[AMath 383] HW #5 Hints
There have been a few questions about determining the infection rate
parameter beta for problem #1. I want to make a few comments about this.
Some of you have correctly determined that the model will grow
approximately exponentially near N=0. Ignoring the recovery rate
momentarily, you might want to estimate dI/dt=beta S I (but with
S=N-I), and then find the linear approximation for I near 0. This will
yield one approach to estimating beta.
Another approach that I originally anticipated people to use was to use
a similar approach to our estimates of per capita birth and death
rates. When the number of infected individuals are small, each infected
individual will give rise to an average of 10 new infecteds in a day.
So, how would we mathematically get the per capita infection rate from
the formula? (This is like a birth rate for the infecteds, or could be
thought of as a death rate for the susceptibles.)
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.
Best wishes,
D. Brian Walton
AMath 383 Instructor
Guggenheim 408C
(206) 685-9298