Chapter 9, Problem 9.7P

Epidemiology. 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 contracting smallpox spores:

$\text{H}\stackrel{k_1}{\to }\text{I}\text{(P9-7}\text{.1)}$

or the person may become ill through contact with another ill person:

$\text{I}\text{+}\text{H}\stackrel{k_2}{\to }\text{2I}\text{(P9-7}\text{.2)}$

The ill person may become healthy:

$\text{I}\stackrel{k_3}{\to }\text{H}\text{(P9-7}\text{.3)}$

or the ill person may expire:

$\text{I}\stackrel{k_4}{\to }\text{D}\text{(P9-7}\text{.4)}$

The reaction given in Equation (P9-7.4) is normally considered completely irreversible, although the reverse reaction has been reported to occur.

1. (a) Derive an equation for the death rate.
2. (b) At what concentration of healthy people does the death rate become critical? (Ans.: When [H] = (k₃ + k₄)/k₂.)
3. (c) Comment on the validity of the PSSH under the conditions of Part (b).
4. (d) If k₁ = 10⁻⁸ h⁻¹, k₂ = 10⁻¹⁶ (people · h)⁻¹, k₃ = 5 × 10⁻¹⁰ h⁻¹, k₄ = 10⁻¹¹ h⁻¹, and H_o = 10⁹ people, use Polymath to plot H, I, and D versus time. Vary k_i and describe what you find. Check with your local disease control center or search online to modify the model and/or substitute appropriate values of k_i. Extend the model, taking into account what you learn from other sources (e.g., the Internet).
5. (e) Apply one or more of the six ideas in Preface Table P-4, page xxviii, to this problem.