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Beating bird flu with maths

Marianne Freiberger

Brief summary

Bird flu poses a risk to wild animals, poultry, and also humans. This article describes how mathematical models can be used to understand and slow the spread and explores some of the challenges involved.

Last month the World Health Organisation (WHO) announced the death of a man in Mexico from bird flu. It's rare that the disease infects humans but when it has in the past, the death rate among known cases has been high — over 50% according to the World Health Organisation.

The news follows nearly three years of bird flu making continual headlines. Since the summer of 2021 the disease has killed tens of thousands of wild birds in the UK alone.  "The range of wild birds affected in the UK has been broader and infections have resulted in more severe disease as well — for example, we have seen a big decline in great skua populations," says Ed Hill, a disease modeller at the University of Warwick and member of the JUNIPER consortium of modellers

Other species and parts of the world have been affected too. At least 24,000 sea lions died in South America in 2023 in regions that hadn't previously been exposed to the particular strain of the virus. Cats, foxes, bears, otters, dolphins, and even mice are just some of the mammals that have also been infected.  As of May 2024 cases of bird flu have been reported in all continents, including Antarctica, with its abundance of sea birds now at risk.

The virus has also managed to infiltrate domestic animal populations.  In 2022 death by disease or culling of over 1.5 million turkeys in the UK led to a shortage of fresh birds for Christmas. Alarmingly, the US is currently seeing a wave of bird flu spreading through cattle herds. To date cattle farms in twelve states are affected and it seems that the virus has developed the ability to spread from cow to cow.  To date three humans who have been in contact with cows have tested positive. Luckily, they only experienced relatively mild symptoms.  

The wave of disease that has swept the world over the last few years is largely down to the H5N1 strain of bird flu, though the death in Mexico was caused by  H5N2 — it was the first known human case of this strain.

The problem facing epidemiologists working to understand and stem the spread is that there's a lot we don't know. Crucially, we don't know exactly how prevalent the disease is in wild birds, how it spreads amongst them and what is the risk of spill-over to other species. "There are severe data gaps," says Mike Tildesley, another JUNIPER member at the University of Warwick.

Despite this lack of information, epidemiologists in the UK are currently ramping up their efforts. Even in the absence of high-quality data, mathematical models can identify major risk factors and give qualitative predictions. Results from different modelling approaches, especially when viewed collectively, can help build the evidence needed to inform decisions about what kind of interventions to take.  They can also help us decide what data needs to be made available and how urgently — this is important information because collecting data, for example by testing wild and domestic animals, is expensive.

Learning from the past

The usefulness of early modelling efforts was demonstrated back in 2006, when a strain of H5N1 also swept across the globe. At that point in time, the UK had been left relatively unscathed. No cases among poultry had been detected and only one in wild birds (a swan). To be prepared for the worst case, however, the Department for Environment, Food and Rural Affairs (Defra) commissioned six modelling groups to see what might happen if the virus did enter the poultry flock.


Wild birds, including kittiwakes, have been heavily affected by bird flu.

"The question was whether the control measures in place at the time [would be] sufficient to control a future outbreak," says Kieran Sharkey, a Professor of Applied Mathematics at the University of Liverpool who was involved with one of the modelling groups. The measures included culling of birds on infected farms, as well as increased biosecurity and surveillance in affected regions.

Using data gathered through surveys, Sharkey's group based their model on a network designed to resemble the British poultry industry. The network linked individual farms if there were contacts between them that could lead to transmission.  This could happen if farms were geographically close, but also if they were visited by the same lorries taking birds to slaughter or delivering feed, or if farms belonged to the same parent company with employees travelling between them.  Farms were also classified to account for other factors that may impact the risk of transmission, for example the species of birds they housed and whether those birds were kept for their eggs or their meat.

The model depended on a number of parameters, reflecting the nature of the disease and operations within the industry. Their values were estimated using information available from the scientific literature and from experts in the field. They included parameters that measured the chance of a particular type of contact — via slaughterhouse or feedmill, for example — leading to transmission. These parameters were estimated using information available from the scientific literature and from experts in the field. 

With the structure of the network and parameter estimates in place, it was then possible to simulate the spread of the disease assuming the infection had entered a single farm. To reflect the element of chance involved in any epidemic, a simulation would pick some parameters at random, for example the time it would take an individual farm to become aware of an outbreak and notify the authorities. Running simulations with different values for the main parameters helped to identify the most important risk factors and also gauge to what extent the results depended on the amount of uncertainty in the values of the parameters.

Using this approach Sharkey and his colleagues found, reassuringly, that the vast majority of outbreaks would likely stay confined to the farms they started in. Farms breeding ducks for their meat, however, stood out. "There was a risk of wider outbreaks propagated and sustained by [these farms]," says Sharkey. This, he suggests, is down to the fact that ducks (like geese) don't tend to get horribly ill and die with H5N1. It could therefore take a while for a farm to notice an infection in their flock, giving the disease the time to spread. These results remained robust, even when the main parameters in the model were varied. (You can find out more about the model and results of this work in this paper.)

Other modelling groups came up with similar results to those found by Sharkey and his team. "[We used] different modelling approaches, but we all ascertained that the risk of widespread infection was pretty low. We had regular meetings in Defra throughout that time where these [issues] were discussed, and a consensus built up [that control measures at the time were relatively effective]."

As far as domestic ducks and geese were concerned, Defra later decreed that these birds should proactively be tested should they find themselves in a region affected by the virus.

