Please answer all parts of question 1 and 2

Exercise III: Pandemic modeling Assume that SUE) describes the number of people aected by a disease after 3 days. Different assumptions can be made about the spread of the disease, giving rise to different equations and conclusions. 1. First assume that the number of people newly aected by the disease each day is proportional to the number of people carrying the disease. This gives rise to the equation S'(t) = ESE) where PS > 0 is some constant. (a) Solve this equation. (b) Say how long it takes to infect half of the whole world (4 billion people), if 10 people are infected on day 0, and h} = 10%. Give both the exact value, and an approximation rounded to the nearest integer (use a calculator). (c) Compute lim SUE), and explain why this is an issue with the model. t}-+oo 2. A better model is given by the equation where M is the total population. The second factor describes the fact that the more people are infected, the fewer healthy people there are to infect (a) Solve this equation. To avoid issues with absolute values, you can assume that it is \"obvious\" that 0 g y(t) S M for all t Z 0. (1)) Say how long it takes to infect half of the whole world (4 billion people), if 10 people are infected on day 0', and K. = 10%. Give both the exact value, and an approximation rounded to the nearest integer (use a calculator). Compare to Question 1.(b). (c) Compute lim S(t), and compare to Question 1.(c). t}+oo