APRIL 15 — The inaugural International Day of Mathematics was celebrated on March 14, 2020 at the time when the world is fighting Covid-19. In conjunction with the celebration, this article introduces and highlights the importance of mathematics to the world, in the context of Covid-19 pandemic.

As Galileo Galilei asserted, "The laws of Nature are written in the language of mathematics.” Mathematicians often speak of the elegance and beauty of mathematics when describing reality. Being an applied mathematician, I am always more interested in what mathematics can do to solve real problems using mathematical models.

To help the understanding of the readers, this article is written using the Q&A format. (Bear with me, it’s quite a long read, but I promise there is no mention of "flatten the curve”).

1. What is a mathematical model?

A mathematical model of a real-world system is a description of the behaviour of the system using mathematical concepts. Models allow us to understand systems, discover and explain patterns and predict the effects of planned changes. For example, weather forecasts are made using models that simulate the atmospheric flow over the coming days.

All models involve a simplified representation of reality, not reality itself. Therefore, expertise is crucial in deciding what to include and what to omit. There is a famous quote attributed to George Box, "All models are wrong, but some are useful.”

One simple example is Google Maps. If it says your journey is 40 minutes, nobody believes it will take exactly that long, but having that information is much more useful than having nothing at all.

2. What are the purposes of developing mathematical models?

In reality, during an outbreak, we only ever see one version of the outbreak.

We rarely see the full picture at first, and this is where mathematics is important. Mathematics is essential for understanding the extent of illness and spread of Covid-19 and can assist us to determine what to do next.

Having helped us understand the past and present of an outbreak, mathematical models can be used to simulate or predict what might occur in future, under different conditions.

What this means is that we can assess our intervention measures in advance to know whether they will be effective or not. This can save us a great deal of money and, more importantly, many lives as well.

Models are at their most useful when they can reveal or quantify something which is not obvious. For instance, during this outbreak, several unprecedented measures were introduced by China in late January, such as citywide travel restrictions and school closures.

Mathematicians are now investigating to what extent (if at all) such interventions have reduced transmission.

3. What are the advantages of mathematical model over physical model?

There are several advantages. A mathematical model can be employed to predict the results of experiments we have yet to conduct or forecast the future (for example, weather forecasting).

The models can be used to carry out experiments which are too difficult, costly, unethical or under conditions that may not be reproduced physically. For example, there would be an outrage if the government risk the lives of a certain group of Covid-19 patients to assess the effectiveness of a certain intervention measure.

4. How reliable are mathematical models in predicting the future of this crisis? What are the limitations and how to address them?

The model predictions are very valuable but are not infallible. Those who put too much trust in mathematical models are demonstrating as much of a misunderstanding of mathematics as those who do not trust mathematical models at all. Mathematical modelling is only as good as the data fed into the computers.

Some parameters must be entirely assumed. Theoretically, you have to change, not just the parameters, but also the structure of a model for every new disease coming along. Furthermore, there is also a lot that models do not capture.

For example, most models do not factor in the anguish of social distancing or whether the public obeys movement control order.

Therefore, the forecast of the models comes with huge caveats because of estimations and assumptions that have to be built into the calculation, given such limited data and how much is still unknown about Covid-19.

The mathematical models are not meant to be crystal balls predicting exact numbers or dates. For example, according to the Malaysian Institute of Economic Research (Mier)’s projection, Malaysia would have 4,087 new Covid-19 cases by March 31.

However, that turned out to be incorrect as Malaysia has only recorded 2,766 cases by then (although the difference might also be due to the lack of tests).

However, this does not mean we should not employ mathematical models, we definitely should. There are ways to address these limitations. Firstly, we should work very closely with clinicians and real public health people, to help us in the development and validation of the model and the interpretation of the results.

Secondly, to minimise the impact of incorrect assumptions and incomplete data, mathematicians usually perform hundreds of different simulations, using different sets of slightly perturbed input parameters. This ‘sensitivity analysis’ attempts to prevent model outputs changing significantly when a single input changes.

Finally, to avoid over-reliance on one model, policymakers typically consider several models. For example, the United Kingdom (UK) government consulted several modelling groups, such as teams at Imperial College London and the London School of Hygiene and Tropical Medicine.

5. What are real-world examples in which mathematical models are used?

What people often do not realise is that mathematics is genuinely useful and applicable in so many different areas. For centuries, mathematics has been used to solve problems in astronomy, physics and engineering (just to name a few).

But now mathematical medicine and biology have become among the fastest emerging research areas in mathematics.

Now, mathematical modelling does not get much more policy-relevant than this. Governments around the world are now relying on forecasts of mathematical models to assist their decision-making during this crisis. However, Covid-19 is not the first infectious disease mathematicians have modelled.

Mathematical models have proved useful in responding to several outbreaks: SARS in 2003; swine flu in 2009; Ebola in 2014 are some recent examples.

6. What sort of questions about COVID-19 that can be answered using mathematical models?

Mathematical models can address many questions about COVID-19, these are some examples (note that different models might give different answers):