---
title: "SIR Model"
type: concept
related: [Sir Model]
source: https://www.jemoka.com/posts/kbhsir_model/
confidence: high
status: active
---

The SIR Model is a model to show how diseases spread.
Susceptible &ndash; # of susceptible people Infectious &mdash; # of infectious people Removed &mdash; # of removed people Compartmental SIR model S =&gt; I =&gt; R [ =&gt; S]
So then, the question is: what is the transfer rate between populations between these compartments?
Parameters:
\(R_0\) &ldquo;reproductive rate&rdquo;: 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) \(D\) &ldquo;duration&rdquo;: duration of the infectious period \(N\) &ldquo;number&rdquo;: 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&rsquo;s able to infect, and \(I\) are the number of people doing the infecting.
And so therefore&mdash;
\(\dv{S}{T} = -\frac{SIR_{0}}{DN}\) \(\dv{I}{T} = \frac{SIR_{0}}{DN}\) \(\dv{I}{T} = \frac{I}{D}\) 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 (&ldquo;how much are you going to reproduce&rdquo;) of the strategies are determined by&mdash;
\(f_{A}(x_{A}, x_{B}) = ax_{A} + bx_{B}\) \(f_{B}(x_{A}, x_{B}) = cx_{A} + dx_{B}\) 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&rsquo;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}
