RE: [AMath 383] HW #5 Hints
The second approach gives an estimate that will be larger than the
first approach. Essentially, this arises from the fact that the
exponential growth model assumes that each individual that becomes
infected has part of the day under consideration to infect others. But
the second approach neglects these secondary infections. The
exponential growth approach (first) leads to a smaller estimate of the
rate because the burden of infecting is shared with early contacts. But
the second approach requires each individual to account for the full 10
new infecteds, leading to a higher rate estimate.
--
Did you mean to say: "But the *first* approach neglects the secondary
infections."
At first you say that the second approach gives an estimate larger than
the first approach because infected people are able to inflect more
people in the same day. I understand that part. But why does the
second approach neglect these seconardy infections? I believe I
understand what you are saying. So, when we look at a per-capita growth
of an infection we are looking at one individuals effect on the number
of effected. Thus for N* that is close to 0 we are able to get an
infection rate that is close to log(10) for 1 day (per capita).
However, when we look at the first method, we look at the entire
population infecting the entire population. But I don't know if I
completely understand the first method that you have described.
I determine an equation as you described for dI/dt where there is no S.
Then I wanted to estimate a rate of growth of infection that is close to
y'=10 at very small values of I. I took the derivate of dI/dt and then
determined the value at I''(0) and then plugging in N=10000. So I got
an infection rate (0.001).
When I attempt the method we used when viewing per-capita growth I
determined the derivate but instead when I received the I'', I said that
e^[I''(0)] and then solved for 10 = e^[I''(0)]. Which yields a value
smaller than I received with the first method.
I don't believe that I'm a performing the operations correctly. In
either case I think I don't understand the first method in enough clarity.