Our lives have been severely restricted in recent months. We are assured that the constraints have been imposed following “the best scientific advice”, but what is the nature of this advice? Among the most important scientific tools used for guidance on the Covid-19 outbreak are mathematical models [TM188; or search for “thatsmaths” at irishtimes.com ].

A model is a set of mathematical equations that can simulate physical, social or biological systems. Mathematical models provide valuable guidance for decision makers and allow us to measure changes, understand observations and predict future behaviour.

**IEMAG**

The Irish Epidemiological Modelling Advisory Group (IEMAG), established in March, provides statistical and mathematical modelling support and advice to the Chief Medical Officer and the National Public Health Emergency Team (NPHET). The members include leading experts from academia and from several government agencies. Prof Philip Nolan, President of Maynooth University, is chairman of the Group.

IEMAG gathers data and monitors the characteristics of COVID-19 in Ireland and internationally. It develops mathematical and statistical models to forecast the evolution of the outbreak, gauges the effect of public health interventions in advance and evaluates probable scenarios for numbers of new cases over time. Responses and interventions are informed by modelling expertise, as well as by public health advice and experience from international guidance and evidence. Data from outbreaks already seen in other countries are used to calibrate and tune the models and guide decisions.

**A Suite of Models**

I spoke in a Zoom session with Professors Brendan Murphy (UCD) and James Gleeson (UL), who outlined the suite of models used by IEMAG to answer vital questions about the pandemic. The central model of IEMAG is a population-based SEIR model. This divides the population into groups that are susceptible (S), exposed (E) to the virus, infectious (I) or recovered (R). The number of people in each group changes over time and is governed by a set of differential equations.

In the IEMAG model, the infectious group is further divided into six sub-groups depending on the path of the infection. Thus, there are nine coupled equations which must be solved together on a computer. The interactions between groups depend on about a dozen factors known as parameters, for example, the incubation period, rate of transmission, fraction of cases that self-quarantine and so on. The results depend sensitively on these parameters. A subcommittee of IEMAG was established to examine the numerous parameters and to determine the most probable values under varying conditions.

**Modelling Uncertainty**

To allow for uncertainty, the model is run for a range of parameter values, giving a probable distribution of predicted values. This is reminiscent of ensemble prediction for weather forecasting. However, people are more fickle than parcels of air: while human behaviour does not much influence the weather, it has major consequences for the evolution of an infectious outbreak. Thus, the precision of pandemic modelling lies somewhere between weather prediction (good) and economic forecasting (poor).

As a result of interventions, there has been a very significant suppression of the spread of Covid-19 in Ireland. The figure, using data up to 16 May, clearly shows the impact of interventions on the reproduction number. There were marked drops on 13-14 March, when schools and universities closed, and 28 March, when the major lock-down started. The reproduction rate of the virus, the R-number, has remained below 1 since then.

The number of deaths per day has been greatly reduced following mitigating steps. However, if measures are relaxed too soon, an R-number above 1 could cause a dangerous peak in infections. If it were allowed to go above 2, the much more extreme consequences would be difficult to cope with.

**Human Response**

The Irish people have generally observed government restrictions to contain the Covid-19 pandemic. It is clear that interventions have greatly reduced the impact of the virus. Without these restrictions, we would be dealing with a catastrophe.

Have we definitely passed the peak? What steps must be taken to avoid another surge? How long will the virus remain a problem? If a vaccine becomes available, what is the best strategy for using it? There are many uncertainties, but IEMAG is using the best available tools to get the most probable answers to such questions.

IEMAG has made several technical reports available online for the benefit of scientists throughout Ireland. These describe the dynamic and statistical models, explain how data from Ireland and elsewhere are used to determine the model parameters and outline the range of methodologies used.

The work of IEMAG shows the benefits of combining expertise from diverse fields in which different approaches complement and reinforce each other. The modelling work is a wonderful example of blending the techniques of applied mathematics and statistics, worthy of recognition by the Irish public. It is really refreshing that the Irish authorities are listening with respect to the scientific experts. Let us hope that this continues.

**Sources**

Irish Epidemiological Modelling Advisory Group to NPHET: Technical Notes.