Catch up / SIR and the spread of disease

10.7

1) main points

Chapter 10.7 works with modeling the spread of disease. The book uses a model called the SIR model which stands for S-susceptible, I-Infected and R-recovered. In differential equation form this comes out to dS/dt= - (Rate susceptibles get sick) = -aSI. We use the constant a, which is proportional to S and I because we assume that if there are more susceptible people around or more infected people around there will be more encounters. It is also negative because we assume that the rate people get sick will eventually decrease. But we must also account for the infected people that are removed from so we add create a differential equation pertaining to I:

dI/dt = (rate susceptibles get sick) - (rate infecteds get sick) = [aSI - bI].


Next we assume that because the recovered are no longer susceptible they increase so we have:

dR/dt = bI

We know that the total population is not changing so S+I+R= Pop so therefore once we know S and I we can calculate for R, so we are only concerned with dS/dt and dR/dt.

We know that a is a measure of how infectious the disease is and with the knowledge that if I=1 and S=762 we see that dS/dt = apprx. -2. With this info we can solve for a and we get a= .0026

When we use a phase plane to plot the information we see how I increases until S= 192 which is the called the threshold population and is the maximum amount of susceptible people to not have an epidemic. In general this number is b/a. This means that the epidemic could have been avoided if all but 192 people were vaccinated

2) Challenges

This model incorporates everything we have learned about differential equations and put it together so a good knowledge of everything is required. My direct understanding of phase planes/slope curves is minimal so sometimes it is hard for me to follow exactly how they come to graph but I do feel like I can adequately interpret them.


3) Reflections

This obviously has amazing real world applications, especially from the preventative standpoint because it can give a relative quota of vaccines to give to a population. I hope to explore this in a bit more detail because it seems like a really useful in regard to disease prevention.