This article was originally published on October 18, 2021. It was last updated on June 28, 2022.

Written by Prof. Hans Bachor & Ruqiyah Patel, Australian Academy of Science

'Modelling' is in the news all the time during the pandemic. Politicians now cite ‘modelling’ as the basis for their decisions regarding health and economics.

An established form of modelling takes place in fashion shows, on the catwalk. It plays an important role in marketing–selling new clothes or cars. That is not what we need here!

*Mathematical modelling* is the ability to analyse and describe a complex system, and to use this knowledge for forecasting and control. Mathematical modelling is at the core of many aspects of our modern lives: part of the way our technology works, part of the way we communicate, and also part of our social and economical wellbeing. The ability to predict what will happen next in our societies, and to plan accordingly, is fundamentally important for achieving a stable future.

**Modelling a pandemic**

The most prominent example of mathematical modelling right now is the global pandemic, with all its human suffering. For almost two years we have seen this reported daily through numbers, data, statistics and forecasts. Mathematical models have been created by leading health experts using the latest science and data. Predictions based on these models are the basis for introducing restrictions, lockdowns and all of the other decisions that have affected our lives. The modelling has been a key part of our efforts to control the pandemic.

Let us have a look at the data. A pandemic is global, and each country has a different history. Here is an overview from May 2022 showing the number of infections across each continent, and the high number of COVID-19 casualties.

*Left: The waves of infections and the effect on different continents from March 2020 until May 2022. The pandemic started with one common wave that spread rapidly across the globe. Next waves hit Asia, Europe, U.S. & Canada and Africa at different times. The very large spike early in 2022 shows the arrival of the Omicron variant. Data from John Hopkins, chart by Spiegel May 2022.*

*Top right: Map of total COVID-19 deaths/100,000 people at September 2021. Examples are: U.S.A. = U.K. = 200, GER =112, AUS = 4.4 NZ = 0.55. Data from John Hopkins, chart by Dan Polansky September 2021.*

*Bottom right: The rate of excess deaths for a select group of countries. In some nations the COVID-19 waves have temporarily increased death to 150% (by an extra 50%) from previous years. The blue regions show fewer deaths than normal, largely due to suppresion of flu. You can see that Australia is pretty unaffected until early in 2022. Data from The Economist, chart by Spiegel May 2022. *

To understand the pandemic better let us have a closer look at just three countries: Australia (AUS = green line), the United Kingdom (UK = blue line) and Germany (GER = red line) in comparison to the whole world (World = orange line). Here is a plot of the deaths per 100,000 people that are attributed to COVID-19 since March 2020 in each region. The y-axis of this plot uses a logarithmic scale–discussed further below and here.

*Source: Tagesspiegel, 4 December 2021*

We see the onset of the pandemic in March 2020. The dramatic increase in infections and deaths started within a few weeks in many countries. Travellers carried the virus from country to country. Nobody was prepared, the health systems were overwhelmed and we had to learn how to reduce the infections. Models predicted a very quick rise in cases. It took at least two months of learning, of understanding the virus and the way it spreads, to start reducing case numbers. We needed new measures: better hygiene, social distancing, eventually masks and PPE, and most dramatically in Australia, travel restrictions.

## Responses to the pandemic

These interventions worked. People are infectious for two weeks and most people will recover soon after. The spread of the virus can be contained. The lockdowns in the three countries were in April-May 2020 and casualties peaked. The rates of deaths declined in Europe for the northern summer, while we had the second wave in Australia after our winter in August & September. With much travel occurring in Europe a second wave arrived there, and this had been predicted by mathematical modelling. The question was: when and how to reintroduce the lockdowns again? This intervention was not popular and thus was delayed. The second wave was bigger and longer. In parallel the pandemic spread to many more countries.

Finally at the end of 2020 science had found a new solution. In record time vaccines were developed and tested—and this had a dramatic positive effect in the UK and Germany, and led to a strong decline in infections and deaths, while some restrictions were still in place. The UK reached a minimum in June 2021 and celebrated Freedom Day. Germany reached a minimum in August. Australia was still shielded internally with almost no COVID-19 cases.

