2. (SARS control) Consider the following model for the spread of Sudden Acute Respiratory Syndrome (SARS):
R0(t,v)=5(1−v) t . 1+t
R0 is the average number of new people a single patient infects, v is the fraction of population that is vaccinated, and t is the average number of days an individual is infectious while remaining amidst the population (i.e. before quarantine).
(a) What are plausible pre-quaratine times? What are possible population-level vaccination rates? And what, subsequently, is a plausible average number of new infections a single patient could cause?
(b) How does the number of new infections change when you change the vaccination rate and keep pre-quarantine time the same? How does the number of new infections change when you change pre-quarantine time but keep the vaccination rate the same?
(c) Suppose t = 4 and v = 0.7. Is increasing vaccination rate or deceasing pre-quarantine time more effective in controlling transmission? What is the most effective ratio of changing the two together? Why might the units of t and v matter when interpreting your answer?
(d) Suppose t = 4 and v = 0.7. Suppose that given some amount of prior spending, every additional 100 million dollars spent on SARS detection/early quarantine reduces t by approximately 0.5, and every additional 100 million dollars spent on vaccine distribution increases v by aproximately 0.05. Finally, suppose that the government budget newly allocated to SARS control is 200 million dollars. Thus, v and t both depend on the ratio p of that budget that is put into detection verus vaccination, i.e. the detection budget is 200p and the vaccination budget is 200(1 − p). What are the rates at which pre- quarantine time and vaccination rate each depend on p? Hint: You should not be taking derivatives of the functions of p that are given. You should be giving new equations in terms of p for each of t and v.
(e) Based on your answers to (d), how then does R0 depend on p?
(f) Based on your answers to (c) and (e), what then is the optimal ratio p for controlling transmission?