Viral Mathematics - Problem Set 1 Solutions

Only problem 3 is kind of hard because it needs Bayes Theorem. The rest of the problems are just high school mathematics. 

Problem 1 

10% of the population are infected with a virus. You see 10 people in a park. What is the probability that at least 2 of them are infected with the virus?

Problem 2

A new virus is spreading in the community exponentially. At the end of day 1, 10 people in the community are infected. At the end of day 4, 2000 people are infected. How many days does it take for the virus to infect 100,000 people.

Problem 3

A diagnosis test for a virus infection is being developed. It is estimated that 0.1% of the population are infected. The new test correctly detects the infection with 99% accuracy. i.e., given a randomly infected person in the population the test will give a correct positive result 99% of the time and a false negative result 1% of the time. Given a non-infected (I.e. healthy) person, the test will correctly give a negative result 99% of the time and a false positive result 1% of the time. If a random person from the population has a positive test result, what is the probability that the person has the infection?

Problem 4

The reproductive number of a virus infection is defined to be the average number of other non-infected persons to which the virus spreads from an infected person. Herd immunity is said to be achieved in the population when the effective reproductive number falls to 1. Suppose a new virus starts out with a reproductive number of 2.5. What percentage of the population must be infected to achieve herd immunity?