1. Consider that a new coronavirus, SARS-Cov-3, surfaces and causes an epidemic of a highly infectious disease, COVID-Ql. Let S: be the size of the susceptible group and It be the size of the infected group. The Pzer vaccine for COVE19 was modied to be affective against COVID 21. The vaccine is called the z vaccine, which has a price p. The demand for the 2: vaccine depends on its price and on the prevalence of the disease (L); that is z = z{ p, Ir) . Unfortunately, there is no recovery 'om the disease, but thankfully, no deaths either. People who are infected stay infected and continue to mix with susceptibles. Thus, there is no removal 'om the model. The number of infections in any period I is 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, 3*]. The number of infections in any period I is therefore as}, z(p,r,)s, =[,31, z(p,1,)]s, The epidemic is suiciently short in duration that we ignore births and deaths in the model. We can visualize the model as follows: [I51 - zip-[HS {3] Write out the system of partial differential equations that dene this model; that is, dedI and dlfdt. {b} What condition must be true, in terms of the model specied in part (a), if the population is in a steady state? That is, what must ddt equal? What is the infection rate compared with the vaccination rate in this case? {c} Suppose the rate of vaccination is given by z=z(p,r,) =kap+41, where k is a scalar constant. Solve your solution in part (b) for I (this is the steady-state level of I, which the textbooks calls I'). (d) Interpret A. What does it mean? {e} Assume 2L :5 ,6. How does I change when the price of vaccine 2 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