Giardia,
enteric viruses
– Drinking water patterns: RDD survey
– Water treatment: B. subtilis, somatic coliphage
Approaches for Risk Estimation:
Risk Assessment Model
Approaches for Risk Estimation: Risk Assessment
Model
Model
Cryptosporidium
Giardia
Viruses
1. Source water
Concentration
(organisms per liter)
(Normal Mean (SD)*)
2.68 (24.20)
0.93 (3.00)
0.40
0.40
0.48
3.84 (0.59)
3.84 (0.59)
0
3.5 (2.93)
4 (2.93)
0.094 (0.42)
0.094 (0.42)
0.094 (0.42)
4. Dose Response §
: 0.004078
: 0.01982
,: 0.26, 0.42
5. Morbidity Ratio#
0.39
0.40
0.57
Recovery rate
1.06 (2.24)
2. Treatment efficiency
(logs removal)
Sedimentation and
filtration (Mean (SD)*)
Chlorination (Mean (SD)
3. Water Consumption
1.99 (0.52)
in liters (mean (SD) ‡)
Approaches for Risk Estimation: Risk
Assessment Results
Overall risk estimate: 14 cases/10,000/yr
Table 2. Summary of risk estimates (cases/10,000,yr)
Cases of Illness
Mean
Percentile
(2.5, 97.5)
Cryptosporidium
2.1
(0.8, 3.5)
Giardia
Enteric viruses
(disinfection = 4
log removal)
Enteric viruses
(disinfection = 4
log removal)
3.4
(0.6, 15.5)
8.4
(0.2, 18.7)
0
(0, 0.2)
Approaches for Risk Estimation:
Comparison/Conclusions
Table 3. Comparison
of risk assessment and intervention trials
Risk Assessment
Intervention Trials
Not relevant
Low
Indirect
Direct
Pathogen inclusion
Few
Many
Model Specification
Adds uncertainty
Not relevant
Transmission
processes
Can be included*
Only in a limited
way
Distribution System
effects
Can be included*
Was included
Examining alternative
control strategies
Yes
No
Expense
Low
High
Time
Fast
Slow
Sensitivity
Causal evidence
*
Was not included in this study
Microbial Risk Assessment
• Two classes of risk assessment models
– Individual-based
– Population-based
• Individual-based estimates
– Risk estimates assume independence among
individuals within the population
– Chemical risk paradigm
– Focus is on direct risks
– Probability of disease for a given individual
– This probability can be either daily, yearly, our
lifetime.
Microbial Risk Assessment
• Chemical risk paradigm
–Hazard identification, exposure assessment,
dose response, risk characterization
N 
• Model structure
P  1  (1  ( ))

where P = probability that a single individual,
exposed to N organisms, will become infected
or diseased.
• Exposure calculation:
N  V1  co  e  kt t 10  kd d
Microbial Risk Assessment
Alternative framework: risk estimates at the
population level allow for the inclusion of indirect
risks due to secondary transmission
Transport to other water sources
Agricultural
Runoff
Drinking
Water
Animals
2°
Trans.
