**Published on** June 14th, 2013 |
*by Jo*

# Predicting the period of epidemic outbreaks – Part 1

* Dr Andrew Black* is an ARC Research Associate in the School of Mathematical Sciences at the University of Adelaide.

*Dr Joshua Ross*is a Senior Lecturer in Applied Mathematics, within the same School. In this blog, Andrew and Josh will tell us about applications of mathematics and statistics to infectious disease epidemiology.

One of the interesting properties of many diseases is that cases tend to follow a cyclic pattern through the years. We live in a relatively lucky time because most diseases have vaccines and children are vaccinated early. Up until the middle of the 20^{th} century this was not the case. Childhood diseases such as measles and whooping cough used to show very pronounced patterns in the number of cases from year to year. For measles there would be a large spike in cases one year followed by a year with only a few cases. Whooping cough cases display a different pattern and these patterns change dramatically after the onset of mass vaccination. How do we understand this cyclic behaviour; what causes it and how does it change with time?

This is a problem where mathematical modelling has been very useful in understanding this effect more fully and predicting the periodicity of these cycles. The effect is best understood with an analogy. This sort of periodic behaviour in the number of cases is intrinsic to the system in the same way that a pendulum or swing oscillates. We can write down equations for the numbers of infected at a given time in a very similar way to how we would describe the motion of a pendulum. Basic physics tells us how the weight and the length of the pendulum determine the period of the oscillations. For a disease, the infectivity and the length of the infectious period as well as the mean lifetime of an individual affect the period of oscillations in the number of cases.

In fact we can take this analogy even further. The oscillations in a swinging pendulum will eventually settle down; this is due to air resistance, which creates a damping force. We see a very similar affect in the equations for the prevalence of a disease; solving the equations shows that the oscillations in the number of cases should eventually die down so we would see a constant number of cases throughout the year. One of the questions mathematical epidemiologists have had to answer is why do we see sustained oscillations in the number of cases, when the equations tell us that the oscillations should settle down?

As with a pendulum the oscillations can be maintained by forcing. We can identify two main types of forcing: the first are external forces; the second are harder to understand and are related to randomness. We will discuss the first and leave the second for another blog.

Seasonal changes are one of the best examples of a form of external forcing on an epidemic system. Influenza is probably the best known example at the moment. Cases of flu tend to peak in the winter months and die down in the summer. One factor in this is that it is colder during the winter so people tend to stay indoors more. Thus there is more contact between people and more chance for flu to spread. These yearly patterns are known as seasonal. For measles and whooping cough the external seasonal forcing is due to the aggregation of children in schools during term time.

How the natural dynamics of the disease respond to this forcing is important. As with a pendulum, just a small amount of forcing can be enough to create very pronounced oscillations in the numbers of infectious cases. In the next blog we will discuss how randomness can also be seen as a form of forcing and lead to large oscillations.

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