"You can use math to do that?"

So, after a short hiatus (due to sketchy internet connections), I thought I'd continue along with some posts relating to my current position as a math grad student. Now that you've had a small taste of mathematical biology, you might be asking yourself "That sounds really interesting and stuff, but how do you actually use math for that?". This question comes up a surprising amount when it comes up that I'm an applied math person, and have done or seen projects on epidemic spreads or why individuals cooperate with one another. Most people just don't see the connection between mathematical modelling and "real life" problems. So, as a primer on this, let's talk about mathematical modelling.

The main idea behind mathematical modelling is to try to describe a problem or situation using mathematics and then gain insight from the maths that you've come up with. Think of it as translating the problem into another language, a language that you can work with to find a solution. Once you have this solution, you can just translate it back and see what it says. The goal is to make sure that the translation in either direction is good, and that nothing gets lost or distorted. Usually, this is the toughest part: coming up with a good model to begin with (and when I say model, I mean a set of expressions describing the problem, not Tyra Banks or Christine Brinkley). Once you get a solid model, usually you can analyze it shed some light onto the problem at hand. Perhaps this is a good time to lead by example.

Suppose I'm a biologist, and I am studying bacteria in laboratory settings. In between the lavish late nights I enjoy in such a profession, I want to understand what influences bacteria population growth or extinction. Here, there's at least two ways I can look how bacteria grows

1. Collect data from the bacteria in petri dishes, and make some plots with it.
2. Create a mathematical model based on what I know about bacteria.

Of course, I could do both, which would really better my understanding of bacteria population growth. But, since I'm a biologist that loves mathematical models, I'll try the second approach.

First, I'll need some variables to work with. Suppose that P(t) is the population of the bacteria at time t. I want to find an expression for what P(t) for any general time t, so the goal is to find a mathematical model that represents this.

Now, I'll need to make some assumptions.

• Assume the population changes at regular, discrete time intervals (for example, the population only changes every hour, and not all the time). This may seem a bit strong ("bacteria reproduce all the time, like a million times a minute!" you say) but this is part of translating the bacteria problem into math. I have to make some kind of simplification or else I won't have anything to work with.
  
• Also, I assume that the only way the population can change is by bacteria reproducing or dying. This means that I can't just "add in" bacteria once I've started, nor can they just "appear" out of nowhere. They have to be the result of reproduction.
  
• Finally, I'll assume that bacteria reproduce and die directly proportional to the population size. In layman's terms, this means that the more bacteria there are, the more will reproduce or die at any time step.

So, with these assumptions in place, I can now develop a model. The simplest model has the form

P(t+1) = P(t) + births - deaths

which means that my population at the next time step is just the population at the previous time step plus the increase from births and less the decrease from deaths. In population dynamics, many, many models start out with "births - deaths".

If assume that the birth and death rates are just a constant proportion of the population at time t, then I can turn this into

P(t+1) = P(t) + bP(t) - dP(t)

where b is the constant birth rate, and d is a constant death rate, and both are fixed constants. What these mean is that at each time step a proportion b of the population reproduces (remember, these are bacteria, so let's just say that they divide to reproduce), and another proportion d die.

So, there is it, I've come up with a mathematical model of bacteria growing. I've now translated the problem of how bacteria grow into mathematics. But that's a lot of work for one day for mathematical biologist, so in my next post, I'll look at working with this model to get some results out of it, and translating those back into real answers.