The spread of infectious diseases like influenza are often modeled using the following autonomous differential equation: dI/dt = βI(N − I) − μI where I is the number of infected people, N is the total size of the population being modeled, β is a constant determining the rate of transmission, and μ is the rate at which people recover from infection.
(a) Suppose beta = 0.02, N = 7000, and mu = 3. Find all equilibria. (Enter your answers as a comma-separated list.) I = 0, 6850
(b) For the equilibria found in part (a), determine whether they are stable or unstable. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) stable I = 6850 unstable I = 0
(c) Leaving the parameters unspecified, what are the equilibria of the model in terms of beta, N, and mu? (Enter your answers as a comma-separated list.)
(d) The epidemiological quantity R_0 is calculated by rearranging the instability criterion for the equilibrium I = 0 to obtain an inequality of the form R_0 > 1. Write R_0 in terms of beta N, and mu. R_0 =