One of the best developed ways to predict how changing inputs to a complex system will change its probable outputs is to simulate the behavior of the system. Modern simulation modeling software environments (such as MATLAB/SIMULINK®, or STELLA/ITHINK® for continuous simulation, and SIMUL8® for discrete-event simulation) make the mechanics of simulation model building and use relatively straightforward. Stochastic simulation risk models have been developed for business, engineering, biological, social, and economic systems. (Agent-based simulation models have also been developed for complex social and economic systems, but this chapter focuses on continuous simulation.)
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Appendices
Appendix A: Proof of Theorem 1
The following succinct proof is due to Professor William Huber (see the acknowledgments at the start of the book).
For k > 1, the w’s satisfy recursive relationships
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(*) w k = a k,k–1 w k–1 and
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(* *) 1/w jk–1 + a kk /w jk = a jj /w jk for all j between 1 and k–1.
These are immediate from the definitions; (* *) is just a rearrangement of the relationship (a jj – a kk )w jk–1 = w jk . For k = 1, the theorem is trivial. For k > 1, we calculate
Apply (*) to the first term and distribute a kk over the second term to obtain
Finally, upon factoring out w k , collecting the coefficients for each exponential, and applying (* *) to the first k–1 of them, the derivative of z k (t) becomes recognizable:
This demonstrates that z(t) satisfies the system of equations. It remains to show that it also satisfies the initial condition. For k = 1, the value is z 1(0) = 1*[exp(a 11 *0)] = 1, as desired. For larger values of k, we need to show that
The partial fraction expansion of 1/w 1k + … + 1/w kk is a sum whose terms are in the form u ikj /(a ii – a jj ). By inspection, we obtain
with the products extending over all l between 1 and k but skipping i and j. The products are subtracted, not added, because (a ii – a jj ) appears in w ik while (a jj – a ii ) = –(a ii – a jj ) appears in w jk with the opposite sign. Evidently, the limiting value of u ikj as a ii and a jj become equal is zero, because the two products approach a common finite value. Thus, (a ii – a jj ) is a factor of u ikj , implying 1/w 1k + … + 1/w kk has a “removable singularity” on the set a ii = a jj . Since i and j were arbitrary, we conclude that z k (0) has no singularities at all and therefore is really a polynomial. The proof is finished by observing that z k (0) approaches zero whenever any a ii becomes arbitrarily large, which for a polynomial can occur only when it is identically zero. QED.
Appendix B: Listing of ITHINK™ Model Equations for the Example in Figure 11.3
Compartment M
M(t) = M(t – dt) + (flow_FM) * dt
INIT M = 0
flow_FM = F*(bFM + delta_FM)
Compartment F
F(t) = F(t – dt) + (flow_PF + proliferation_F – flow_FM) * dt
INIT F =
INFLOWS to Compartment F:
flow_PF = P*switch?*(bNP + deltaNP) + P*(1 – switch?)*(bPF + deltaPF)
proliferation_F = F*(bF + delta_bF)
Compartment P
P(t) = P(t – dt) + (flow_NP – flow_PF) * dt
INIT P = 0
INFLOWS to Compartment P:
flow_NP = N*(1 – switch?)*(bNP + deltaNP) + N*switch?*(bPF + deltaPF)
OUTFLOWS:
flow_PF = P*switch?*(bNP + deltaNP) + P*(1 – switch?)*(bPF + deltaPF)
Compartment N
INFLOWS to Compartment N:
N(t) = N(t – dt) + (growth – flow_NP) * dt
INIT N =
growth = if (TIME < 20) then 100/20 else 0
OUTFLOWS:
flow_NP = N*(1 – switch?)*(bNP + deltaNP) + N*switch?*(bPF + deltaPF)
FORMULAS AND PARAMETERS
bF = 0.08 {0.08}
bFM = 0.00008
bNP = 0.00006
bPF = 0.05
deltaNP = bNP*RNP*exposed?
deltaPF = bPF*RPF*exposed?
delta_bF = bF*RF*exposed?
delta_FM = bFM*RFM*exposed?
end_time = 60
exposed? = if ((TIME >= start_time) and (TIME <= end_time)) then exposure_factor else 0
exposure_factor = 1 {fraction of saturation exposure intternal dose}
M_x_100 = M*100
RF = 0.9
RFM = 2.19
RNP = 2 {2}
RPF = 0.2 {0.2}
start_time = 20
switch? = 0
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Cox, L.A. (2009). Determining What Can Be Predicted: Identifiability. In: Risk Analysis of Complex and Uncertain Systems. International Series in Operations Research & Management Science, vol 129. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-89014-2_11
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DOI: https://doi.org/10.1007/978-0-387-89014-2_11
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