Chapter 15

Assuming that immediately after a 2 gm injection of penicillin into the vascular pool, the penicillin concentration throughout the pool is \(200 \mu \mathrm{g} / \mathrm{ml}\), what is the estimated volume of the vascular pool? How does this estimate compare with the blood volume of an adult of about 5 liters and a serum volume of 2.75 liters?

Solve the following systems. a. \(\begin{array}{rl}A_{0} & =1 \quad A_{t+1}=0.52 A_{t}+0.04 B_{t} \\\ B_{0}=2 & B_{t+1}=0.24 A_{t}+0.4 B_{t}\end{array}\) b. \(\begin{aligned} A_{0} &=1 \quad A_{t+1}=0.3 A_{t}+0.9 B_{t} \\ B_{0} &=0 \quad B_{t+1}=0.2 A_{t}+0.6 B_{t} \end{aligned}\) c. \(\begin{array}{rl}A_{0} & =3 & A_{t+1}=0.3 A_{t}-0.5 B_{t} \\ B_{0} & =2 \quad B_{t+1}=0.2 A_{t}+0.1 B_{t}\end{array}\) d. \(\begin{array}{rl}A_{0} & =2 A_{t+1}=0.5 A_{t}-0.1 B_{t} \\ B_{0}=3 & B_{t+1}=0.1 A_{t}+0.3 B_{t}\end{array}\) e. \(\begin{aligned} A_{0} &=4 \quad A_{t+1}=3.0 A_{t}-0.5 B_{t} \\ B_{0} &=5 \quad B_{t+1}=2.0 A_{t}+1.0 B_{t} \end{aligned}\) f. \(A_{0}=0 \quad A_{t+1}=0.42 A_{t}+0.04 B_{t}\) \(B_{0}=1 \quad B_{t+1}=0.24 A_{t}+0.38 B_{t}\) g. \(\begin{aligned} A_{0} &=0 \quad A_{t+1}=0.62 A_{t}-0.08 B_{t} \\ B_{0} &=-1 \quad B_{t+1}=0.48 A_{t}+0.18 B_{t} \end{aligned}\) h. \(\begin{aligned} A_{0} &=1 \quad A_{t+1}=3 A_{t}+6 B_{t} \\ B_{0} &=4 \quad B_{t+1}=2 A_{t}+4 B_{t} \end{aligned}\) i. \(\begin{aligned} A_{0} &=1 \\ B_{0} &=3 \quad B_{t+1}=0.5 A_{t}-1.00 B_{t} \\\=& 0.1 A_{t}+0.30 B_{t} \end{aligned}\) j. \(\begin{aligned} A_{0} &=3 A_{t+1}=0.3 A_{t}+0.6 B_{t} \\ B_{0} &=4 B_{t+1}=0.6 A_{t}+0.3 B_{t} \end{aligned}\)

Find the roots to the characteristic equation, \(r^{2}-p r+q=0\) and write the solutions for \(A_{t}\) in the following systems. a. \(\begin{array}{rlrl}A_{0} & =10 & A_{t+1} & =0.8 A_{t}+0.2 B_{t} \\ B_{0} & =0 & B_{t+1} & =0.1 A_{t}+0.7 B_{t}\end{array}\) b. \(\begin{array}{rlrl}A_{0} & =0 & A_{t+1} & =0.6 A_{t}+0.3 B_{t} \\ B_{0} & =5 & B_{t+1} & =0.2 A_{t}+0.7 B_{t}\end{array}\) c. \(\begin{array}{rlrl}A_{0} & =1 & A_{t+1} & =0.26 A_{t}+0.04 B_{t} \\ B_{0} & =1 & B_{t+1} & =0.06 A_{t}+0.24 B_{t}\end{array}\) d. \(\begin{aligned} A_{0} &=2 & & A_{t+1}=& 1.04 A_{t}+0.16 B_{t} \\ B_{0} &=3 & & B_{t+1} &=0.24 A_{t}+0.96 B_{t} \end{aligned}\) e. \(\begin{array}{rlrl}A_{0} & =20 & A_{t+1} & =0.86 A_{t}+0.04 B_{t} \\\ B_{0} & =10 & B_{t+1} & =0.06 A_{t}+0.84 B_{t}\end{array}\)

