Consider a total population N(t) containing n(t) diseased individuals. Suppose thaton average the members of the diseased population n(t) infect undiseased individualsat a rate α per diseased individual, per unit time, per undiseased individual, and thatthey die off at a rate one per time τ per diseased individual (τ is usually known asthe lifetime).
a) If the population is initially all infected (noting that this means n(t) = N(t)since there is no recovery from this disease), write a differential equation for thesubsequent evolution of N(t) and solve it, given that the initial population is N0 attime t = 0.
b) If the initial total population is N0, and the initial number of diseased individualsis n0 < N0, find a differential equation for the total population N(t) of this system.[Hints:]
Can anyone help here? I literally have no idea how to begin with either of these questions and feel I am missing something really obvious. I appreciate the help :-)