Back to the present

Eighteen years on from 2006 we find ourselves facing the same disease, but in very different conditions. "Over the last two years we have had significant outbreaks of H5N1  influenza in wild bird populations, spreading very effectively, and several incidents in the poultry flock," says Sharkey. Investigations suggest that outbreaks in poultry farms so far have come almost exclusively from wild birds. The question now is how this is going to affect the industry and what it means for the risk of spill-overs to humans.


Ducks don't tend to get very ill with bird flu.

While conditions and questions have changed, the original modelling approaches still remain useful. Sharkey and his team are currently waiting for information from the Animal Plant Health Agency (part of Defra) to see to what extent the nature of the industry has changed since 2006 in order to update their network model. "Our initial stab at this would be to regenerate our old model, put in the new network data, and modify the parameters of transmission — because even though it's the same virus something has changed since 2006.  It is now spreading effectively in wild birds in the UK, whereas before it was not." 

A crucial change will be the assumptions on how the virus gets into the poultry flock. Because back in 2006 the prevalence of the disease in wild birds was low,  modellers assumed that only a single farm would initially become infected from the wild. Now that prevalence in wild bird populations is much higher, the assumption will be that the infection can enter multiple farms at the same time. Information on just how high the outside pressure of infection is on poultry farms will come from risk maps produced, for example, by the British Trust for Ornithology. Simulations will then provide insights into how the disease is likely to spread, combining incidence from wild birds with the transmission mechanisms of the industry.

In addition to existing modelling approaches, there is also a new tool which wasn't as available back in 2006: artificial intelligence. Machine learning algorithms can spot patterns in data that humans are hard pressed to find. Feeding the available bird flu data to a suitably designed algorithm may help explain the spread we observe, both in wild birds and within the industry. 

Sharkey and his team are planning to pursue both approaches. At the same time, other researchers in the UK are getting to work on their own models. An example is an approach that is being developed by Chris Davis, a postdoctoral researcher working with Hill and Tildesley, which which focuses on the geographical spread of potential outbreaks.

All modellers are faced with challenges surrounding the availability of data, particularly concerning the prevalence of the disease in wild birds. Information here comes mostly from people finding dead and sick birds in the wild — if an infected bird doesn't get very ill or die, then the infection will be missed. 

While high quality data is essential if we want precise predictions of how the disease will spread, the hope is that a multi-pronged approach involving several modelling groups will build crucial evidence to help policy makers with decisions. "It's difficult to wade through imperfect data, but [modelling] has historically proved pretty useful for informing policy and understanding what's going on," says Sharkey. "It's not just about the output of a model, but the process of modelling. It means that you have focussed discussions about the underlying mechanisms with industry experts and policy makers, and this leads to a coherent understanding."

Cattle and people

Current modelling efforts in the UK focus mostly on the poultry flock. Since there's no movement of cows between the US and the UK, there's no risk of the virus entering UK cattle herds from that direction. If cows in the UK do become infected through other routes, then there are existing models of infectious diseases of cattle that could be adapted to consider the spread of avian influenza in the cattle industry.


Cattle in the US have now also been affected by bird flu.

A main challenge here will again be data. If an outbreak among cattle isn't detected early, for example through the testing of milk, then not only will many cows fall sick, but researchers will also miss the opportunity to track the initial spread of the disease to gain the information they need to adapt their models.

As far as the risk to humans is concerned, Sharkey hopes that models designed to understand the spread among poultry will provide important clues. The idea is to derive risk factors for different sectors of the industry. Seeing how many people in a sector regularly come into close contact with the animals, researchers may then be able to estimate the chance of human infection. At the same time, researchers elsewhere are addressing the human risk from different directions, for example by seeing if the genetic makeup of a strain can help us predict whether it's likely to spill over to humans.

When it comes to negotiating the human-animal boundary, experts suggest, a unified approach is essential. "There's a need for integration between human and animal health bodies," says Tildesley. "This is a clear case where a collaborative approach between, for example, the UK Health Security Agency (UKHSA) and Defra is crucial." Hill points to the One Health approach advocated by WHO, which recognises the link between the health of humans, animals and ecosystems.

The ultimate threat to human health will come if the virus gains the ability to spread between people. But as we've seen in recent years, respiratory diseases can spread quickly. Should bird flu indeed become a human disease, then the sophisticated models that have been developed for seasonal flu and COVID-19 will kick into action and play a crucial role in stemming the spread.

About this article

Ed Hill is a Warwick Zeeman Lecturer in the Mathematics Institute at the University of Warwick and a member of the Zeeman Institute for Systems Biology & Infectious Disease Epidemiology Research (SBIDER). He has research interests addressing interdisciplinary problems in epidemiology that involve the dynamics of behaviour, with applications to public, veterinary and plant health policy.

Kieran Sharkey is a Professor of Applied Mathematics at the University of Liverpool. He specialises in networks and complex systems and works on applications in epidemiology and evolution.

Mike Tildesley is a Professor of Infectious Disease Modelling in the School of Life Sciences and the Mathematica Institute at the University of Warwick and a member of SBIDER. His research focuses upon the development of models of infectious diseases and their utility as predictive tools.

Marianne Freiberger, Editor of Plus, interviewed Hill, Sharkey and Tildesley in May 2024.

This article is part of our collaboration with JUNIPER, the Joint UNIversities Pandemic and Epidemiological Research network. JUNIPER is a collaborative network of researchers from across the UK who work at the interface between mathematical modelling, infectious disease control and public health policy. You can see more content produced with JUNIPER here.

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