SIR model.There is an infectious disease for which a vaccine exists. The government considers subsidizing the vaccine. Derive how a change in the price of vaccines changes the steady state infected population,I, in an SIR model - - in which the demand for vaccine is affected by the size of the infected population. (I.e., derivedI/dp , which indicates the vaccine subsidy effectiveness affecting the size of the infected popoulation or prevalenceI)

(a) Recall the formula in the SIR model reflecting the size of the susceptible population in steady state :

(eq.1)

Take the derivative of Equation (eq 1) with respect top. [Hint:Remember thatvtis a function of both the price of vaccines and the number of infected people,pandI. Remember too thatIwill change withp, so the derivative ofvtwith respect toprequires the use of the chain rule. By contrast, the birth rate b, the infectivity rate , and the death rate of the susceptible group dSare all constants and don't change with the price of vaccinesp.]

(b) Recall the steady state population of the susceptible groupSfound in Equation 21.8. Does it depend on the price of vaccinesp?

(c) Using this fact aboutdS/dp that you derived in the last question, simplify theexpression you derived in the first part of this question.

(d) Now solve fordI/dp When the prevalence elasticity of demand for the vaccinedv/_dIis high, is the steady-state size of the infected population more or less affected by changes in the vaccine price,p, compared to when the prevalence elasticity of demand for the vaccine dv/_dIis low?

(e) Will a vaccine subsidy will be more or less effective (dI/dp higher or lower) if infectivity is high? Explain the intuition for this mathematical result.