Don't panic!

Mathematical models could predict how diseases and viruses will spread - and hopefully prevent them from doing so, explains E&T.

If anyone still needed it after the SARS and avian flu outbreaks in recent years, the current swine influenza pandemic has provided a graphic illustration of how quickly and widely a virus can spread around the world these days. From the first reports of cases in Mexico in mid-March, it has taken less than two months to spread throughout North America, much of Europe and as far south as New Zealand.

Although the swine flu scare seems for now to have been contained, such events always bring to mind the Spanish flu pandemic of 1918-19, which is estimated to have killed upwards of 40 million people worldwide. Inevitably, such comparisons create widespread panic and cause untold damage to the global economy, at a time when it is already seriously weakened by the worldwide recession. Air travel, tourism, and meat imports and exports for example are all badly affected across the globe, as people cancel or delay their leisure plans and countries tread a cautious path through their food sourcing.

All this is despite the fact that the travel restrictions initially recommended by the World Health Organization have now been relaxed, and that the virus is not spread through meat. It seems that, at this point, fear of the virus is causing more harm than the virus itself.

So, clearly, if we could predict the spread of a disease or virus then we could take action appropriate to the level of its threat to our health - and the health of the overall economy. A mathematical model would be the obvious way forward here, but what should be equally clear is that building such a model is no simple matter. Different diseases spread in different ways, and viruses mutate, so a model of how one pandemic progressed cannot be applied to subsequent pandemics.

The difficulty is that diseases and viruses don't spread in the same way as the kind of physical systems that engineers are familiar with, so there's no physical law that can be applied to them. So there are many different methods of modelling.

Deterministic and stochastic models

That said though, there are actually just two fundamental types of mathematical model in epidemiology - deterministic and stochastic.

Deterministic models assume all contact rates between each infectious and susceptible person are equal, and that a new infection will result from each of these contacts at a given rate. They're useful for estimating some important characteristics of epidemics, such as the conditions under which they occur and the rate at which they grow.

Stochastic models, by contrast, build in random variables and use probabilistic factors to determine the rate of infection - factors that deterministic models ignore. They depend on the chance variations in the risk of exposure and other illness dynamics.

The choice of model depends largely on the size of the population and the desired degree of precision. For a large population, or when little precision is needed, there's practically no difference between the two types, so a deterministic model is usually chosen because it's comparatively straightforward - although even then, some researchers say they should be used with caution. But in small populations, even down to the household level, a stochastic model is needed to give enough precision to outweigh the model's extra complexity.

Yet no model will work without data; and the more data the better because it means a simpler model can be used. This being the real world, however, models frequently have to work with less than perfect data, in terms of robustness and completeness, so it's common for epidemiologists to use a variety of models in a combined effort to achieve a consensus.

One key quantity common to all types of model is the basic reproduction number of a disease - R0. This is the number of people a person with a disease will infect. If it falls to a level of '< 1' then the infection dies out; if it's '> 1', there is an epidemic; if it is '= 1' then the disease becomes endemic, remaining in the population at a consistent rate.

But there's a problem with this, as Jaideep Ray, a researcher at Sandia National Laboratories in California, explains. "Characterising diseases requires observations of real outbreaks and then building computer models around them, so a disease's R0 is drawn from historical outbreaks of it. But even for a known disease it varies wildly from outbreak to outbreak - for smallpox, for example, it's been observed to vary between 3 and 17."

So when making predictions, people generally use a range of R0 values to bracket what the disease will behave like. "Once an outbreak starts, people collect data, come up with a provisional R0 - after, say, about a week - and use that for predictions too," he says. "They then continuously refine this provisional R0 and, sooner or later, this becomes a better way of doing things than using a range of values from historical outbreaks."

Ray looks at the problem in inverse, inferring the R0 - or other parameters that can be worked into an R0 - from data or observations of an ongoing outbreak. At the moment his inference system supports smallpox but he says it can easily be extended to other diseases.

Social networks

"There's a caveat though," he says. "If you're dealing with communicable diseases, social networks become a key factor, and it's hard to make inferences about these networks.

"Communicable diseases spread faster through people in closer proximity. For example, close family members of an infected small child would have a higher probability of contracting the disease than someone who lives elsewhere.

"But a person might come in contact with many more people, say at work. These 'social connections' are not observed, but nevertheless play a huge part in spreading the disease. So if a disease's parameters are to be inferred, these unobserved social connections have to be either postulated or inferred. That's where the trouble starts."

