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MathGroup Archive 2004

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50391] Re: [mg50375] Problem with a system of equations describing an exposure to lead...
  • From: DrBob <drbob at bigfoot.com>
  • Date: Wed, 1 Sep 2004 01:49:38 -0400 (EDT)
  • References: <200408311028.GAA18673@smc.vnet.net>
  • Reply-to: drbob at bigfoot.com
  • Sender: owner-wri-mathgroup at wolfram.com

First of all, it's easier to check the answer if all quantities represent present levels (including cumulative urine). Secondly, you have a sign wrong on the right-hand-side of the equation for bl[t] (I think). Third, using small l in a variable name is a bold and (to me) annoying choice. I'd never do it unless it was a name like "bell" or "line" or "blood" in which it's easy to see what's meant. I wasted a few minutes typing in b1 in the RSolve statement where I needed bl, and getting puzzling errors.

Here's a table that shows no loss in total lead:

eqns = {urine[t] == urine[t - 1] + a blood[t - 1],
  bone[t] == b blood[t - 1] + (1 - c) bone[t - 1],
  blood[t] == (1 - a - b) blood[t - 1] + c bone[t - 1],
  urine[0] == 0, bone[0] == 0, blood[0] == lead};
soln = First@Block[{lead = 1, a = 1/4, b = 1/3, c = 1/10},
         RSolve[eqns, {urine[t], bone[t], blood[t]}, t]
         ];
k = 10;
TableForm[Table[Flatten[{#, Total@#}] &[Through[{urine, blood, bone}@t] /.
     soln] // N, {t,
       0, k}], TableHeadings -> {Range[0, k], {urine, blood, bone, lead}}]

By the way, you don't need RSolve for this unless you want a closed-form solution:

Clear[urine, bone, blood]
urine[0] = 0;
bone[0] = 0;
blood[0] = lead;
urine[t_] := urine[t] = a blood[-1 + t] + urine[-1 + t]
bone[t_] := bone[t] = b blood[-1 + t] + (1 - c) bone[-1 + t]
blood[t_] := blood[t] = (1 - a - b) blood[-1 + t] + c bone[-1 + t]
k = 10;
TableForm[Table[Flatten[{#, Total@#}] &[Through[{urine, blood, bone}@
     t] /. soln] // N, {t, 0, k}], TableHeadings -> {Range[0,
       k], {urine, blood, bone, lead}}]

Bobby

On Tue, 31 Aug 2004 06:28:57 -0400 (EDT), 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.  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).  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?  I am using Mathematica 5.
>
> Thanks in advance.



-- 
DrBob at bigfoot.com
www.eclecticdreams.net


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