The SIR model has proved incredibly useful in predicting the evolution of certain categories of infectious outbreaks and serves as our introduction to modeling systems with ODEs.

We will have our first lab next class. It will apply the SIR model to a zombie epidemic as presented in an article published by a University of Ottawa researcher and his students. Their compartment model divides the population into three classes: susceptible ($S$), zombie ($Z$), removed ($R$).

Susceptibles may die from natural causes (at a rate of $\delta$) or from a zombie attack. The susceptible population can only increase through new births (at a rate of $\Pi$). They may become infected through contact with zombies. Zombies themselves can only die by being defeated by a human (for some examples of how this may be done see Zombieland). The zombie population can increase through infection of susceptibles and by resurrection of the dead.

The coupled system of ODEs that describe the model is shown below where $\alpha$, $\beta$, and $\zeta$ have the same meaning as in the lecture notes.

\[
\begin{align}
\frac{dS}{dt}=&\Pi S-\delta S - \beta SZ\\
\frac{dZ}{dt}=&\beta SZ - \alpha SZ + \zeta R\\
\frac{dR}{dt}=&\delta S + \alpha SZ - \zeta R
\end{align}
\]

Task 1

To start, the timescale of a zombie epidemic appears to be short and it is reasonable to ignore birth and natural death rates altogether.

On a piece of paper, rewrite the simplified equations and redraw the SIR model.

Add an incubation period

In the aforementioned model, individuals go straight from susceptible to zombies without an incubation period. We should include some exposure period during which an infected individual is not yet a zombie. To do so, we will need a new class of individuals: the latent ($L$).

Task 2

Redraw the SIR model to account for latency and revise the set of equations.

How to kill a zombie

There is one major problem with the model: it assumes that everyone in the removed class can be resurrected as a zombie. Obviously, humans have no chance of eradicating zombies if they can be continually resurrected. Destroyed zombies should not be able to come back to "life" (at least according to current research).

Task 3

Modify the SIR model to be able to account for "destroyed" zombies and write the corresponding set of differential equations.