In his 1972 book, Natural History of Infectious Disease, esteemed microbiologist Sir Macfarlane Burnet made a bold prediction about the risk that infectious diseases posed to humanity:
“The most likely forecast about the future of infectious disease is that it will be very dull”1
There was reason to be optimistic: the scientific community was revelling in the successful vaccination campaigns that had all but wiped out polio and smallpox. Coupled with the discovery of new classes of antibiotics, it really did seem that infectious pathogens posed little threat to humanity. This positivity did not last, though. The HIV/AIDS crisis hit just over a decade later and an increasingly globalised society, increased movement of people and worsening climate emergency has left us more susceptible to major outbreaks of infectious diseases than ever.
Indeed, there have been a number of epidemics in the past twenty years that had potential to spread globally: SARS in 2002, the H1N1 ‘swine flu’ epidemic in 2009, the Ebola virus outbreak in West Africa in 2013 and the emergence of further novel coronaviruses, responsible for the ongoing outbreak of MERS and the developing pandemic spread of COVID-19, both respiratory diseases. As society becomes increasingly vulnerable to the epidemic spread of infectious diseases, it has become vital to understand how these diseases emerge and spread.
As the world comes to grips with the ongoing pandemic spread of SARS-CoV-2, the recently emerged coronavirus causing the respiratory syndrome COVID-19, mathematicians and epidemiologists are playing just as vital a role as scientists in the laboratory. Answers to where the virus will spread, how many will become infected and the best action to stem the flow are likely to come from lines of code, rather than laboratory cultures. Mathematical modelling will allow for patterns to be observed, common systems to be understood and accurate predictions to be made.
Researchers need to know some key things to mitigate the spread of infectious diseases:
- The pathogenesis – how does the pathogen cause disease?
- Relevant environmental factors – where does the pathogen thrive?
- The social structure of a contagion – how does the pathogen spread between individuals?
Modellers use a method called Network Theory to study how links between individuals contribute to the spread of a disease. The very features of modern society that make us vulnerable to these diseases also provide epidemiologists with more data than ever before. From countless social media interactions to the tracking of mobile phone location data, it is easier than ever to begin mapping contact between individuals on a large scale. All this data can be mapped into a network of social contacts. Two types of network exist: static networks, which simply map whether individuals have been in contact, and temporal networks which include an estimation of when individuals were in contact.
Once contact networks are established, statisticians can apply a model of disease transmission to that network. One such model is the SIR model, in which all individuals in a population are sorted into three categories: susceptible (S), infected (I) and recovered (R). By applying various parameters, including the probability of infection, the number of infectious individuals a susceptible person contacts and the rate of recovery, the model can predict the spread of a disease in a population.
The animation below demonstrates how the SIR model can be used to predict spread in a population. A value of 1 (dark red) indicates a person is recovered, a value that is larger than 0.5 (orange, yellow and green) indicates a person is infected, and a value that is less than 1 (all colours other than red) indicates a person is susceptible. In this case, an infected person can spread the disease to the eight people around them, with people on the border as an exception.
These models are useless without applying them to a specific group of people. In order to make accurate predictions, models need to be made specific to the population for which public health strategies need to be designed. This approach was a crucial part of the fight to curtail the outbreak of Ebola virus in Western Africa in 2014. Many researchers took on the task of modelling the spread of disease. One team, led by Srinivasan Venkatramanan, pioneered an ultra-modern method of setting up a ‘synthetic population’ on which to apply their mathematical models3. His team combined data detailing population density, demographics, residence and workplace locations to build a fully simulated society. Each individual was given a set of activities, which allowed the modelling team to build a network of social contacts. Disease spread models, such as the SIR model, can then be applied to the synthetic population to produce an estimation of how an infectious agent will spread.
So, since relatively robust models exist, what does the future hold for the current SARS-Cov-2 pandemic? Unfortunately, epidemiologists and statisticians can’t tell us quite yet. A model is only as good as the data fed into it. We simply do not know enough about the pathogenesis of COVID-19 and they way it behaves, including the incubation period and the infectivity of the virus, to produce an accurate model.
John Edmunds OBE of the Centre for the Mathematical Modelling of Infectious Diseases at the London School of Hygiene & Tropical Medicine, explains that even establishing the case fatality ratio – the proportion of infected people that die because of the infection – is not easy:
“If you just divide the total numbers of deaths by the total numbers of cases, you are going to get the wrong answer.”
John Edmunds OBE, Centre for the Mathematical Modelling of Infectious Diseases at the London School of Hygiene & Tropical Medicine
This is certainly the case in the current pandemic. In recent days, the World Health Organisation has quoted fatality rates in the range of 2 – 3.4%. Though the WHO briefings should be taken highly seriously and it is likely that SARS-CoV-2 has a higher mortality rate than influenza, the states rates are highly inflated due to the true number of cases being much higher than reported.
The case fatality rate is a key part of the mathematicians’ arsenal, as it can be used to estimate further characteristics of the disease:
Say a patient dies on a given day, 15 days after first presenting with symptoms. If medical professionals can reliably say that the disease has a 1% case fatality rate, then it means that 100 people were infected on the same day as the person that died. Adding in that, say, cases double every five days, we can then say that there will be 800 true infectious per deceased individual.
As the pandemic progresses and researchers can access more and more data, more models are being produced. The largest model to date does not predict outright the number of infectious that can be expected worldwide, but does provide valuable insight into the dynamics of the disease4. The most important lesson to take away from the model is that early and aggressive intervention is crucial. Patterns of spread suggested that only a handful of infectious individuals could lead to a major outbreak if introduced into a susceptible population:
| Number of infectious individuals introduced | Chance of causing a large outbreak |
| 1 | 17-25% |
| 4 | 50% |
| 10 | 80% |
The analysis also demonstrated the effect of aggressive restrictions on public behaviour. It estimated that the effective reproductive number (RT) – the average number of secondary cases generated by per infectious case – of SARS-CoV-2 was between 1.6 and 2.6 before flight restrictions were implemented in China. After the restrictions on 23 Jan, the RT was 1.05. Given that viruses require an RT greater than 1 to sustain themselves, it is clear that modifying our behaviour will significantly reduce the ability of SARS-CoV-2 rapidly spread further.
The Effective Reproduction Number (RT) is defined similarly, but unlike R0 is not a static value. The RT can change due to natural events or measures taken to curb spread, such as social distancing. Tracking the RT is therefore useful in determining whether public health strategies are working.
Mathematical models of disease are now a crucial tool to predict and curb the spread of infectious agents. Complex computations are now responsible for estimating numbers of infections, informing public health strategies and predicting at-risk populations. We have learnt much from recent outbreaks and our models of SARS-CoV-2 are rapidly growing and improving. It is difficult to have patience during a pandemic; clear and rapid response is required to prevent infections and deaths soaring. The production of models will come and may well be the key to ending the pandemic, but models are only as good as they data we provide them with. Thankfully, with scientific community galvanised and collaborating as never before, the data, and the models they feed, are getting stronger by the day.