Food
Recreational Waters
or
Wastewater reuse
Microbial Risk Assessment
Eisenberg et al. AJE 2005
• Transmission pathways
– Example: a Cryptosporidium outbreak in Milwaukee Wisconsin, 1993
• Competing hypotheses on the cause
– Oocyst contamination of drinking water influent coupled with
treatment failure
– Chemical risk paradigm may be sufficient (still need to consider
secondary transmission)
– Amplification of oocyst concentrations in the drinking water
influent due to a person-environment-person transmission
process
– Chemical risk paradigm cannot address this potential cause of
the outbreak
A model of disease transmission:
The SIR model
• Mathematical modeling of a population where
individuals fall into three main categories:
– Susceptible (S)
– Infectious (I)
– Recovered (R)
• Different individuals within this population can be
in one of a few key states at any given time
–
–
–
–
Susceptible to disease (S)
infectious/asymptomatic (I)
infectious/symptomatic (I)
non-infectious/asymptomatic; recovered (R)
• A dynamic model: individuals are moving from state
to state over time
The SIR model: key details
There are two sets of variables:
• Variables describing the states people are in
– S=susceptible
– I=infectious
– R=non-infectious/asymptomatic
• Variables describing how many people are
moving between these states (parameters)
– Example: γ=Fraction of people in state R who
move to state S
The SIR Model
g
S

I
W
•
•
•
•
•
•
•
•
•
d
R
ENVIRONMENT
S: Susceptible
I: Infectious (symptomatic+asymptomatic)
R: Non-infectious
W: Concentration of pathogens in the environment
β: Infection rate due to exposure to pathogen
δ: Fraction of people who move from state I to state R
γ: Fraction of people who move from state R to state S
Solid lines: Individuals moving from state to state
Dashed lines: Pathogen flows between individuals in different states
The SIR Model: slightly different
version
g
a
X
0+ (W)
Y
λ
W
(ρ)
ρ
Z
μ
σ
D
δ+μ
The variables
• X: susceptible
• Y: infectious/asymptomatic
• Z: non-infectious/asymptomatic
• D: infectious/symptomatic
• W: concentration of pathogens at the source
The SIR Model: slightly different version
(cont)
g
a
X
0+ (W)
Y
λ
W
(ρ)
ρ
Z
μ
σ
D
The parameters
• ρ: fraction in state Y who move to state D
• α: Fraction in state Y who move to state Z
• σ: Fraction in state D who move to state Z
• γ: Fraction in state Z who move to state X
• δ: Fraction in state D who die
• μ: Fraction who die of natural causes
• λ: Numbers of pathogen shed per
infectious/asymptomatic individual
δ+μ
Dynamic Modeling of Disease
Transmission: an example
dX
dt
 a  gZ  X   0 X   (W ) X
• Remember: a derivative is a rate of change
• X= the population of individuals susceptible to a
disease
• dX/dt = rate of change in the susceptible population
• The equation describes individuals moving in and out
of the susceptible population
• Each variable represents some number of individuals
moving
– into the susceptible population (+) from some other group,
– out of the susceptible population (-) to some other group
Dynamic Modeling of Disease
Transmission: an example
dX
dt
 a  gZ  X   0 X   (W ) X
• a= number of susceptible individuals migrating into the
population
• γZ =number of non-infectious/asymptomatic individuals
migrating back into the susceptible population
• μX =Fraction of susceptible individuals who drop out of the
susceptible population because they die of natural causes
• β0X =number of susceptible individuals who become infected
and drop out of the susceptible population
• β(W)X =number of susceptible people becoming ill due to
pathogen exposure and drop out of the susceptible population
Analysis of Disease Transmission
Models
• Traditional approaches to evaluating
dynamics models are qualitative
– Stability analysis, threshold estimates (Ro),
qualitative fits
– Statistics rarely used to analyze output
• Methodological goal to obtain public
health relevant estimates of the outbreak
– Need to modify traditional statistical
techniques to address models with large
number of parameters, sparse data, and
collinearity
Analysis of Disease Transmission
Models
Likelihood
 Traditional likelihood methods
– Difficult to find maximum likelihood point in highly parameterized models.
– Confidence intervals are often not possible in complex likelihood spaces
 Profile likelihood is an alternative option
– Fix a subset of the parameters across a grid of values.
– At each point in the grid the remaining parameters are maximized.
Bayesian techniques
 Practical for combining outbreak data with existing information about parameters.
