# Maths, viruses and zombies

The TV show “The Walking Dead” did a lot of favours to zombies. The idea of zombies and zombie apocalypses wasn’t really something that was discussed extensively before this show came about. But there was one movie that pre-dates this show that really saw the start of the zombie craze, and that was “I am Legend” with actor Will Smith. In this movie, Will Smith depicts a man who is the sole human survivor in New York, where everyone has become nocturnal zombies.

Now you might be asking, what does this have to do with mathematics? and viruses? Well in the movie “I am Legend”, a virus known as an oncolytic virus, has turned the human population into zombies. Why do we care about this virus? Well it’s because oncolytic viruses are actually a really exciting type of virus. These viruses are a novel cancer treatment that are in numerous clinical trails, with one virus even approved for treatment of melanoma.

And here’s where the maths comes in…

While exciting, this treatment has been struggling for many years to be successful in the majority of patients so mathematicians have developed a basic way to model this therapy and have been working to understand it’s pitfalls.

Oncolytic viruses can be modelled simplying using a system of three ordinary differential equations (ODEs) that capture the four main steps of the virus life cycle:

Step 1: a virus infects a cell

Step 2: the virus makes new copies of itself inside the cell

Step 3: the virus kills the cell and escapes and releases all the new copies it made

Step 4: Repeat Steps 1 to 4

Now consider $V$ is the number of viruses, $S$ is the number of susceptible (uninfected) cancer cells, and $I$ is the number of infected cells then we can capture the action of these oncolytic viruses mathematically with

$\frac{dV}{dt} = pI-\eta V,$

$\frac{dS}{dt} = -\beta S V,$

$\frac{dI}{dt} = \beta S V-d I$

See the schematic summarising this below:

This system here models a population of virus particles $V(t)$ that are infecting a pool of uninfected cells $S(t)$ and making infected cells $I(t)$. Step 1 through to Step 4 are all included in the model. The infection of viruses into cells is given by $\beta SV$ and the death of an infected cell is given by $dI$. The produce of new viruses from the infected cell is $pI$ and the general clearance/decay of viruses is given by $\eta V$.

This extremely simple model has been used by mathematicians throughout the world to predict ways of improving the efficacy of this treatment.

While exciting we still have a lot more than can be done in this space mathematically.

Maybe you can think of ways to improve this model and help improve the effectiveness of this treatment – whilst also making sure we don’t all end up as zombies?

Have a go at simulating the ODE system yourself.