The SIR-style diagram in Figure 21.8 demonstrates the dynamics of a new disease called fictitia that is sweeping the nation. Fictitia is different from the disease we diagrammed in Section 21. 2 in a number of ways:

• There is no vaccine for fictitia and no way to recover from it, so there is no recovered population.

• Fictitia renders patients sterile, so the birth rate is proportional to the size of the susceptible population, not the whole population.

• Fortunately, there are some costly self-protective activities h that people can take to avoid contracting fictitia. Their rate of use depends on both the cost of the activities p and the prevalence of the disease It. Specifically, the rate of use h(p, I_t) = C − αp + γI_t for constant C.

a. Given these assumptions about the fictitia epidemic, write down the two laws of motion for S_t and I_t in terms of S_t, I_t, C, p, α, β, γ, and the λ_s.

b. What conditions must be true if the populations of infected and uninfected people are in steady state (that is, they do not change over time)?

c. Assume γ = 0. Interpret this assumption.

d. Now assumeγ > β. Interpret this assumption.

e. Assuming γ > β, solve for I∗ as a function of C, p, α, β, γ, and the λs.

f. How does a change in the price of p affect the steady-state level of I∗?

Transcribed Image Text:

Birth Susceptible Ad St Atst [31,-h(p.)] St Infected Ad, It Death Figure 21.8. The SIR model space for fictitia. St 4 Abst [3,-h(p.)]S, Р h(p. 4₂) size of the susceptible group size of the infected group number of births number of infections price of costly self-protective activities rate of costly self-protective activities death rate of susceptible group - death rate of infected group Ads Adi