Consider that a new coronavirus, SARS-Cov-3, surfaces and causes an epidemic of a highly infectious disease, COVID-21. LetStbe the size of the susceptible group andItbe the size of the infected group. The Pfizer vaccine for COVID-19 was modified to be affective against COVID- 21. The vaccine is called the z vaccine, which has a pricep. The demand for the z vaccine depends on its price and on the prevalence of the disease (It); that isz=z(p,It).

Unfortunately, there is no recovery from the disease, but thankfully, no deaths either. People who are infected stay infected and continue to mix with susceptibles. Thus, there is no removal from the model.

The number of infections in any periodtis determined by a rate of transmission. Susceptible (S) and infected (I) people mix homogenously with a contact rate that is proportional to the concentrations of either population group; that is,S*I. The number of infections in any periodtis therefore

S Iz(p,I)S= Iz(p,I)Stt ttt tt

The epidemic is sufficiently short in duration that we ignore births and deaths in the model. We can visualize the model as follows:

(a)Write out the system of partial differential equations that define this model; that is, dS/dtand dI/dt.

(b)What condition must be true, in terms of the model specified in part(a), if the population is in a steady state? That is, what must dI/dtequal? What is the infection rate compared with the vaccination rate in this case?

(c)Suppose the rate of vaccination is given byz=z(p,It=)k+pIt

wherekis a scalar constant. Solve your solution in part(b)forI(this is the steady-state level ofI, which the textbooks callsI*).

(d)Interpret. What does it mean?

(e)Assume > . How does I change when the price of vaccine z changes?

(f)Do you believe a subsidy would be effective in this epidemic? That is, would a subsidy result in a lower number of infectives, all else equal?