# Matthew Roberts

Research Fellow

## Organisation

University of Bath

## Research summary

Mathematically, a tree is a network in which there is only one route between any two points. Think of a family tree: if I know my parents’ genes, then I can get no more information about myself by looking at my grandparents or cousins, because every path from them to me goes via my parents. This property of tree structures is very useful, and has applications as diverse as epidemics, solid state physics and telecommunications networks.

Modelling epidemics is a difficult task because unlikely events can drastically change the outcome. For example, the probability that a deadly disease reaches the UK might be small, but if it happens then millions of people could die. One way to simplify the problem is to look at the average number of people in the UK that we expect to be infected.

Averages, however, can be misleading. If I roll a die, the average result is 3.5. But I certainly won’t roll a 3.5, and I have as much chance of a 6 as a 3 or a 4. Epidemics are more extreme: if on average 60 people in the UK are infected, it could be that there is a one in a million chance of the whole population being wiped out, and otherwise no-one contracts the disease. The average behaviour is not a good guide.

This is of course a well-known phenomenon; averages remain a valuable tool, and the fact that an infection network looks like a tree helps even when averages do not. What is less well understood is how spatial variations – for epidemics, how disease spreads more rapidly in some areas than others – can amplify the effect. If the epidemic mentioned above takes root in the UK, it is likely to reach many other countries from there; so if we take the average of 60 infected people in the UK as typical, then a global pandemic appears likely, when really the probability might only be one in a million. My research is about finding ways to solve mathematical problems like this, where we need to keep track of all the extreme events that seem unlikely but could cause chaos.

## Grants awarded

#### Spatial dependence in branching structures

Scheme: University Research Fellowship

Dates: Oct 2016 - Sep 2021

Value: £437,741.50