However, with the arrival of the Delta variant the situation changed. Infections and hospitalisations rose as predicted by the latest modelling at the time and Australia entered its largest COVID waves and lockdowns. This has happened again with the arrival of the Omicron variant. Now we are wondering about the future: how will the pandemic continue into 2022 and beyond, for the world and for us in Australia? Can we travel again, see family, study abroad, explore other countries, behave as in 2019?

We know in Australia through modelling that we need a minimum rate of vaccination before we can progress. That means we need to compare different scenarios: using the models to forecast what would happen, assuming we are vaccinated or not, have lockdowns or not, stay at home or not, attend major events or not, travel or not, etc.

These alternative scenarios apply to all of us, but they depend on our individual behaviour. While the spread of the virus is from person to person, the models are statistical and describe what will happen to us together, as a population. We learned that single events, one infected person spreading it to a group, can start a major outbreak, but these events are impossible to predict. The models that are used for predictions test what certain rules, like staying at home, isolation, and travel restrictions would contribute to reducing the infections and deaths. It is up to us to follow the rules.

You might ask: should we trust the modelling? What are the limits of modelling? Is there one best model? Are there simple models we can understand? We can learn about mathematical modelling through some examples and through analogies.

## Examples of mathematical models in everyday life

Mice plagues are a good starting point. They frequently come after a good rainy year and a good harvest and we know that the population of mice can almost explode overnight.

A mouse plague is not a wave of mice marching through the countryside. It is the dynamic growth of the population due to an abundance of food. The reproduction of mice and the short time between generations, measured in days not years, creates an exponential growth, doubling again and again. This is followed by the collapse of the population due to starvation or diseases or our intervention. Over time, this looks like a wave of mice.

To what level will the mice population grow? How long will the plague last? That depends on several facts: how fast do the mice breed and with how many offspring do they have, how much food is available, the weather at the time, the action of humans. Once starvation or disease takes over the population halves again and again. The plague comes to a stop.

These features are at the core of a* model*: we have a mathematical description of the processes, here the breeding of new generations and of starvation. And we have to include a number of assumptions, such as the type of mice, the type of food available and what the humans will do. Together these allow us to predict the situation day by day and helps to plan our action to control the situation.

We use such *modelling* of natural processes all the time, for example in the form of weather forecasts. They have become remarkably reliable 2-3 days in advance, thanks to plenty of good data, a clear understanding of the atmosphere and much computing power. Experts at the Bureau of Meteorology (BOM) continue to perfect this. The forecasts are used successfully to warn us of impending disasters and trigger preventative measures. We can now look ahead a few days into the future weather for specific locations with good precisions. However, the precision is rather low looking weeks ahead.

Scientists, including at the BOM, have extended this forecasting to the whole globe and to decades into the future. We can predict general trends in the change of the climate. We cannot predict individual events, like a specific heatwave or cyclone, but we can very reliably predict long term trends such as sea level rise, global average temperatures for different parts of the globe and the impact on nature and on us, on food production and health. Climate modelling is warning us of these dangerous trends. The models allow us to predict the impact of global human activities, of the sum of the impacts we all individually create. And we know what actions we should take.

*Models* are used in many technical applications in our daily lives. Think of the electronic stability control in your car, now compulsory in Australia. This unit uses sensors to record data about the movement of your car and predicts what will happen next using a mathematical model. If necessary, it will intervene using the brakes. This is all happening faster than most drivers could react. The unit, which uses the model and data, guides the car to a stable situation.

We are using similar systems, with active or passive control, everywhere. An interesting example is the management of a nuclear power station. Here we control and stabilise the effects of reproducing radioactive decay of the atoms in the fuel. Uncontrolled reproduction could lead to a chain reaction and disaster. However in a reactor this is controlled through clever design and machinery to keep the radiation at a safe and steady rate. In this way a reactor in power station safely creates heat and ultimately electricity.

One future idea is to extend the control in cars to autonomous driving in the real world, with other drivers, cyclists, pedestrians and kangaroos around. This is difficult to control and not yet reliable, particularly when it comes to predicting what a roo might do. We see a limitation of mathematical models: it is difficult to predict the behaviour of animals and people. We can see trends, but we fail in reacting to individual events.

We also face problems when modelling and predicting the behaviour and actions of groups of people, for example by forecasting the future of the stock market, which involves predicting the response of stock traders, consumers, decision makers etc. The difficult part here is that the assumptions of how people will react will change all the time. For each step in time the situation and the forecast itself influences the behaviour. These dependencies frequently exceed the capacity of the mathematical models.