 Modifications required to deal with collinearities
Model 1
Goals:
To examine the role of person-person (secondary)
transmission
To estimate the fraction of outbreak cases
associated with person-person (secondary)
transmission
Cryptosporidium Outbreak - Model
Diagram
IA(t)
r
S(t)
Susceptible
 + S
E1
p
E2
p
... p
Ek
Latently Infected
Infectious
(asymptomatic)
p
d
1r
IS(t)
Infectious
(symptomatic)
W(t)
Environmental
Transmission
S: Susceptible
W: Concentration of Pathogens in the Environment
IS: Symptomatic and Infectious
IA: Asymptomatic and Infectious
R: Immune/ Partially Protected
Solid: Individual Flows from State to State
Dashed: Pathogen Flows
R(t)
Removed
Analysis - Model 1
Monte Carlo Markov Chain (MCMC) was used to
generate a posterior distribution.
Two step procedure was used to address collinearities
of the parameter estimates
– MCMC at profiled points
– Second MCMC on draws from first MCMC
Cumulative incidence, I1, was produced by a random
draw of the posterior
Cumulative incidence, I0, was produced by first setting
bs=0 then obtaining a random draw of the posterior.
The attributable risk associated with secondary
transmission was I1- I0
Risk Attributable to
Secondary Transmission
10% , 95% CI [6, 21]
700
600
Frequency
500
400
300
200
100
0
0
0.1
0.2
0.3
Percent attributable risk
0.4
0.5
Model 2
Goal:
To examine the role of personenvironment-person transmission
To estimate the preventable fraction due to
an increase in distance between wastewater
outlet and drinking water inlet
Examine preventable fraction as a function
of transport time parameter, d
– Where d is a surrogate for the potential
intervention of moving the drinking water inlet
farther from the wastewater outlet
Cryptosporidium Outbreak- Model
Diagram
IA(t)
r
S(t)
Susceptible
 + S
E1
p
E2
p
... p
Ek
Latently Infected
Infectious
(asymptomatic)
p
d
1r
IS(t)
R(t)
Removed
Infectious
(symptomatic)
W(t)
Environmental
Transmission
S: Susceptible
W: Concentration of Pathogens in the Environment
IS: Symptomatic and Infectious
IA: Asymptomatic and Infectious
R: Immune/ Partially Protected
Solid: Individual Flows from State to State
Dashed: Pathogen Flows
Analysis - Model 2
• Estimate the likelihood for different values
of d, ranging from 1 - 40 days.
• Estimate the attack rate (AR) for the MLE
parameters
• Estimate the AR for different values of d,
keeping all other parameters constant at
their MLE values.
• Plot PFd = 1 - ARMLE / ARd
Profile Likelihood of the Delay
Parameter
MLE for the time between contamination of sewage and
exposure from drinking water tap was 11 days (95%
CI [8.3, 19])
-2450
Log Likelihood
-2455
-2460
-2465
-2470
-2475
-2480
0
5
10
15
20
Days
25
30
35
40
Preventable Fraction As a
Function of Delay Time
Predicting the public health benefits of moving
the drinking water inlet
0.9
Preventable Fraction
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Days
30
35
40
45
Conclusions
• Secondary transmission was small.
– Best guess is 10%, most likely less than
21%
– Consistent with empirical findings of
McKenzie et al.
– Kinetics of the outbreak in Milwaukee
were too quick to be driven solely by
secondary transmission
Conclusions
• Person-water-person transmission as the
main infection pathway has not been well
studied
– Few data exist that examines personwater-person transmission
– Studies have demonstrated a correlation
between cases of specific viral serotypes
in humans and in sewage
– Provides information on a potentially
important environmental intervention
Conclusions: Methods
Analyzing disease transmission models using
statistical techniques
Allows inferences about parameters that are
interesting and relevant
– Can get at posterior distribution that allows for
calculation of relevant public health measures
Requires the modification of existing techniques
– Profile likelihood to deal with large numbers of
parameters
– Bayesian estimation techniques to address the
co-linearity.
Conclusions
Risk assessments should use models that
can integrate relevant information
Health data
–Epidemiology
–Basic biology
Environmental data
–Water quality
–Fate and transport
Need a population perspective
–Model-based approach