So I was given a question about spread of disease:
A virus is spreading through a city of 50,000 people who take no precautions. The virus was brought to the town by 100 people and it was found that 1000 people were infected after 10 weeks. How long will it take for half of the population to be infected?
I did some digging and discovered these 2 logistic equations
) $y'= k y (1-\frac{y}{M})$
) $y' = k y (M-y)$
where,
$k$ is a constant
$y$ is the infected people
$M$ is the total population
I understand the concept of how the first equation is formed more or less from several websites explained. However, I don't know much about the second equation but I have seen it in some similar questions. So, what I am confused about is what are the conditions in using these two equations or differences between these two equations and which one is better for the problem mentioned? (If possible, I hope someone could explain more about the second equation.)
P.S. This is my first time posting and sorry for the plain equation font, I'll try to figure out how MathJax works slowly but surely. Thank you in advance.
Actually, those two logistic growth equations are the same. Your first eqn. is $$ y'=ky(1-\frac{y}{M})=ky(\frac{M-y}{M})=\frac{k}{M}y(M-y) $$ which is the second equation with a different $k'=k/M$.