Editing Compartmental models (epidemiology) (section)

===The SIR model with vital dynamics and constant population===

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading.<ref name="Beckley"/> The model with mass-action transmission is:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I  - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

:<math> R_0 = \frac{ \beta}{\mu+\gamma}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

:<math> R_0 \le 1 \Rightarrow  \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>
:<math> R_0 > 1 , I(0)> 0 \Rightarrow  \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.