# More than a right or wrong answer: how mathematical modelling aids our understanding of biological problems

Undergraduate maths student Lydia Wren teamed up with mathematical biologist Dr Alex Best on an undergraduate research placement. Together, they ran simulations to apply novel parameters to a classic model of infectious disease.

### Mathematical biology for disease modelling

By applying mathematical and statistical models to real-world biological problems, scientists can investigate the development and behaviour of different biological systems. This is what mathematical biology aims to do.

Much of the work done by mathematical biologists centres around modelling the movement and behaviour of populations to aid conservation efforts and to understand the co-evolution of different species. However, a large part of mathematical biology also focuses on disease modelling, something which the Covid-19 pandemic has brought to the forefront.

One of the most basic mathematical descriptions of how disease spreads throughout a given population is the Susceptible-Infectious-Recovered (SIR) model. According to this model, a population experiencing disease can be divided into three compartments where individuals are either susceptible (or otherwise healthy), infected, or recovered (i.e. they have had the disease and are now no longer infectious). The SIR model is a building block of all infectious disease models.

But like all models, the SIR model has limitations. One limitation shared among many classic disease models is that they are only very simplified simulations of what actually happens in reality. Many of these classic disease models are based on two extremes and simulate scenarios where either all individuals are infected by those far away from them, or by those right next to them. Dr Alex Best, a lecturer of mathematical biology at the University of Sheffield, identified this problem.

“In the instance of disease infection, your chances of getting infected aren't going to depend on how many people in a far away place are infected, it's going to depend on how many people in your own city and personal social networks are infected,” Alex said

### Exploring a topical field

Alex’s own PhD focussed on tackling this issue of extremes within the standard models used to investigate both evolution and the behaviour of ecological systems. To do this, he developed a sliding scale of scenarios which could then be applied to biological models, allowing for the simulation of more complex situations. For example, Alex could simulate how a certain biological state or factor would spread throughout a population if 80% of interactions were between near neighbours, and 20% were between those further afield.  It was this work that inspired him to apply this research to epidemic models too.

“When I got to thinking about the standard epidemic models, I realised that nobody had actually done this work just for the basic human SIR model. I teach undergraduates about the classic SIR model in lectures, so it seemed like a great bit of work for an undergraduate to do,” Alex said.

That undergraduate came in the form of Lydia Wren. As a student on the mathematics integrated masters course, Lydia was looking for an exciting research project to do in the summer before her final year.

Lydia said: “I took the mathematical biology module in my third year, and that really interested me. I was definitely excited that I could use my skills for something that was useful in the real world.”

Lydia had already worked with Alex during the mathematical biology module, and joined him again for a placement she obtained through the Undergraduate Research Internship Programme (UGRI). The UGRI programme gives students in the school of mathematics and statistics the opportunity to complete paid internships working with University mathematicians on research into novel mathematical problems.

The UGRI programme already has an excellent track record of producing novel and impactful student-led research, with a number of students having been selected to present their work at mathematics conferences. In the case of Lydia’s internship, the research she and Alex collaborated on was published as a study in the Bulletin of Mathematical Biology.

Alex said: “I had this idea of something I was interested in looking at, but wouldn't have had the opportunity to follow through with it without help from a placement student. I was able to give Lydia the code I already had, but it was up to her to finish it off and turn it into the precise model we needed.

“She ran all the computer simulations and did a really great job, and by the end of the internship we clearly had something new and interesting to say, so we wrote it up as a paper.”

In running their model, Alex and Lydia found that as interactions move from being completely global through to being completely local, the response in terms of peak infections and total infections does not respond linearly. Instead, their model showed a saturating effect, whereby the size of the peak and total infections didn’t significantly differ between scenarios in which all interactions occur between far away individuals compared to those where a large proportion of interactions are between near neighbours.

Alex said: “We found that when you've got mostly near-neighbour interactions, the infection ends up blocking itself. But for this to happen you have to have high proportions of local interactions, around 80 to 90 percent, before you see big decreases in the peak and total infections.”

From analysing these results, Alex and Lydia were able to think about how the model could be used to guide the application of a disease control strategy. They found that, due to the saturating effect of global interactions, measures to restrict travel within a population would need to be implemented very early on in an epidemic.

Since graduating from her Mathematics MMath course, Lydia has now begun her career as a junior software analyst at healthcare IT company Medisoft. It’s a role in which the analytical and communication skills she gained during her UGRI project prove very useful.

“It really helped me develop a level of independence. It might sound silly, but the most useful thing I learned though doing the placement was just how to ask for help when I was stuck. When I started the internship, it took me a while to figure out I could just ask Alex if I was struggling with something and he was really supportive. I've found that skill to be really useful in my career as well.”

Working on this project as a student also gave Lydia an appreciation of how mathematics can be used to find solutions to real life problems.

Lydia said: “I’m not a biologist, but doing this project in the midst of a pandemic definitely encouraged the interest I had in that area. The idea that the research we were doing could be applied to the restrictions we were under at the time made it come to life as we were translating real-world situations into models.”

Written by Louise Elliott