For the trigonometrically strong. Show that $$ A_{t}=\rho^{t} \sin t \theta \quad \text { solves } \quad A_{t+2}-p A_{t+1}+q A_{t}=0 $$ where $$ p^{2}-4 q<0, \quad \rho^{2}=q \quad \text { and } \quad \theta=\arccos \left(\frac{p}{2 \rho}\right) $$ You may wish to know that \(\sin (x+y)+\sin (x-y)=2 \sin x \cos y\)

Find the solutions to a. \(w_{0}=3 \quad w_{1}=1\) b. \(w_{0}=0 \quad w_{1}=0\) c. \(w_{0}=2 \quad w_{1}=1\) d. \(w_{0}=1 \quad w_{1}=1\) $$ \begin{aligned} w_{t+2}-5 w_{t+1}+6 w_{t} &=0 \\ w_{t+2}+8 w_{t+1}+12 w_{t} &=0 \\ w_{t+2}-6 w_{t+1}+8 w_{t} &=0 \\ w_{t+2}-5 w_{t+1}+4 w_{t} &=0 \end{aligned} $$

Define the matrices $$ A=\left[\begin{array}{rr} 1.2 & -0.6 \\ 0.4 & 0.2 \end{array}\right] \quad B=\left[\begin{array}{ll} 1.2 & 0.4 \\ 0.8 & 0.4 \end{array}\right] \quad C=\left[\begin{array}{rr} 2.5 & -2.5 \\ -5.0 & 7.5 \end{array}\right] \quad L=\left[\begin{array}{rr} 0.8 & 0 \\ 0 & 0.6 \end{array}\right] $$ a. Compute the characteristic roots of \(A\). b. Use pencil and paper to show that \(B \times C=I\). c. Use pencil and paper to show that \(B \times L \times C=A\). d. Use pencil and paper to compute \(L^{2}\). e. Use pencil and paper to show that \(B \times L^{2} \times C=A^{2}\). f. Show that \(B \times L^{5} \times C=A^{5}\).

Solve for \(B_{t}\) For the initial conditions and iteration equations: \(A_{0}=2 \quad A_{t+1}=0.3 A_{t}+0.1 B_{t}\) \(B_{0}=5 \quad B_{t+1}=0.1 A_{t}+0.3 B_{t}\) a. Eliminate \(A_{t}\) by subtraction \(\left(0.1 A_{t+1}-0.3 B_{t+1}\right)\). b. Explain why \(B_{t+2}=0.1 A_{t+1}+0.3 B_{t+1}\). c. Use the two equations from parts a. and b. to write $$ B_{t+2}-0.6 B_{t+1}+0.08 B_{t}=0 $$ d. Suppose that for some number, \(r, B_{t}=C r^{t} .\) Show that \(C \neq 0, r \neq 0\) imply that \(r^{2}-0.6 r+0.08=0\) e. Find the roots of \(r^{2}-0.6 r+0.08=0\). f. Show that for any two numbers, \(C_{1}\) and \(C_{2}\), $$ B_{t}=C_{1} \times 0.2^{t}+C_{2} \times 0.4^{t} \quad \text { solves } \quad B_{t+2}-0.6 B_{t+1}+0.08 B_{t}=0 $$ g. Compute \(B_{1}\). h. Show that $$ B_{t}=1.5 \times 0.2^{t}+3.5 \times 0.4^{t} $$

Find solutions for both \(A_{t}\) and \(B_{t}\) satisfying $$ \begin{array}{lll} \text { a. } & A_{0}=10 & A_{t+1}=0.50 A_{t}+0.2 B_{t} \\ & B_{0}=0 & B_{t+1}=0.15 B_{t}+0.7 B_{t} \\ \text { b. } & A_{0}=5 & A_{t+1}=0.6 A_{t}+0.1 B_{t} \\ & B_{0}=10 & B_{t+1}=0.2 A_{t}+0.7 B_{t} \end{array} $$

Suppose at the beginning of the study there are 24,950 susceptible people, 50 infected people, and no recovered/immune people, and let \(\beta=0.00002\) and \(\gamma=0.2\). Compute the values of \(S_{t}, I_{t},\) \(R_{t}\) for 70 days. a. When is the epidemic at its height? b. Do all of the people get sick? c. Repeat the computation for \(\beta=0.00006\) and only 12 days. You will find that all of the people will get sick. What is the least value of \(\beta\) for which all people get the flu? The event that everyone gets sick is a property of our discrete model, and perhaps a peculiar property. You will show in Exercise 18.5 .11 that in the continuous model analogous to Equations 15.4 it never happens that everyone gets sick.