## Modelling and controlling a pandemic

Mathematical models for **pandemics** describe the reproduction and spread of the virus in a community. This is based on the propagation of infection from one person to others. We have statistical models for whole communities averaging over many individual cases. On average, does one contagious person infect one other person, fewer, or more? If each contagious person infects exactly one other person, the infection will spread steadily. If each person infects less than one other person, the pandemic will eventually disappear. That is our goal.

The models are based on the properties of the COVID-19 virus which we have learned in 2020, of new variants like Delta and Omicron, and the properties and availability of vaccines right now. They factor in the impact that restrictions, such as wearing masks, hygiene, physical distancing and lockdowns, will have on the spread of the virus. And this depends on the behaviour of people. One might think that it is possible to model a pandemic completely, using clever mathematics. But the outcome is too sensitive on the assumptions like the mobility, living and working conditions of the population, the quality of the health system, the exact measures imposed in the lockdowns, the rate of vaccination and the response to misinformation. All these are difficult to estimate and vary in time.

For this reason we see so many different ways the pandemic has evolved in different countries. Each nation has different goals, rules and different behaviours. Each culture shows different reactions to the rules. Each government has different ways to enforce rules, and this is clearly visible in the way this COVID-19 pandemic has spread across the globe.

Still, when a new wave sets in, the models are very useful to alert us that action has to be taken. We want to know: can we slow down the reproduction and spread of the virus? How strong does the intervention have to be? When will infections decline? Models are used in all countries to predict what could happen. They show how quickly the virus would reproduce and spread in the population if no action is taken and how big the effect of an intervention could be.

It is wise to use several models, or at least one model with a range of assumptions on human behaviour, and to correct the assumptions dynamically. For example a series of large events, like footy finals or concerts, can change the outcome. But so could good weather bringing people more indoors or outdoors. As we have seen, modelling the behaviour of people is one of the biggest challenges and is left to experts. But we can directly see some properties of the modelling.

## Understanding simple models: exponential growth and decline

One very simple mathematical model is that of exponential growth, stable situation and exponential decline, the three phases of a situation controlled by a single parameter.

Let’s take a detour. Imagine you are sitting down at a wobbly table and you want to make it stable. All you can use is a single sheet of paper. You know that by folding the paper enough times, you can create a thick pad to place under one of the legs of the table. Each step of folding the paper will double the thickness, again and again. Until you stop, or you can’t fold the paper any more. The succession of thicknesses might be: 0.125 mm, 0.25 mm, 0.5 mm, 1.0 mm, 2.0 mm, 4.0 mm, 8.0 mm. In just six steps you have created an 8 mm thick pad. This is all controlled by one core number: the increase in thickness in each time step.

For the population of mice, or the number of radioactive decays or the number of photons, or infected people we have **the reproduction number R**.

**is how much the population changes in each time step. By what factor does it increase, step by step. This can be used to describe the folding and unfolding of paper, the breeding and starvation of mice and many other examples in real life.**

*R*For the pandemic, ** R** states how contagious the situation is. For

**>1 we have exponential growth, for**

*R***=1 the situation is stable, for**

*R***<1 we have an exponential decline. It is easier to test data for exponential growth when the vertical axis is on a logarithmic scale, as in the examples shown here. Exponential growth and decline show as a straight line on a logarithmic scale—and we can test data to see if this simple model fits. See the reSolve article**

*R***for more information.**

*COVID-19 - Predicting the future*In many practical cases ** R** varies with time. For the paper folding we have

**=2 as long as we fold step by step,**

*R***=1 when we do nothing and**

*R***=0.5 when we unfold. For the mice plague, the supply of food will affect the reproduction number**

*R***. The population P(n) will increase exponential, then stabilise and finally decline. For the technical systems such as the car, we are actively controlling the machine to maintain**

*R***=1, that means a stable situation.**

*R*For the **pandemic,** the recent wave of infections in NSW is a good example. Starting with a single case we see the exponential rise with an ** R** value much larger than 1 and the number of cases almost doubling every week. Eventually the lockdowns worked, the infections peaked and now there is a decline in cases:

**<1. Together we have controlled the reproduction number, and thus the spread of the virus.**

*R*

*Source: The Guardian, 14 October 2021 *

The simple model of exponential growth has some strengths. The diagram above for the pandemic is plotted on a logarithmic scale. AUS, GER and the UK show a behaviour that is almost linear up and down, which means an exponential rise and fall with values of ** R** fixed for weeks or months.

