SIR Model — Jemoka Knowledge Base

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}