Chapter 14

The SIR model.

Here we look at the SIR model discussed in this chapter. This is used in connection with infectious diseases such as chicken pox and COVID-19 (but not the common cold or seasonal flu) for which individuals who survive the disease obtain long-term immunity.

Consider the following subpopulations:

(Note that $r(t)$ increases over time, $s(t)$ decreases over time since survival from exposure confers immunity, and $s + i + r = 1$.)

The SIR model is the system of ODE’s: \begin{align} \dfrac{ds}{dt} &= -\tau s(t)i(t), \\ \dfrac{di}{dt} &= (\tau s(t) - \gamma) i(t), \\ \dfrac{dr}{dt} &= \gamma i(t). \end{align} with transmission rate $\tau > 0$ (lower-case Greek tau) and recovery rate $\gamma > 0$ (lower-case Greek gamma) each having dimension 1/time (e.g., per day). This is non-linear due to $s(t)i(t)$ in two of the equations.

Let $R_0 = \tau/\gamma$. This ratio is a dimensionless natural quantity for describing the dynamics. The interpretation of $R_0$ is the average number of new infections caused by an infected person; it is called the basic reproduction number (it is not a “rate”, since it is dimensionless). For example, the deadliest second of the four waves of the 1918 flu had $R_0 = 3.8$.

To play with the visulization yourself, change the parameters, and choose your initial conditions by dragging the purple point. Observe that the three curves $s,i,r$ stay nonnegative.

Hamiltonian systems: the bead on a ring, revisited.

We now return to the bead on the rotating ring (with constant angular velocity $\omega > 0$) that we analyzed in the previous chapter. Recall that this is described by ODE system \begin{align} \frac{d}{dt} \begin{bmatrix}\phi(t) \\ \rm{w}(t)\end{bmatrix} = \mathbf{f}(\phi(t), \rm{w}(t)) \end{align} with $\mathbf{f} \colon \mathbf{R}^2 \to \mathbf{R}^2$ defined by \begin{equation}\mathbf{f}(\phi,{\rm{w}})=({\rm{w}}, -(g/r)\sin\phi+\omega^2\sin\phi\cos\phi)=({\rm{w}}, -(g/r)\sin\phi+\ (\omega^2/2)\sin(2\phi)).\end{equation}

The bead on the rotating ring is a Hamiltonian system. Let \begin{align} E &=\frac{1}{2}{\rm{w}}^2 -\frac{g}{r}\cos\phi +\frac{\omega^2}{4}\cos(2\phi) \\ & =\frac{1}{2}{\rm{w}}^2 -\frac{g}{r}\cos\phi + \frac{1}{2}(\omega \cos \phi)^2-\frac{\omega^2}{4}, \end{align} then $$\pd{E}{{\rm{w}}}={\rm{w}},\,\,\,-\pd{E}{\phi}=-\frac{g}{r}\sin\phi+\frac{\omega^2}{2}\sin(2\phi).$$

Here, same as in the previous chapter, without loss of generality, we choose $r$ so that $g/r = 1$ with unit 1/time².

In the graph below we plot level sets of $E$. To play with the visulization yourself, change the angluar velocity $\omega$ of the ring, choose which level set $E=c$ you want to see (shown in black), choose your desired display options, and see how the phase portrait changes. You can also choose your desired initial condition by dragging the blue point on the graph. The stationary points of the form $(2k\pi,0)$ are shown as red dots, the stationary points of the form $((2k+1)\pi,0)$ are shown as green dots, and the stationary points of the form $(\pm \operatorname{arccos}(g/(r\omega^2))+2k\pi,0)$ are shown as blue dots. Observe how the level set varies.