A new stable state for epidemic dynamics

We recently discovered a previously unknown stable state in the well-known Susceptible-Infected-Susceptible (SIS) model from theoretical epidemiology.

The SIS model is a way to model the spread of diseases in a population. The idea is that individuals start off able to catch the disease (“susceptible”), and are connected by a social network that defines how they will interact with others in the group. When some disease-carrying (“infected”) individuals are introduced into the group, they infect susceptible with whom they interact with some probability, making these people infected. Individuals stay infected for some time, passing on the infection to other susceptible individuals, until they eventually recover and become susceptible to re-infection. (A variant on the SIS model, SIR, has infected people “recovering” and becoming unable to being re-infected.) SIS can be used as a simple mathematical model of diseases like colds and flu.

Some of the questions we typically ask of a model are:

  • For different virulence of the disease, does it stay in the population and become endemic, or does it flare as a pandemic before dying out?
  • How does the way people interact affect the behaviour of the disease? Are some social situations worse than others?

Saray Shai and Simon Dobson recently studied the case in which individuals are separated into two populations with a small number of links between them, as might be found between two countries with occasional travellers. They also modified the basic SIS model so that people could disconnect themselves (avoid meeting) people who are infected, as people might do in the case of disease infection. They then simulated the model over a wide range of networks and parameter values, as well as exploring it mathematically.

The disease either dies out or infects everyone
The disease either dies out or infects everyone
SIS with third stable state
An extra stable branch appears in the dynamics

In most cases, a disease introduced either dies-out or infects everyone (the diagram on the left). However, under certain circumstances something else occurs: the disease becomes resident in the population near the contact points between the two networks, but does not break out into the wider population — shown by the third stable branch on the diagram right. What seems to be happening is that people near the contact points (for example who live in ports) become re-infected by frequent contact with infected individuals, while people living elsewhere have the opportunity to disconnect themselves from infected individuals and so avoid spreading the disease to the general population.

What this shows is both the sensitivity of disease outbreaks to infectiousness and public response, but also suggests some limits to the effectiveness of both quarantine and disease-avoiding behaviour. Certainly this isn’t enough to suggest new policy responses, but it does show the ways in which complex systems and extensive data-driven simulation can be used to help study epidemic dynamics.

More details are in the paper:

Saray Shai and Simon Dobson. Coupled adaptive complex networks. Physical Review E 87, 04812. 2013.