Chapter 6.6, Problem 12P

The equations presented in Section 6.1 for modelling lead uptake, subsequent exchange between tissue compartments, and removal from the human body are

$\begin{array}{l}{x}_1^{\text{'}}=\left(L+k_{12}{x}_2+k_{13}{x}_3\right)-\left(k_{21}+k_{31}+k_{01}\right)x_1\\ x_2^{\text{'}}=k_{21}{x}_1-\left(k_{02}+k_{12}\right)x_2\\ x_3^{\text{'}}=k_{31}{x}_1-k_{13}{x}_3.\end{array}$

Employing the methods of Section 6.6, find the solution of this system subject to the initial conditions $x_1\left(0\right)=0,x_2\left(0\right)=0$ and $x_3\left(0\right)=0$ using the parameter values $k_{21}=0.011,k_{12}=0.012,k_{31}=0.0039,k_{13}=0.000035,k_{01}=0.021,k_{02}=0.016$, and input function

$L\left(t\right)=\{\begin{array}{ll}35,\hfill & 0\le t\le 365,\hfill \\ 0,\hfill & t>365.\hfill \end{array}$

Solve the problem in two stages, the first over the time interval $0\le t\le 365$ and the second over the time interval $t\ge 365$. Evaluate the solution for the first stage at $t=365$ to provide initial conditions for the second stage of the problem. You may wish to use a computer to assist your calculations. Check your results by comparing graphs of the solutions with the graphs in Figure $\mathrm{6.1.4}$ or by comparing with an approximation obtained by using an initial value problem solver.