Risk Analysis Workshop: Risk of Infection From Drinking Water Containing  Cryptosporidium - A Simplified Model

Introduction

The relationships between the quantities that determine the Dose in assessing the risk of infection from drinking water contaminated with Protozoan Microparasites (eg. Cryptosporidium) is given by:

Dose = Conc . I . 10 ^{- DR} . V

Where:

Conc = Concentration of pathogenic micro-organisms.

I = Fraction of the detected pathogens that is capable of infection (viability).

DR = Removal or inactivation efficiency of the treatment process, expressed as its Decimal Reduction factor (DR=0 – nothing removed, DR=1 reduces the concentration by a factor of 10).

V = Daily individual consumption of unboiled drinking water.

Dose-Response

When the dose is known, the probability of infection by Cryptosporidium oocysts may be calculated with the dose-response relation of the exponential model:

P_inf^* = 1 - e ^{- r . Dose}

Where:

P_inf^* = Probability of infection

r = Dose response parameter if r = 0 the probability of infection is 0.

Variability and Uncertainty

The quantities in the equations may vary around a mean value as a result of purely natural processes that cannot be eliminated but additionally there may be uncertainty concerning the mean value which arises from lack of knowledge.

For example, the concentration may in fact vary according to a Poisson distribution but in addition we may be uncertain as to the value of the mean to use in the Poisson function.

In principle we should treat these two aspects which each contributing to the final distribution separately.

However, for this simple case we will model the overall variability and uncertainty with a single probability distribution.

Exercise

For this exercise we will start with sample data.

Firstly to model the distributions from the sample data:

Load the Excel sheet cryptoexample.xls

Open the sheet labelled Simulation

To prepare the model the following headings have been set up in your worksheet:

Define the Distributions

Undertake a Best Fit analysis to suggest the best distribution for the sample data

Select Fit Distributions to model the input data and choose an appropriate user defined distribution and customize it by specifying the relevant parameters according to your expert judgement.

Note that the concentration data Domain is Discrete.

Consider which distribution you need and note your reasons.

Once decided select Write to Cell and select the output cell on the Simulation tab for the @Risk function.

This will set the cell to contain samples from the distribution you choose when the simulation is run.

Repeat the process for each variable –

Viability (Continuous domain)

Volume ingested (Continuous domain)

Remember to include a sensible minimum truncation limit for Volume Ingested

Assume that we have information from a theoretical model that the parameter DR is normally distributed about the mean 0.6 with a standard deviation 0.02.

Right-click the first cell in the DR column and select Define Distribution from the @RISK menu.

Select the normal distribution and set the mean and standard deviation.

Truncate the distribution (Tr. min) at 0.

Click Apply.

For the moment we will assume that the Dose-response parameter (r) is know to be 0.66, with no uncertainty – enter this value in the appropriate cell.

Enter the Formulae and Select the Output Cells

Enter the appropriate formula for Dose in the first cell of the Dose column.

Add this Output from the @RISK menu.

Enter the formula for the probability of infection in the first empty cell of the Probability of infection column.

Add this  Output from the @RISK menu.

Before Running the Monte Carlo Simulation

Before running, check the inputs and outputs are correctly defined.

This can be done using the @RISK menu item which has been added to the Excel Tool bar @RISK->Model->List Outputs and Inputs.

Choose the simulation settings using the menu item @RISK->Simulation->Settings.

You can choose the number of iterations (this is the number of samples taken from the distributions – choose 50000 for now) and the number of simulations (the number of times the whole process is repeated choose 1 for now).

There are other options but accept the defaults for the rest.

Run the simulation.

Either, in the @RISK model window select Simulation->Start, or, from Excel select @RISK->Simulation->Start.

The results appear in a new @RISK window with summary statistics displayed for the outputs and the inputs.

@RISK also creates some report sheets (depending on how you set up the Report Settings in the steps.

a. Determine the values of Dose and Probability of Infection that will not be exceeded in 95% of cases.

b. Using the appropriate icon on the Graph Window fit a distribution to the simulated output data for Dose.

c. Discuss the possible fits and in particular the affect on the 95% value each distribution predicts.

What will this mean for any potential receptor?

Feedback

Concentration Organisms per Litre

Viability

Volume Ingested

Dose

Probability of Infection

Dose Fit

CD/JS/DPG @Risk 5.5 Version 2.0 December 2011