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Published on August 23rd, 2013 | by Jo

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# Predicting the period of epidemic outbreaks – Part 2: The impact of randomness

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.

In our last blog we talked about why many endemic diseases display a cyclic behaviour, with numbers of cases going up and down over a time scale of years. We identified two main causes of this periodicity: external forces and randomness. In this blog we’ll explain the second of these—we’ll discuss why epidemics are a random process and how this intrinsic randomness can in turn lead to large oscillations in the numbers of infected from year to year. This turns out to be an intriguing phenomenon that is observed in many different biological systems from the chemical reactions inside cells all the way up to ecosystems.

First we need to understand the way in which epidemics are random and what this means. In building a model of an epidemic process we have to make a number of assumptions and simplifications. Reality is very complicated—so complicated that we could never hope to know all the factors necessary to know with complete certainty what will unfold in the future. By this we mean that we could not hope to predict who will get sick, and at what time, exactly.

Consider how we might calculate how long a person is infectious for—the period of infectiousness. To determine this we would have to know about the complex interplay between the body’s immune system and the disease. This is very challenging, requiring sophisticated models in its own right, but we can work out the average duration of infectiousness and even the distribution of the infectious period. Most people will have close to the average duration, but others will have significantly longer or shorter periods. Any individual infectious period will appear random, but we can say something about the probability of seeing such an infectious period. In reality it isn’t random—it just appears so because we don’t know enough about the underlying processes. The same logic applies to the infection process; we can’t say exactly who will be next to be infected with certainty, but we can say who is more likely to be the next case. For example, it’s much more likely to be a family member than a stranger because there will be more contacts.

These random events introduce a realistic aspect of noise into our models. This noise turns out to be vitally important because it can excite the natural dynamics of the system. We can understand this by returning to our analogy with a pendulum. All oscillatory systems have a resonant frequency. If we apply forcing to the pendulum at the resonant frequency, then the oscillations can get very large. The same sort of principle applies to our epidemic models, except it is the noise that is intrinsic to the model that provides the forcing. This means the oscillations we observe from this type of process look a bit different. A forced pendulum oscillates at just one frequency whereas the number of infected within these epidemic models oscillates with a wide range of frequencies, but some are more pronounced than others.

In these two blogs we have described the main two types of forcing that can affect epidemiological systems. The actual patterns we observe for a given disease will depend on the relative strengths of these two factors, which in turn depend on properties of the diseases themselves and the demographics of the population. Historically, cases of measles have been shown to be very sensitive to external forcing due to the opening and closing of schools. In contrast, the patterns of whooping cough were much more sensitive to noise as talked about above. Knowing these differences is important when we come to think about control policies. For example, it is possible that implementing a vaccination scheme might lower the overall level of infection, but also change the period of the oscillations, ultimately making it harder to completely eradicate a disease.

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