I have the following question from a past paper which I seemed to be rusty on and just wanted to go through it:
A virus spreads slowly through a population of animals. Initially no animals are infected, but after 10 days 25% of the population in infected. Let denote the proportion of the population that is infected. Then
(i) Solve the differential equation to get an expression for P(t).
(ii) Find the value of the constant k.
(iii) How long will it take for 75% of the population to become infected?
This seems wrong as at , no animals should be infected, but the equation I derived gives a value of 2. What have I done wrong?
Also, why don't I have a coefficient in front of the term? I have also never seen a constant next to the normal term; is that normal?
I will wait till we have figured out part (i) before I attempt the next two parts.
Just out of curiosity, how did you spot to switch the sides of kt and ln|1-P|? After considering the situation presented by the question, it seems we are dealing with an exponential growth problem which would suggest I need to find a formula with a +k value. Typically I would try to get it into the form of +k for growth and -k for decay. I can see now with the final answer that it makes sense, but how would you spot it before getting to the end?
So, whenever I have a problem where the term being multiplied by the constant k in the instantaneous growth/decay rate equation:
is negative, I look to find a P(t) term with a -k exponent and vice-versa, regardless of whether the question is about growth or decay?