*This content is part of our Contagious maths project, and builds on the mathematical models we built in part 1 , part 2 and part 3. You can find all the content for Contagious maths here.*

### Connections

At the absolute heart of modelling epidemics is how we are connected to each other. "Infection transmits between two people," says Julia Gog, a disease modeller at the University of Cambridge. "So it's about who spends time with who." In the previous part, we explored the ideas about how some people have more contacts, some have fewer, and that in turn shapes the number of people they can go on to infect. "Now we can refine this further and think in more detail about how people are connected in space," says Julia.

For example, if we're thinking about pandemics we're thinking on an international scale and how people move between countries. At the next scale down we can consider a single country and the geography of the towns and cities and how they are connected to each other. "Zoom in a bit further and we start to get to the bits of our lives we recognise," says Julia. "How we are arranged in households, maybe how we go to work or school, and how these different [households] are connected up." And finally, think about where you are right now. If you're in a classroom, a workplace or a cafe, you'll probably be sitting at chairs and tables in some sort of set spatial arrangement.

All of these patterns can be represented in the models we use to understand how epidemics spread on different spatial arrangements. And there's only one way to find out what is going to happen... give it a go!

### Try it out for yourself!

You can use the lovely interactivity from part 3 again to explore how a disease might spread in different spatial arrangements. Just a reminder before you get started, it's helpful to know the following information:

- The circles represent the people in the population - they start out with no immunity to the disease.
- White circles represent people who are extremely cautious – they will expose no other people to the disease while infected.
- Yellow circles represent people who are mildly cautious – they will expose one person to the disease while infected.
- Blue circles represent people who are more carefree and trying to live their life as normal – they will expose two people to the disease while infected.
- A spikey red circle represents someone who is infected - they might infect people nearby in the next generation.
- A purple square represents someone who has recovered from the disease and are immune to it.
- For higher mean values of
*R*there will be some "super-spreaders" in the population, who are represented by green circles. In this model we have decided that super-spreaders infect 5 people.

Now you can change the layout of the interactivity by clicking on the purple cog and choosing an option. How does a disease spread differently when people are evenly spaced out in a grid? What about when they are arranged in satellite cities that are linked together, or when they are in several suburbs connected to a main city?

As well as changing the layout, as before you can also use the slider to change the value of *R* for your population. There's also some other parts of the model you could explore too. If you click on a person you can choose to infect or immunise them, and if they are already immune you can make them vulnerable again. This allows you to explore what happens when people are vaccinated, and when the immunity they have gained wanes.

### Moving on...

You've now had a chance to see how different spatial arrangements impact the spread of the disease. But there's one very important thing we haven't yet taken account of. Here's Julia again.

Our connections to other people don't stay constant: "We're not like trees just next to the same people all the time!" says Julia. We move and that movement means we come into contact with different people. Locally we go to school, or work and then back to home. On a bigger scale people travel between countries. "Just think about your day and how you move around, how you come into contact with different groups of people through the day. This [would] allow infection to transmit between these different groups."

An example of this would be in a major city with lots of villages and towns on the outskirts. If an epidemic starts in one of those towns, sooner or later it will probably get into the city because of someone commuting in to visit or for work. "People move and that allows the epidemic to move," says Julia.

"We can incorporate all of this into our models, but as soon as you've got people moving around the dynamics are getting very funky indeed," says Julia. Really interesting patterns can arise and it's not obvious what is going to happen. As researchers, the approach of Julia and her colleagues in this situation is to just run their models a lot of times. "Maybe we will change some things, for example who gets infected first, the spatial arrangement or how people are moving. But by running it a lot of times you get an intuition, a sense of how epidemics spread in space when people are moving."

### Get moving!

Our friend, Oscar Gillespie, from NRICH, has created another interactivity for you to explore how people's movements might affect the epidemic. It is very similar to the interactivity above, with a key difference: the people, the circles, will now roam around the space. A red spiky star represents an infected person - they will move around while they are 'incubating' the disease, during the time between generations. By clicking on the purple cog you can select different situations: people mingling at a party, commuting, or visiting each other at their houses.

### Using maths to work on real life problems

We hope you've enjoyed having a chance to explore epidemics and the mathematical models used to understand them. "There's a lot of similarities between what you've been doing and how we work here in the university on research," says Julia. "Something you've probably noticed is we will never get to the final, perfect model."

Julia and her colleagues start off with a model that is relatively simple and concise. Then, when they spot something different from how a disease spreads in reality, or they need some different element because of a particular problem they are working on, they add further refinements to their model. "Research is like that, we keep going. There's no - we have got the right answer and we're done. It's a process that just keeps evolving."

Hopefully this series of videos has also given you a sense that using maths to tackle real world problems, such as helping to control epidemics, is very exciting. "For us in this area of research it is exciting to be able to contribute to the world in this way," says Julia.

We also hope this series has helped you give you a sense of what life is like as a mathematician working in this area. "We're not just sitting there with our calculator number crunching with these models," says Julia. "We've got to think about a lot of different aspects of how diseases work, and how populations work. You end up having to think and work with others in different disciplines. So there's a very collaborative element to this kind of applied maths."

Julia may be based in the maths department at the University of Cambridge, but she also needs to work with people in the medical school, with experts in how people move and mix, and in the vet school if she's working on animal diseases. And of course Julia also works with pure mathematicians when she needs theoretical help with her research.

"And of course, with something like infectious disease modelling I work with colleagues across a whole range of universities in the UK," says Julia. "And [when] we're facing one problem across the globe I work with colleagues in very many countries. So it's a much bigger research problem than just one mathematician sitting at their desk with a pen and a pencil."

Finally, we thought you might like to find out about the questions Julia, and some of her colleagues, are currently exploring in their own work. Find out in the final part of *Contagious Maths*.

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These Contagious Maths resources were developed and written by Julia Gog and the MMP team, including both NRICH and Plus, and funded by the Royal Society’s Rosalind Franklin Award 2020. We have tailored these resources for ages 11-14 on NRICH, and for older students and wider audiences on Plus.
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