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?
\(\dfrac{dI}{dt} = I(\beta \dfrac{S}{N} - \gamma )\)
\(\dfrac{dI}{dt} = 0\)
\(R_0 \dfrac{S}{N} = 1 )\)
\(\dfrac{S}{N} = \dfrac{1}{R_0} )\)
\(\dfrac{\beta }{\gamma } > \dfrac{S}{N}\)
What proportion need vaccination?