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Application of multi-trial Monte Carlo simulations to the estimation of
associated with the use of wastewaters for unrestricted crop irrigation

Unrestricted irrigation refers to the irrigation of all crops, including vegetables and salad crops eaten uncooked.

Risks to human health due to wastewater-borne pathogens (such as rotavirus, Campylobacter, Cryptosporidium) are estimated by quantitative microbial risk analysis (QMRA) using the scenario of regular consumption of wastewater-irrigated lettuce, which was first described by Shuval, Lampert and Fattal (1987).

Dose–response equations

Two 'dose–response' equations are used (as described in reference #2 above):

1. The exponential d–r equation, which is used for protozoan pathogens (Cryptosporidium and Giardia):

PI(d) = 1 – exp(-rd)                                   

where PI(d) is the probability of an individual, or the probable proportion of a community, becoming infected after ingesting a single dose of d protozoan pathogens, and r is a dimensionless ‘pathogen infectivity’ constant. The value of r is 0.0042 for Cryptosporidium and 0.0199 for Giardia.

2. The beta-Poisson d–r equation, which is used for viral and bacterial pathogens:

PI (d) = 1 – [1 + (d/N50)(21/a – 1)]-a                                             

where a is a dimensionless pathogen infectivity constant and N50 is the median infective dose (ie, the number of pathogens required to cause infection in 50% of those exposed). Values of N50 and a for selected excreted viral and bacterial pathogens are given in the Table below.







Vibrio cholerae












The above equations are for the ingestion of a single dose of d pathogens. However, the consumption of wastewater-irrigated lettuce (or any other crop eaten raw) occurs many times per year, and we need to be able to estimate the annual risk of infection resulting from ingesting d pathogens on n occasions per year. The equation for this annual risk of infection [PI(A)(d)] is:

PI(A)(d) = 1 - [1 - PI(d)]n                                             

where PI(d) on the right-hand side of the equation is, as before, the risk of infection from a single exposure to the pathogen dose d, and n is the number of days in a year on which a person is exposed to this single dose d.  For the consumption of wastewater-irrigated crops n would be 365 if the crops were eaten every day, but it could obviously be less – for example, 365/2 if the crops were eaten every second day.

Estimation of the single dose d and calculation of annual risk of infection

Shuval, Lampert and Fattal assumed that:

● people ate 100 grams of lettuce on each of 150 days per year,

● the wastewater contained 1 pathogen per 105 faecal coliforms (FC),

● there was a 3-log die-off of the pathogens between harvest and consumption

and they found that, after irrigation, 10.8 ml of wastewater remained on 100 g of lettuce.

They were thus able to determine the number of pathogens per 100 g of lettuce at the time of consumption for various levels of wastewater treatment expressed as numbers of FC per 100 ml of treated (or untreated) wastewater. Knowing this single dose d and the number of times per year people ingested it (150 days in their example), they were able to calculate the risk of infection per person per year.


This procedure allows each parameter in the dose–response equations and in Shuval et al.'s assumptions and measurements to vary within a specified range. For example, 10–15 ml of wastewater could be assumed to remain on 100 g of lettuce after irrigation; there could be a 1–10 (or 0.1–1) pathogen per 105 FC, a 2–3 (or lower) log die-off between harvest and consumption; and so on. The values of r, N50 and a in the dose–response equations can vary by ± any desired percentage.

The Monte Carlo QMRA program selects at random a value for each parameter from within its specified range, and then determines the risk of infection per person per year for the pathogen being considered. This is repeated any required number of times, usually 10,000 times – the program does all these 10,000 repetitions in about two seconds.

Download the Monte Carlo QMRA program for unrestricted irrigation (you may need to right click and choose "Save Target As ...")

Read how to enter parameter value ranges and how to do a multi-trial program run


The World Health Organization, in the third edition of Guidelines for Drinking Water Quality, accepts as 'tolerable' a risk of infection (not disease) from drinking fully treated drinking water of:

10-3 per person per year

ie, it is considered acceptable if one person per 1000 in a community is infected in any 12-month period by a waterborne pathogen. For an individual this level of annual risk means that he or she has a chance of becoming infected once every 1000 years – or, as we live much less than this, we have a 70-year lifetime infection risk from drinking fully treated drinking water of about 1 in 15.

The same tolerable risk of 10–3 per person per year is adopted for unrestricted wastewater irrigation. We thus ensure that eating wastewater-irrigated salad crops and vegetables uncooked is as safe as drinking fully treated drinking water.

: E-mail either Professor Duncan Mara or, for questions about the Monte Carlo programs, Dr Andrew Sleigh.

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