Herd Immunity: Let denote the reproduction number (i.e., the average number of new infections caused by a person, infected with a virus, in the early stages of an outbreak in a fully susceptible population).

Let v denote the fraction of the population with immunity (these are people vaccinated, people who have recovered from the infection, and people who just have immunities naturally). We have .

Let . At the extremes, if , then ; and if , then . Herd immunity is the value of v such that . That is, the reproduction number falls to less than 1, which means that the disease eventually dies out naturally.

a. Solve (in terms of the reproduction number ) for the value of v at which herd immunity occurs. That is, what condition must v fulfill in order to achieve herd immunity?

b. In Column A in Excel, enter the reproduction numbers 1,2,3,.,15. In Column B, calculate the values of v at which herd immunity occurs. Use the graphing tool in Excel to draw a graph with the reproduction numbers in Column A on the horizontal axis and the herd immunity values on the vertical axis.

kindly Pickup if you solved both part's other don't..