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Numerical approach to Cryptosporidium risk assessment using reliability method

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Abstract

A previously developed Cryptosporidium transport model is solved numerically to investigate the transport and interactions between Cryptosporidium, water and surface sediment and to estimate the risk of surface water contamination by Cryptosporidium. The primary objective of this study is to expand the work of Yeghiazarian (Ph.D. dissertation, Cornell University 2001)where the analytical solution of the Cryptosporidium transport model was obtained for a simple case of specific attachment of Cryptosporidium oocysts to fine soil particles wherein some parameters have zero values. However, some studies have shown several cases where these parameters are not zero. This necessitated further study to generate a solution to the complete Cryptosporidium transport model. Utilizing the finite difference method, the Cryptosporidium transport model is solved numerically for the general case of a system with any parameter values. Previously, first- and second-order reliability methods (FORM and SORM) were employed for risk assessment using analytical transport results (Yeghiazarian, Ph.D. dissertation, Cornell University, 2001), but in this work, FORM and SORM are applied to the numerical solution of the Cryptosporidium transport model to estimate the risk of Cryptosporidium contamination in surface water. The risk of surface water contamination is estimated by the probability that the Cryptosporidium concentration in surface water at a given time and location exceeds a safety threshold. The numerical solution is interfaced with the general-purpose reliability code, CALREL, to estimate the probability of failure on one hillslope. The sensitivity of system reliability to process parameters is reported.

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Appendices

Appendix

Analysis of the stability of the numerical solution

The numerical approximation of Cryptosporidium concentrations in Eq. (8) has truncation errors at each iteration. In terms of stability analysis, the error at any iteration in time should not be amplified but is attenuated as the computation progresses. One way to investigate the stability is to see if the errors in Eq. (8) are decreased at each recursive step. Equation (8) can be written with an explicit error term as:

$$ {\left[ {\begin{array}{*{20}c} {{\varepsilon _{1} (x,t + 1)}} \\ {{\varepsilon _{2} (x,t + 1)}} \\ {{\varepsilon _{3} (x,t + 1)}} \\ {{\varepsilon _{4} (x,t + 1)}} \\ \end{array}} \right]}\,= {\user2{A}}^{-1} {\user2{B}} {\left[ {\begin{array}{*{20}c} {{\varepsilon _{1} (x,t)}} \\ {{\varepsilon _{2} (x,t)}} \\ {{\varepsilon _{3} (x - 1,t)}} \\ {{\varepsilon _{4} (x - 1,t)}} \\ \end{array}} \right]} $$
(23)

where ε is the errors that come from the finite difference approximation.

In order for the errors in Eq. (23) to be attenuated at each iteration step, the absolute value of the eigenvalue of the matrix A −1 B, |λ|, should be <1.

The matrices, A and B are composed of parameters of the Cryptosporidium transport model and Δt. Therefore, we can calculate the eigenvalues of (A −1 B) by plugging in all parameter values of the transport model. If we plug in the parameter values of the loamy sand case in Table 1, the eigenvalues can be obtained as follows:

$$ \lambda = {\left[ {\begin{array}{*{20}c} {\begin{aligned}& \lambda _{1} \\& \\ \end{aligned}} \\ {\begin{aligned}& \lambda _{2} \\& \\ \end{aligned}} \\ {\begin{aligned}& \lambda _{3} \\& \\ \end{aligned}} \\ {{\lambda _{4}}} \\ \end{array}} \right]} = {\left[ {\begin{array}{*{20}c} {\begin{aligned}& - \frac{{{{- 4}}{{.4919}} + \Delta \,t}}{{4.4919 + \Delta \,t}} \\& \\ \end{aligned}} \\ {\begin{aligned}& - \frac{{{{- 2560}} + \Delta \,t}}{{{{2560}} + \Delta \,t}} \\& \\ \end{aligned}} \\ {\begin{aligned}& - \frac{{{{- 0}}{{.2505}} + \Delta \,t}}{{{{0}}{{.2505}} + \Delta \,t}} \\& \\ \end{aligned}} \\ {{- \frac{{{{- 0}}{{.2061}} + \Delta \,t}}{{{{0}}{{.2061}} + \Delta \,t}}}} \\ \end{array}} \right]} $$

When Δt is zero, all the eigenvalues are equal to one. When Δt goes to infinity, the eigenvalues approaches to negative one. Figure 5 shows the relationship between the eigenvalue λ3 and Δt. The range of the eigenvalue is from one to negative one, thus |λ| is <1. Consequently, the numerical scheme for Cryptosporidium transport model is stable.

Fig. 5
figure 5

Eigenvalue with respect to Δt

The accuracy of the numerical solution of the Cryptosporidium transport model has been compared with the analytical solution for the simple case (Yeghiazarian 2001) by Park (2003).

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Park, Y., Yeghiazarian, L., Stedinger, J.R. et al. Numerical approach to Cryptosporidium risk assessment using reliability method. Stoch Environ Res Risk Assess 22, 169–183 (2008). https://doi.org/10.1007/s00477-007-0105-6

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