Skip to main content
Mathematical physics

Mathematical physics

A few long jumps can make an epidemic

05 Nov 2014

As recent events have demonstrated dramatically, one intercontinental aeroplane flight can turn a regional virus outbreak into a global health event. Now, two physicists have used a computer model to show that the precise frequency of such long-distance jumps has a strong influence on the rate at which infections spread. The researchers’ simulations can also describe the spread of invasive species, genetic mutations within a population, and even rumours.

Throughout most of history, diseases, genetic mutations and species have usually spread relatively slowly, because individuals normally move only short distances in their lifetimes. This kind of spreading creates what University of California, Berkeley, physicist Oskar Hallatschek calls “wave-like” or “pancake-like” growth, whereby a population radiates outwards from a central core in a roughly circular fashion. The Black Death of the 1300s, for instance, spread this way, moving across Europe at between 300–600 km per year.

Global hitchhikers

Today, however, a pathogen or potentially invasive plant can easily hitchhike across a city, a continent or even an ocean, in just a day or two. The individual can then seed a new population, which can itself launch additional long-range jumps to new territories. Seeking to predict mathematically how epidemics and invasions spread, scientists have developed computer simulations in which such long-range dispersal events happen continuously but at very low rates.

But Hallatschek and his colleague, physicist Daniel Fisher of Stanford University, realized that for real organisms, these hitchhiking events do not occur continuously but rather in discrete steps, like a aeroplane flight or boat trip. The researchers wanted to know how these rare, random events affect the overall rate of spread of a disease or mutation. “We were shocked that this was not understood,” Hallatschek says. “This was our motivation.”

Home and away

To discover more, the researchers created a model of the world as a 2D grid. A simulation begins with one “infected” individual at a single point on the grid. During one time step, an infected individual has a certain probability of moving and thereby infecting another individual at another point on the grid. In general, an infected individual is more likely to move to a nearby point than to a faraway one. But in different runs of their model, the physicists varied the exact probability distribution of jumps of different distances.

 Our model was purposely made so that it is as simple as possible but still sort of nontrivial, so that we can obtain some relevant insights
Oskar Hallatschek, University of California, Berkeley

The above video shows the time evolution of one such simulation. The epidemic begins in the centre of the grid and spreads out in space. Long-range jumps that mimic events such as air travel are shown to cause new outbreaks of the disease.

Hallatschek and Fisher found that the spreading rate depended sensitively on the probability of a long-distance jump. If this probability was low enough, the spread was slow and pancake-like. At higher probabilities, however, enough individuals seeded new growth far away from the original population that the overall rate of spread increased dramatically, resembling that of a metastasizing cancer.

Travel networks ignored

Hallatschek notes that his team’s model cannot predict the spread of a real disease like Ebola or swine flu. These diseases spread along human travel networks, which make jumps between certain points – from Monrovia to Dallas, for example – far more likely to occur than an equidistant jump from one isolated rural area to another. The researchers’ model also does not contain specific details such as infection and recovery rates that would allow the researchers to predict the spread of a particular disease.

But Hallatschek notes that more complicated models containing such details are no better at forecasting the course of epidemics. Disease spread is idiosyncratic and strongly influenced by random events and initial conditions, which makes it very hard to predict. Opting for a very simple model allowed Hallatschek and Fisher to avoid these problems. “Our model was purposely made so that it is as simple as possible but still sort of non-trivial, so that we can obtain some relevant insights,” Hallatschek says.

“I think it’s an excellent paper,” says Dirk Brockmann, a physicist at Humboldt University Berlin. He applauds Hallatschek and Fisher for combining theory with numerical results to provide broad insight into how diseases, organisms and mutations spread. “The next step would be to see natural systems where you may observe this,” Brockmann says. “It would be great to see if there’s empirical evidence that this sort of thing is going on.”

The research appears in the Proceedings of the National Academy of Sciences.

Related events

Copyright © 2024 by IOP Publishing Ltd and individual contributors