Susceptible, Infectious, or Recovered models (SIR)

Introduction

Assumtions

Generally, k states, markov chain?

Assumptions are:

+ Lasting immunity + No births/other deaths

Components:

+ \(S(t)\) - Susceptible + \(I(t)\) - Infected + \(R(t)\) - Removed (recovered or died)

Proportion of people who recover each period - \(\gamma\).

Each period, infected can transmit to \(\beta\) people. Total of \(I\beta\).

Not everyone susceptible though, so \(I\beta \dfrac{S}{N}\)

Dynamics:

+ \(\dfrac{dR}{dt} = \gamma I\) + \(\dfrac{dS}{dt} = -\beta I \dfrac{S}{N}\) + \(\dfrac{dI}{dt} = \beta I \dfrac{S}{N} - \gamma I\)

Note that \(dfrac{dR}{dt} + dfrac{dI}{dt} + \dfrac{dS}{dt} = 0\)

We can then work out \(\dfrac{dI}{dS}\)

\(\dfrac{dI}{dS} = \dfrac{\dfrac{dI}{dt}}{\dfrac{dS}{dt}}\)

\(\dfrac{dI}{dS} = \dfrac{\beta I \dfrac{S}{N} - \gamma I}{-\beta I \dfrac{S}{N}}\)

\(\dfrac{dI}{dS} = \dfrac{\beta S - \gamma N}{-\beta S}\)

\(\dfrac{dI}{dS} = - 1 + \dfrac{\gamma }{\beta }\dfrac{N}{S}\)

We can then work out \(\dfrac{dS}{dR}\)

\(\dfrac{dS}{dR} = \dfrac{\dfrac{dS}{dt}}{\dfrac{dR}{dt}}\)

\(\dfrac{dS}{dR} = \dfrac{-\beta I \dfrac{S}{N}}{\gamma I}\)

\(\dfrac{dS}{dR} = -\dfrac{\beta }{\gamma }\dfrac{S}{N}\)

We can rewrite the infection dynamic:

+ \(\dfrac{dI}{dt} = \beta I \dfrac{S}{N} - \gamma I\) + \(\dfrac{dI}{dt} = I(\beta \dfrac{S}{N} - \gamma )\) + \(\dfrac{dI}{dt} = I\gamma (\dfrac{\beta }{\gamma }\dfrac{S}{N} - 1)\)

This means that outbreak if \(\dfrac{\beta }{\gamma } > \dfrac{S}{N}\)

\(R_0 = \dfrac{\beta }{\gamma }\)

What is steady state?

\(\dfrac{dI}{dt} = I(\beta \dfrac{S}{N} - \gamma )\)

\(\dfrac{dI}{dt} = 0\)

\(R_0 \dfrac{S}{N} = 1 )\)

\(\dfrac{S}{N} = \dfrac{1}{R_0} )\)

\(\dfrac{\beta }{\gamma } > \dfrac{S}{N}\)

Vaccinations

What proportion need vaccination?