(b) A virus sweeps through an island population seeking out people to infect in a manner similar to a blind search method. The virus remains infectious for just 1 day after infection, although the illness lasts much longer.
(i) If the pattern of infection spreads out uniformly from the initial infected case with no particular enhanced concentration at any one place, what search method best describes the growth of the viral infection?
[10%]
(ii) An initial viral outbreak is imported through an asymptomatic islander visiting family abroad. After testing positive for the virus immediately on return from his trip, he ignores advice to isolate and goes to a local supermarket. At the supermarket he inadvertently infects 3 more single men on the island that live alone in 3 different cities on the island. Patient Zero then returns home to isolate, infecting no other people. The three people he has infected return to their cities with no other infections created until they return to their respective homes. The probability of a virus carrier infecting someone they encounter is 1/3. Each of these 3 primary virus carriers encounters on average 27 people per day.
Derive an expression to estimate the total number of people infected on the island, S, given the number of days elapsed, d, and the number of new infections per virus carrier per day, p.
Assume the country has closed its border, there are no other sources of viral infection, and that there are no forms of public health control (vaccines, lockdowns etc) in action. Furthermore, assume infected individuals can only infect other people the day after being infected themselves, and any individual is infectious for over a week. You should also assume there is no interaction between the different outbreaks in the different cities.
[15%]
(iii) Now estimate the total number of people on the island that will be infected after 4 days.
[15%]