So naturally there is a lot of research into modelling human movement and social contact patterns. For example, one recent study, carried out between universities in the US and South Korea, has shown that people tend to carry out day-to-day activities such as going to the bank or shops in clusters, making many short 'jumps' within a cluster and a few long jumps between clusters. And in March the Royal Society in the UK published a paper that shows this clustering can significantly reduce the early growth of an epidemic.

Furthermore, a study conducted across continental Europe found that people have an average of 13.4 contacts per day, and that some contact patterns are common to each country, implying that the spread of a disease would be similar across Europe and in countries with similar social structures. And with the same aim, an online survey (www.contactssurvey.org) has just been launched in the UK by Warwick and Liverpool universities.

There are still more factors that, depending on the disease, may need to be built into a model, such as vertical and vector transmissions, age-structured populations and acquired immunity through vaccinations. While these last two are self-explanatory, vertical transmission is important in the case of diseases such as Aids and Hepatitis B, where children of infected parents can be born with the infection, while vertical transmission applies to diseases transmitted indirectly from human to human, as in malaria spread by mosquitoes, so a malaria model needs to include both species.

Again though we come back to the issue of collecting data, without which no model is viable, and modern technology is playing a growing role here. For example, scientists are using Nasa satellites to help predict outbreaks of diseases such as Ebola and Rift Valley Fever by monitoring changes in factors such as rainfall and vegetation in known high-risk areas. The technology not only helps monitor outbreaks, it also provides information from around the world about possible plague carriers such as insects or rodents. And, because plague is also considered a bioterrorism threat, the scientists can determine the cause of an outbreak. 

Nasa satellites are also targeting malaria, which kills more than a million people every year and infects a further 500 million. It's not clear how much warning the Nasa system gives but Dr Andy Morse, scientist at Liverpool University, says he has developed a climate-based model to predict malaria epidemics up to five months in advance - the earliest such predictions have ever been made.

His model is based on factors including rainfall and health surveillance data, and uses upcoming season forecasts for rainfall to predict unusual changes in the seasonal pattern of disease in Botswana, which is where his team based its study as its climate makes it susceptible to malaria epidemics.

"The risk of an epidemic in tropical countries such as Botswana increases dramatically shortly after a season of good rainfall, when the heat and humidity allow mosquito populations to thrive," he says. "By using a number of climate models, we were able to compose weather predictions for such countries, which could then be used to calculate the severity of an epidemic months before its occurrence."

Satellites are also playing a key role in helping to predict epidemics of water-borne diseases such as cholera, and meningitis outbreaks across the sub-Saharan belt, which often follow in the wake of huge dust storms.

Cholera is an unpredictable and severe problem for developing countries, but the bacterium that causes it has a known association with a crustacean that lives on a type of plankton. Cholera outbreaks have been linked with environmental factors such as sea surface temperature, ocean height and biomass, so a team at the University of Maryland in the US has used remote satellite imaging to track changes in these factors and applied the data to predicting outbreaks.

Meningitis causes seizures and deafness in those it doesn't kill. Outbreaks in the sub-Saharan belt take place after a period without rain, low humidity and a lot of dust in the air, and, although the exact correlation is not yet known, researchers in the European Space Agency's Epidimio project are using satellite data to follow week by week the development of dust storms and the appearance of conditions favourable for an epidemic to start.

Google flu trends and wheresgeorge

Unsurprisingly the Internet is weighing in with prediction tools as well. Google Flu Trends, for example, was launched in late 2008 after the company's engineers discovered a close relationship between the number of people Googling for flu-related topics and those with flu symptoms. The company says Flu Trends makes it possible to estimate possible flu activity in near real-time, and that the estimates, which have been validated against historical data from the US Centres for Disease Control and Prevention, can be up to two weeks ahead of federal government data.

At the moment, however, the system covers only the US, although in light of the swine flu pandemic Google is also running an experimental system for Mexico. In time, the company says it hopes to extend the service to other countries and may include other diseases.

Then there's WheresGeorge.com, an Internet game where users enter the serial numbers from their dollar bills to track their travels around the US and Europe. In an echo of the human movement models discussed earlier, Professor Dirk Brockmann of Northwestern University in the US has used the game to calculate the odds that a given dollar bill will stay within a 10km radius, to create an epidemiological model to predict the course of the swine flu pandemic over the next few weeks, as well as future outbreaks.

Such models are too new to be backed up by historical data, but Prof Brockmann's prediction that there will be only 1,700 or so cases of swine flu in the US by late May matches almost exactly that from another model using a system at Indiana University, which is based on similar assumptions such as a low R0 but which uses a different algorithm.

Like all models these are continually being refined, and there's a constant push for new ones, so you can be sure that as long as there are diseases out there, we'll have mathematical models to predict how they will spread - and hopefully prevent them from spreading.

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