# Covid-19 Vaccination Campaign

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?（）（VcV）, where:

-Iis a cost parameter that incorporates the health risks associated with catching Covid-19;

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-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/（）（VrV）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.