*Source: Tagesspiegel, 4 December 2021*

We see an exponential rise (fixed ** R**>1, see red arrow up) at the start of the pandemic in March 2020 and an exponential decline (fixed

**<1, see green arrow down) caused by reduced contacts in the lockdowns. Australia follows Europe with a second wave after a few months delay.**

*R*Next we see an exponential rise for the second wave in Europe after the northern summer and an exponential decline when vaccinations set in. Finally we have an exponential rise when the Delta variant takes over in 2021.

As the pandemic moves from continent to continent, we have for the whole world more or less ** R**=1 (horizontal yellow arrow). Recently there has been a very slow decline, presumably due to vaccinations. Looking to the future we can see signs for a bigger impact of vaccinations. We can hope for a steadily declining mortality in all countries.

The exponential model is only a rough guide. We have learned that major interventions have a dramatic effect on the reproduction number in a community, and that this can control the pandemic. Our goal is to achieve ** R**=1, that means no further exponential spread of infections. We want to do this at the lowest possible level of infections. Zero COVID is now unlikely. We have a clear mission: keep the infection rate constant.

The good match we see in the data with the simple exponential model could just be a correlation. However, detailed science studies show us how and why actions like distancing, wearing masks, and vaccinations are slowing the spread of the virus.

**What has changed with the introduction of Omicron?**

We know that the Omicron variant spreads faster than the Delta or original variant. Let’s say that the reproduction number ** R **has doubled for Omicron. At the same time, the good news is that because of high vaccination rates, fewer people infected need to go to hospital–let’s say this “conversion” rate is represented as

**= ¼**

*H**,*compared to the Delta variant

**What is the combined effect of the increased**

*.***and the decreased**

*R***?**

*H*The two numbers affect the future quite differently: ** R** affects the exponential growth, while the conversion

**has a linear effect. For a short time, the lower conversion rate**

*H***keeps people out of the hospital, but within a few growth cycles the faster exponential growth will dominate. The probability to end up in hospital for each individual, and thus the number of people requiring care, will grow fast. Such is the impact of exponential growth. This unfortunately describes the present situation in the UK, Denmark, and the Netherlands, among other countries.**

*H*One good weapon to limit this growth is vaccination, including boosters. When 90% or more of the population is fully vaccinated, the effective reproduction number ‘** Romi**’ (

**as it relates specifically to the Omicron variant) is smaller. The UK shows a doubling of infections in 2-3 days–it can be slower in Australia. We only can measure**

*R***once Omicron has started to spread and dominate the infections. The same is true for ‘**

*Romi***’ (**

*Homi***as it relates specifically to the Omicron variant).**

*H*When we focus on hospitalisation in all of Australia, we see a large Omicron spike in February 2022, followed by a decline and then stabilisation at an unfortunate high level. This means that the impact of COVID-19 is not growing and not declining, and the corresponding ** R** factor is actually very close to

**=1. We have reached a more or less stable situation, but unfortunately with a significant number of people in hospital all the time. Stronger action to reduce the spread of the virus would be required to keep people away from hospitals.**

*R*

**Models are central to a safe society**

The pandemic models we hear about in the news and which are being used now are more sophisticated and of high reliability. They include the latest data and medical understanding. They are the best tools available to evaluate possible future scenarios and to make decisions about the restrictions we need to have. We learn which combination of actions is necessary now and in future. We will be able to adapt to new situations, like a new variant or a vaccine or a potential treatment of COVID. It is our individual actions and caution that will create a healthy life with freedom for us all.

Simple models like exponential growth show us why we have rules in a pandemic and how we can best contribute to stopping it. At the same time, let’s appreciate the many other mathematical models and controls that make our daily life safe and pleasant. Models, like those for the change in climate, can guide us to a healthy future.

## Further reading

*The rules of contagion* - why things spread and why they stop. Adam Kucharski Profile Books 2020 ISBN 978 1 78816 472 6. eISBN 978 1 78283 430 4