Humans have migrated for millennia. From the first crossing of the Bering Strait to the Spanish conquest, from British and French colonial expansion to the influx of students to London today, migration has always been a central feature of human life.

Human migration is a sensitive topic which is easily politicised. It is often thought about in the context of international or illegal migration, most frequently from developing countries to developed ones, and as something that needs to be stopped. The debate around migration would surely benefit from more data and mathematical modelling, and from fewer sensationalist media reports that can often present a distorted reality.

Modelling any social behaviour is complicated for many reasons. Firstly, it is impossible to observe all of the people involved or consider all of the reasons why they behave, or stop behaving, in the way that they do. Often this means that we can only spot emerging patterns that arise from collective behaviour. Secondly, there will always be outliers. For example: evidence shows that a person who smokes 20 cigarettes a day is 26 times more likely to develop lung cancer than a person who doesn’t smoke, but clearly, there will always be heavy smokers who remain cancer-free. Observing these ‘lucky’ individuals does not mean that evidence against smoking should be dismissed, but when we analyse social patterns we necessarily consider a general case that will not apply to each and every person.

## Mathematical models of migration

There are many models of migration, each of which aims to capture a different aspect of the phenomenon. This depends on the purpose of the model (the question being considered) and the data that is available. Below, we present some of the most important models in current research.

#### Stochastic migration and cumulative inertia

The act of migration can be considered as a random variable that has a certain probability of occurring. The model of cumulative inertia (developed by Myers, McGinnis and Masnick in the 1960s) considers the probability $p$ that a person moves in a given year, taking into account the number of years $t$ that they have lived in their current location. For some fixed values of $\alpha>0$ and $\beta$, we have

$$p = \alpha t ^{\beta}.$$

By using actual data, Myers *et al. *showed that $\beta<0$, so that as time increases the probability of migration decreases. Thus, a person is more likely to migrate from a city if they recently arrived there.

#### Markov chains and migration

Consider the possible locations in which a person might live. The simplest case has two locations, for instance, the city and the countryside. By looking at the probability that a person moves from one location to the other, the migration flux can be modelled as a Markov chain with a transition matrix $\mathbf{A}$. The interesting part of considering the migration flux in this way is that it gives the stationary distribution of the population: a rough approximation of how many people will live at each location after a long period of time.

There is a stochastic (random) component to the act of migration and the choice of destination. The Markov chain model helps understand the flow of migration and the impact that it has on different cities. For example, hundreds of thousands of people moved to New York City last year but, surprisingly, more people moved away. The Big Apple is not getting bigger!

#### The radiation model

This model also considers a stochastic component of migration, as well as taking into account the population of each location. Migration is modelled as a decision problem in which a person decides whether to move based on, for example, the potential income from a job at their target location. The model can also be used to give an estimate of the number of people who will commute along a given route, or the maximum distance that people are likely to commute.

#### The gravity model

Our final model takes a different approach and is based on physical principles similar to Newton’s law of gravity. The model supposes that the attractiveness of moving from one location to another is proportional to the size of the target location, and inversely proportional to the distance between the two.

Let $X_i$ and $X_j$ be the population of cities $i$ and $j$ respectively, and let $d_{ij}$ be the distance between them. The gravity model of migration says that the expected flux $F_{ij}$ of people moving from $i$ to $j$ is given by

$$F_{ij} = \gamma \frac{X_i X_j}{d_{ij}^{\kappa}},$$

where $\gamma>0$ captures the total flux of migration, and $\kappa>0$ is the impact of the distance between $i$ and $j$.

The gravity model is also frequently used to model trade between two locations, where now $X_i$ measures the economic power of a location instead of the population size. For instance, a farmer who produces milk will want to sell their products in a location that is large enough to guarantee demand, but also close enough to keep transportation costs low.

Although mathematical models can’t give a perfect description of the complex pattern that is observed in reality, they can help us understand the reasons why more people might migrate to or from specific locations. They can also help explain the impact of distance on migration statistics and can be used to forecast, for example, the number of people who will migrate following a natural disaster and where they are most likely to go.

## Hands-on display of the gravity model of migration

The Chalkdust team was invited to produce a hands-on exhibit about migration at Greenwich Maths Time, and we decided to focus on migration within Africa.

To display the gravity model, we painted a large map of Africa onto elasticated fabric. We then placed a weight on each of the 23 largest cities on the continent, with the mass of the weight being proportional to the population of that city.

The largest metropolitan areas of the continent (El Cairo, Lagos and Kinshasa) have several million inhabitants and so were given the heaviest weights, while the lightest weight was attached to Algiers, the smallest city that we considered.

When the map was lifted off the ground, it deformed under the weight of the cities and our ‘population’ of seeds was able to move freely under the effects of gravity. The result was visually striking — the vast majority of the seed-people ended up at the largest cities, a few moved towards the smaller ones and even fewer stayed where they were. This gave a great feel for the workings of the gravity model, and how it predicts that migration and trade are attracted to the biggest cities.

## Migration data

Although estimating the migration flux between two countries is a very challenging task, institutions like the UN work to provide the most accurate data possible. The results are often quite surprising as they can differ from the narrative that is established in the media.

Some relevant facts about international migration:

- In 2015, the number of international migrants was nearly 250 million people.
- It is estimated that there are currently 5 million international migrants who originated in the UK, 4 million from Germany, 3 million from Italy, 2 million from France and 1 million from Netherlands.
- Nearly two-thirds of international migrants are people who move inside their own continent, for example, Europeans who move to another European country.

These somewhat surprising statistics highlight the importance of real data in the debate around migration, and how its use can challenge predominant perceptions that are often based on feeling rather than facts. Migration has always been a central tenet of society, and being able to model it from a scientific perspective based on mathematics and data can allow us to understand patterns, predict trends and design better policies on both a national and international scale.