Controlling SARS Example 9.1.12 used the equation R0sd, vd − 5s1 2 vd d 1 1 d to model the effect of quarantine and vaccination on the spread of SARS, where v is the fraction of the population that is vaccinated and d is the average number of days an infectious individual remains in the population (recall that R0 is the average number of new infections generated by each infected individual). Suppose that d − 2, v − 0.1, and public health authorities are considering a reallocation of resources that will increase v (which decreases R0) but that comes at the cost of also increasing d (which increases R0). The net effect of this reallocation on R0 will therefore depend on how much the reallocation increases each of v and d. Suppose any reallocation results in d increasing twice as much as v. (a) Use directional derivatives to determine whether a small reallocation of resources will be beneficial (that is, whether it will decrease R0). (b) Illustrate your answer to part (a) with a sketch of the directional derivative on a plot of the level curves of R0.