4

P7-8) Consider the application of the PSSH to epidemiology. We shall treat each of the following steps as elementary in that the rate will be proportional to the number of people in a particular state of health. A healthy person, H, can become ill, I, spontaneously, such as by contraction smallpox spores, or he may become ill through contact with another ill person. The ill person may become healthy or he may die. a) Derive an equation for the death rate. Solution:

b) At what concentration of healthy people does the death rate become critical? Solution: k +k rD is infinite at [ H ] = 3 4 . k2 c) Comment on the validity of the PSSH under the conditions of part (b) Solution:

d) Use Polymath to plot H, I, and D versus time. Vary ki and describe what you find. Solution: POLYMATH Results 03-04-2007, Rev5.1.233

Calculated values of the DEQ variables Variable t H I D k1 k2 k3 k4

initial value 0 1.0E+09 0 0 1.0E-08 1.0E-16 5.0E-10 1.0E-09

minimal value 0 4.771E+06 0 0 1.0E-08 1.0E-16 5.0E-10 1.0E-09

maximal value 1.0E+08 1.0E+09 9.46E+08 7.263E+07 1.0E-08 1.0E-16 5.0E-10 1.0E-09

ODE Report (RKF45) Differential equations as entered by the user [1] d(H)/d(t) = -k1*H-k2*I*H+k3*I [2] d(I)/d(t) = k1*H+k2*I*H-k3*I-k4*I [3] d(D)/d(t) = k4*I Explicit equations as entered by the user [1] k1 = 1E-8 [2] k2 = 1E-16 [3] k3 = 5E-10 [4] k4 = 1E-9 Independent variable variable name : t initial value : 0 final value : 100000000 Precision Step size guess. h = 0.000001 Truncation error tolerance. eps = 0.000001 General number of differential equations: 3 number of explicit equations: 4

I H

D

final value 1.0E+08 4.771E+06 9.226E+08 7.263E+07 1.0E-08 1.0E-16 5.0E-10 1.0E-09

The following plot is an example of what happens when changing a kinetic parameter, in this case, changing k4 to 10-7 which causes people to die faster and people to become ill slower since ill people are not around as long to infect others.

I

H

D

e) E.g., write down incorrect stoichiometry or rate expressions. f) E.g., include birth rate.

P7-11) Beef caralase has been used to accelerate the decomposition of hydrogen peroxide to wield water and oxygen. The concentration of hydrogen peroxide is given as a function of time for a reaction mixture with a pH of 6.76 maintained at 30oC. t (min) CH2O2 (mol/L)

0 0.02

10 0.01775

20 0.0158

a) Determine the Michaelis-Menton parameters Vmax and KM. Solution: assume a well mixed batch reactor

50 0.0106

100 0.005

b) If the total enzyme concentration is tripled, what will the substrate concentration be after 20 min? Solution:

c) E.g., consider a complex kinetic rate expression. d) E.g., write down incorrect stoichiometry or rate expressions.