# Mathematical biology – by way of example

Mathematical biology takes many different forms depending on the practitioner. I take mine with one math and two biologys (the so-called “little m, big B”), but others like it stronger (“big M, little b”). Under my worldview, mechanistic models are a tool to analyze biological data; a tool that infuses our knowledge of the relevant biological processes into the analytical framework. That might sound very pie-in-the-sky, and so I’ve made up an example to illustrate what I mean. This example has been constructed so that it doesn’t require any advanced knowledge: if you know how to add and multiply – that’s all you’ll need to answer these questions.

In the example below, the relevant biological processes are described in the section what we know already. You will need to use logical thinking to relate the what we know already section to the data reported in the DATASHEET so that you can answer the questions.

If you have ever wondered ‘what is Theoretical Biology?’ this example helps to answer that question too. Specifically, the required steps to do modelling, as inspired by this example, would be: 1) to write down the information that goes in the what we know already section (you’d refer to these as the model assumptions); 2) to devise a scheme to relate what we know already with the biological quantities of interest (this is the model derivation step); and 3) to report the results of your analysis (model analysis and interpretation).

As you work through this example, think about the types of questions that you are able to answer and how fulfilling it is that careful thinking has enabled us to draw some valuable conclusions. Understand too, that a criticism of mathematical modelling is that, in reality, everything might not happen quite as perfectly as we describe it to happen in the what we know already section. These sentiments capture the good and the bad of mathematical modelling. Mathematical models enable new and exciting insights, but our excitement is temped because these insights are only possible owing to the assumptions that have been made, and while we do our best to make sure these assumptions are good, we know that these assumptions can never be prefect.

If this sounds like fun, then have a go at the example below. If you want to email me your answers, I can email you back to let you know how you did  (see here for my email address).

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INFLUENZA X

A new and unknown disease, Influenza X, has swept through a small town (popn. 100). Your task is to describe the characteristics of the disease. Health officials want to know:

1. How many days are citizens infected before they recover?
2. What fraction of infected citizens died from the disease?
3. What is the rate of becoming infected?

During the epidemic, citizens can be classified into one of these four groups:

• Susceptible
• Infected
• Recovered, or

As is shown in the diagram:

• Only Susceptible citizens can be Infected.
• Infected citizens either Die or Recover.
• Citizens must have been Infected before they can Recover.
• Only Infected citizens die from the disease.
• Once they have Recovered, citizens cannot be re-infected.
• All Infected citizens take the same number of days to Die or Recover.
• During the epidemic no one enters or leaves the city. No babies are born; no one dies of anything other than Influenza X.

During the epidemic all that was recorded was the number of citizens who were Susceptible, Infected or Recoverd on each day and the number of people who had Died up until that point. This information is summarized in the DATASHEET provided at the end of this post. This information is also presented graphically below and you’ll get a better understanding of the data by considering how the graphs and the DATASHEET are related (Question 4).

QUESTIONS

1. Fill in the missing values on the DATASHEET (below).
2. How many days are citizens infected before they recover?
3. What fraction of infected citizens died from the disease?
4. Label the axes on the graphs.
5. The transmission rate of Influenza X is 0.008 (the units have deliberately been omitted). Consider the graphs above and describe how this rate was estimated?
6. How is the unknown quantity from the DATASHEET calculated?

DATASHEET

Some definitions
• If a patient is infected on Day 1 and recovers on Day 4 that patient is infected for 3 days (i.e., Day 1-3 inclusive).
• Infected (cumulative) on Day T means the total number of citizens who have been infected any time from Day 1 to Day T (inclusive). Citizens who subsequently Died or Recovered are included in this number.