Danang city recently had a second outbreak of Covid-19. With a population of 1.1 million people, assume the city decided not to introduce any measure to contain the spread of the virus.

Assume the following model describes the number of infected residents y at time t.

dy =ay(Ky) (1)

dt

Where the following are constraints: 1. is positive

2. K is positive

Given the nature of the virus, assume that when 80 percent of the city's population is infected, herd immunity has been achieved and the virus no longer spreads.

At time t = 0, there are 6 people infected in the city, and 3,000 people infected after 30 days. (a) Solve the differential equation to determine an expression for y(t)

(b) Derive the number of days before fifty percent of the population is infected using the above model

(c) Suggest how your answer might differ if herd immunity is not achieved