Re: Problem with a system of equations describing an exposure to lead...

*To*: mathgroup at smc.vnet.net*Subject*: [mg50398] Re: Problem with a system of equations describing an exposure to lead...*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Thu, 2 Sep 2004 04:34:22 -0400 (EDT)*Organization*: The University of Western Australia*References*: <ch1ke0$ic4$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <ch1ke0$ic4$1 at smc.vnet.net>, Richard Palmer <dickp at bellatlantic.net> wrote: > I have a system of equations > > Eqns={u[t]==a bl[t-1], > bo[t]==b bl[t-1]-c bo[t-1], > bl[t]==(1-a-b) bl[t-1]-c bo[t-1], > u[0]==0, > bo[0]==0, > bl[0]==pb} > > They are supposed to represent a system of what happens when a person > suffers a one-time exposure to lead (pb). At any given time, the person > will pass some portion of the lead that is in the blood and soft tissue > (bl[t]) through the urine (u[t]) while some portion moves to the bone > (bo[t]) and a small portion moves from the bone back into the blood. This > is easily solved with RSolve. Why are you modelling this as a difference equation and not as a differential equation? > However, I am concerned the solution may not > be correct: I believe that for any time t, the amount of lead in the bone > and blood less all lead passed to date through the urine should equal the > level of exposure (pb). If I understand you correctly, you expect that, for any time t, bo[t] + bl[t] + u[t] == pb It is easy to show that this requirement is inconsistent with your system of equations, by turning the equations and initial conditions into a set of replacement rules: eqnrules = { u[t_] -> a bl[t-1], bo[t_] -> b bl[t-1]-c bo[t-1], bl[t_] -> (1-a-b) bl[t-1]-c bo[t-1]}; ics = {u[0] -> 0, bo[0] -> 0, bl[0] -> pb}; For t = 1, the equations are consistent (as you will see from the RSolve solution for numerical values): bo[t] + bl[t] + u[t] /. t -> 1 /. eqnrules /. ics // Simplify However, for t = 2, two applications of the rules reveals the problem: bo[t] + bl[t] + u[t] /. t -> 2 /. eqnrules /. ics // Simplify % /. eqnrules /. ics // Simplify The result is not equal to pb unless the constants are suitably restricted. > Taking the Rsolve solutions and making a table of > u[t], bl[t], and bo[t] for times 0-5 will show the problem (substitute > random fractions for a,b, and c and take pb to be 1). Is the mistake in the > solution or in my formulation of the model? Looks like a model formuation problem to me ... Cheers, Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul

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**Re: Problem with a system of equations describing an exposure to lead...**

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