You were just hired by the CDC to lead the Covid-19 vaccination campaign. The private payoff from getting vaccinated is 7r = -c, where c is cost parameter that incorporates the risk of complications associated with vaccination （vaccines per se are free of charge）. The private payoff associated with not getting vaccinated is 7r = —bJ?（）（Vc — V）, where:
-I）is a cost parameter that incorporates the health risks associated with catching Covid-19;
-7?（）is the basic reproduction number （i.e. how transmissible the disease is）;
- Vc is the critical level of immunity, or vaccination, the population needs to reach to eventually eradicate Covid-19 （i.e. the herd immunity threshold）;
- V is the fraction of the population that is currently immune/vaccinated.
Together, /7/（）（Vr 一 V）n represents the probability of catching Covid-19. For simplicity, assume that the immunity one develops to Covid-19 is permanent, and that catching the disease provides you with the same immunity that the vaccine does, so that V includes both vaccinated and infected individuals.
l.A. [0.5 point] Let S denote the fraction of the population that is susceptible. Find R, the effective reproduction number and find Vc, the herd immunity threshold. How does Vc change as new variants that have higher a R（）become the new dominant strain? Show your work and explain in words.