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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: [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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