The SIR Model is a model to show how diseases spread.
- Susceptible – # of susceptible people
- Infectious — # of infectious people
- Removed — # of removed people
Compartmental SIR model
S => I => R [ => S]
So then, the question is: what is the transfer rate between populations between these compartments?
Parameters:
- \(R_0\) “reproductive rate”: the number of people that one infectious person will infect over the duration of their entire infectious period, if the rest of the population is entirely susceptible (only appropriate for a short duration)
- \(D\) “duration”: duration of the infectious period
- \(N\) “number”: population size (fixed)
Transition I to R:
\begin{equation} \frac{I}{D} \end{equation}
\(I\) is the number of infectious people, and \(\frac{1}{D}\) is the number of people that recover/remove per day (i.e. because the duration is \(D\).)
Transition from S to I:
\begin{equation} I \frac{R_0}{D} \frac{S}{N} \end{equation}
So for \(\frac{R_0}{D}\) is the number of people able to infect per day, \(\frac{S}{N}\) is the percentage of population that’s able to infect, and \(I\) are the number of people doing the infecting.
And so therefore—
- \(\dv{S}{T} = -\frac{SIR_{0}}{DN}\)
- \(\dv{I}{T} = \frac{SIR_{0}}{DN}\)
- \(\dv{I}{T} = \frac{I}{D}\)
Evolutionary Game Theory
Suppose that we have two strategies, \(A\) and \(B\), and they have some payoff matrix:
| A | B | |
|---|---|---|
| A | (a,a) | (b,c) |
| B | (c,b) | (d,d) |
and we have some values:
\begin{equation} \mqty(x_{a} \\x_{b}) \end{equation}
are the relative abundances (i.e. that \(xa+xb\)).
The finesses (“how much are you going to reproduce”) of the strategies are determined by—
- \(f_{A}(x_{A}, x_{B}) = ax_{A} + bx_{B}\)
- \(f_{B}(x_{A}, x_{B}) = cx_{A} + dx_{B}\)
Except for payoff constants \((a,b,c,d)\), everything else is a function of time.
The mean fitness, then:
\begin{equation} q = x_{A}f_{A} + x_{B}f_{B} \end{equation}
Let’s have the actual, absolute number of individuals:
\begin{equation} \mqty(N_{A}\\ N_{B}) \end{equation}
So, we can talk about the change is individuals using strategy \(A\):
\begin{equation} \dv t x_{A} = \dv t \frac{N_{A}}{N} = X_{A}(f_{a}) \end{equation}