3. An Epidemic Model We can model the spread of viruses like COVIDlg using differential equations, with the following simplifying assumptions: 0 The population at any time can be divided into three categories: infected people, susceptible people, and immune people. The population starts with a small number of infected people, and the rest of them are susceptible. In particular, at the start, there are no immune people. 9 Every infected person recovers, and once recovered, is immune from further infection, 0 A susceptible person immediately becomes infected upon contact with an infected person. a The population size is unchanging during the time period of study and is equal to N. Since the numbers of infected, susceptible, and immune people always add up to N , it sufces to model two of these quantities. A virologist wrote down the following system of differential equations in A(t) and B(t) in her notes to model the dynamics of the spread of COVlD-IQ: dA b E = 7NA(t)B(t), l: > 0 078 b E 7 NA(t)B(t) 7 we), is > o. (:1) Unfortunately, the virologist has forgotten which two of the three quantities (infected, susceptible, or immune people) the functions A and B were modeling. Using all the differential equations analysis abilities at your disposal. help the virologist conclusively decide what quantities A and B model. Be sure to justify your answer (in particular, rule out the alternatives). (h) Let N = 1000, and suppose the initial condition is that 10 people are infected and 990 are susceptible and R: 2 0.5. i. Find a I} value so that everyone eventually become infected and then recover. Explain why you know this happens. ii. Find a 5 value so that there is B(t) 5 10 for all time. This is the same as asking for I: such that % S 0. In other words, we want a situation where the number of infected people at any time is kept smaller than the initial number of infected people. New people may become infected but not faster than infected people recover. This would for example help keep hospitals able to manage infected people. iii. What factors do you think aect b? Which of the two parameters is or b dorm the social distancing approach aim to impact