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 2008The following is based on lecture material prepared by Prof. Joe Eisenberg, formerly of the University of California-Berkeley 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

Animals

AgriculturalRunoff

DrinkingWater

2°Trans.

Recreational Waters

orWastewater reuse

Transport to other water sources

Food

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

year

Approaches for Risk Estimation:

Epidemiologic Measures

•Relative Risk (RR) Incidence in exposed groupIncidence 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

Interpretation: There are 6 excess cases of disease per 100 subjects receiving tap water

06.012.018.0 activetapwater IncidenceIncidence

Approaches for Risk Estimation:

Epidemiologic MeasuresAttributable Risk Percent (AR%)

Excess cases in exposedIncidence in exposed

Interpretation: 33% of the cases of disease in the tap water group are due to water

33.018.0

06.0 tapwater

tapwater

Incidence

CasesExcess

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

Interpretation: In the community, 2 excess cases of disease per every 100 subjects in the community

02.012.014.0 activeComm IncidenceIncidence

Approaches for Risk Estimation: Epidemiologic

Measures• Population attributable risk percentage

Excess cases in the communityIncidence in the exposed

Interpretation: 14% of the cases of disease in the community are due to tap water

14.014.0

02.0 tapwater

Comm

Incidence

CasesExcess

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 TrialComparing 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)*)

1.06 (2.24)

2.68 (24.20)

0.93 (3.00)

Recovery rate 0.40 0.40 0.48

2. Treatment efficiency (logs removal)

Sedimentation and filtration (Mean (SD)*)

3.84 (0.59)

3.84 (0.59)

1.99 (0.52)

Chlorination (Mean (SD) 0 3.5 (2.93) 4 (2.93)

3. Water Consumption

in liters (mean (SD) ‡)

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

Approaches for Risk Estimation: Risk Assessment Results

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 3.4 (0.6, 15.5) Enteric viruses (disinfection = 4 log removal)

8.4 (0.2, 18.7)

Enteric viruses (disinfection = 4 log removal)

0 (0, 0.2)

Overall risk estimate: 14 cases/10,000/yr

Approaches for Risk Estimation: Comparison/Conclusions

Risk Assessment Intervention Trials

Sensitivity Not relevant Low

Causal evidence 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 *Was not included in this study

Table 3. Comparison of risk assessment and intervention trials

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

• Model structure

where P = probability that a single individual, exposed to N organisms, will become infected or diseased.

• Exposure calculation:

))(1(1

NP

dktko

dtecVN 101

Microbial Risk AssessmentAlternative framework: risk estimates at the population level allow for the inclusion of indirect risks due to secondary transmission

Animals

AgriculturalRunoff

DrinkingWater

2°Trans.

Recreational Watersor

Wastewater reuse

Transport to other water sources

Food

Microbial Risk AssessmentEisenberg 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 detailsThere 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

• 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

ENVIRONMENTW

S R

I

The SIR Model: slightly different version

The variables• X: susceptible• Y: infectious/asymptomatic• Z: non-infectious/asymptomatic• D: infectious/symptomatic• W: concentration of pathogens at the source• a: number of new susceptible individuals

migrating in

δ+μ

W

X Z0+ (W) (ρ)

Dρ σ

Y

λ

a μ

The SIR Model: slightly different version (cont)

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• β0 : Baseline transmission rate• β : Infection rate due to pathogen

δ+μW

X Z0+ (W)

(ρ)

Dρ σ

Y

λ

a μ

Dynamic Modeling of Disease Transmission: an

exampleXWXXZadt

dX )(0

• 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

exampleXWXXZadt

dX )(0

• 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 ModelsLikelihood

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

S(t)Susceptible E1 E2 Ek

IA(t) Infectious

(asymptomatic)

R(t) Removed

W(t)Environmental Transmission

...

Latently Infected IS(t)Infectious

(symptomatic)

+ S

S: SusceptibleW: Concentration of Pathogens in the EnvironmentIS: Symptomatic and Infectious

IA: Asymptomatic and Infectious

R: Immune/ Partially ProtectedSolid: Individual Flows from State to StateDashed: Pathogen Flows

Analysis - Model 1Monte 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]

0 0.1 0.2 0.3 0.4 0.50

100

200

300

400

500

600

700

Percent attributable risk

Fre

qu

enc

y

0 0.1 0.2 0.3 0.4 0.50

100

200

300

400

500

600

700

Percent attributable risk

Fre

qu

enc

y

Model 2Goal:

To examine the role of person-environment-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

S(t)Susceptible E1 E2 Ek

IA(t) Infectious

(asymptomatic)

R(t) Removed

W(t)Environmental Transmission

...

Latently Infected IS(t)Infectious

(symptomatic)

+ S

S: SusceptibleW: Concentration of Pathogens in the EnvironmentIS: Symptomatic and Infectious

IA: Asymptomatic and Infectious

R: Immune/ Partially ProtectedSolid: Individual Flows from State to StateDashed: 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

0 5 10 15 20 25 30 35 40-2480

-2475

-2470

-2465

-2460

-2455

-2450

Days

Lo

g L

ike

lih

oo

d

0 5 10 15 20 25 30 35 40-2480

-2475

-2470

-2465

-2460

-2455

-2450

Days

Lo

g L

ike

lih

oo

d

MLE for the time between contamination of sewage and exposure from drinking water tap

was 11 days (95% CI [8.3, 19])

Preventable Fraction As a Function of Delay Time

Predicting the public health benefits of moving the drinking water inlet

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Days

Pre

ve

nta

ble

Fra

cti

on

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Days

Pre

ve

nta

ble

Fra

cti

on

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 person-water-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: MethodsAnalyzing disease transmission models using

statistical techniquesAllows 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.

ConclusionsRisk assessments should use models that

can integrate relevant informationHealth data

–Epidemiology–Basic biology

Environmental data–Water quality–Fate and transport

Need a population perspective–Model-based approach