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Corona, ebola and zombies

How can we know if the various measures against infection have an effect? "We have to have confidence in solid mathematical models, although it may contradict the gut feeling," Aksel Hiorth and Roald Kommedal writes in this chronicle.

  • Fig1.png (rw_largeArt_768).png
    Figure 1: Illustration of how the number of infected will grow if each person infects on average 2 or 3 healthy. Such a disease is said to have a reproduction number, R0, of 2 and 3 respectively.
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    Figure 2: An infected (zombie) in a population of 683 people (Dirdal). Every time a human encounter a zombie, you draw a random number between 0 and 1. If that number is less than 0.9 you get infected. If the number is greater than 0.9 the zombie is killed.

"All models are wrong, but some are useful". This quote by George Box is central in this chronicle by Aksel Hiorth and Roald Kommedal. They both teach at Faculty of Science and Technology at UiS. In this text they have written about the Coronavirus outbreak and how to model it, and how students at University of Stavanger get educated for situations like the present outbreak in the master's degree Computational Engineering.

The text has been published in Dagsavisen and forskning.no. Note: Both in Norwegian.

Translated to English below:

Trust the math, not the gut feeling

How can we know if the various measures against infection have an effect? We must have confidence in solid mathematical models, although it may contradict the gut feeling.

In “Debatten” at NRK, Tuesday 17th of March, researcher Gunnhild Alvik Nyborg argued for complete isolation because the corona virus spreads exponentially. This aroused strong reactions.

The figures from the Norwegian Institute of Public Health (Folkehelseinstituttet, FHI) do not indicate that the virus is spreading exponentially, and for the week that followed we can probably say that growth was stable, i.e. linear.

But is it the model Nyborg referred to or the figures from FHI that are not correct, and can we trust that the data gives a realistic picture of the epidemic?

What exactly is exponential growth?

What is exponential growth and why are we talking about this in connection with the spread of diseases?

Figure 1 shows examples of a disease that infects two or three people on average. After five generations, the spread of infection from one person will have led to the infection of 63 and 364 persons respectively.

It was the English priest and demographer Thomas Malthus who first pointed out that populations of people by uninhibited contact would grow "geometrically". His major concern was that contact between people led to reproduction, and that food access to the same population would only increase proportionally, " (…) A slight acquaintance with numbers will show the immensity of the first power compared to the second." The result of such growth would end with hunger, wars and epidemics.

In case of illness, people get healthy (or at worst die), so that after n days the number of infected people becomes significantly smaller. However, the exponential model is only a model for population growth, and other important mechanisms of dispersal are overlooked. It is therefore not suitable for modeling real populations because neither human reproduction nor the spread of the virus will in practice be unrestrained.

Mathematical models to help the brain

But how can we know if various measures against infection have an effect? Is the cure tougher than the disease? And do we decide this from the gut feeling, or are there other ways to figure this out? These are complicated questions that are difficult to answer, and then there are doubts and fears. One way to help the brain is to formulate the problem in mathematical models.

The most popular models used in epidemiology are so-called "compartment models". These models assume that within a population, all people are equally likely to meet and become infected. The assumption is smart because you do not have to model everyone, and the model can be solved in seconds.

Other advantages are that you can get relatively useful estimates without too much information, and they can be used on a large scale. For example, with all of Norway as a compartment. It is somewhat more reasonable to assume that each municipality is a compartment, with the possibility that people can travel between the municipalities. There is one such model used by the Norwegian Institute of Public Health (FHI), where daily data from mobile phones is used to estimate the transfer between the compartments. Do such simple models work?

We can learn from a zombie outbreak

Engineering students at the University of Stavanger learn how to use mathematical models to study various scenarios and effects of drastic measures. The models can provide decision support to government and politicians.

Last year, the students received two projects that dealt with the spread of infectious diseases. First, they were to model a zombie outbreak by studying what would happen in the communities of Sokndal and Dirdal if one person in each community was infected with a zombie virus.

The fate of Sokndal and Dirdal depended entirely on how early the zombie was discovered and the necessary action taken. For Dirdal and Sokndal this took 72 and 48 hours respectively. The difference was that the municipality of Dirdal was "more tolerant of unorthodox behavior", so it took longer to separate the zombie from other inhabitants.

In Sokndal, therefore, about 70 per cent of the population survived, while in Dirdal only 10 per cent.

To demonstrate relevance, students use the same compartment models to study the Ebola outbreak in West Africa. By adapting the models to observed infection data from WHO's websites, it was possible to estimate how the reproduction rate changed over time and one was able to get a very good match between the model and the observed number of infected in Liberia, Guinea and Sierra Leone.

Different measures can be studied with models

The simplicity of the mixed tank models becomes an Achilles heel when studying the effect of different measures because many measures cannot be explicitly modeled. In the next project, we let the students model the movement of individuals and zombies, using Monte Carlo models. This is a class of models that use random numbers to represent that people can make different choices.

Humans and zombies moved randomly around. Every time a human encounter a zombie there was a likelihood of infection, as illustrated in figure 2.

If man was not infected, we assumed the zombie was killed. In this model, various measures can be studied, for example, the people who survived a zombie attack became better at surviving at the next meeting. 

These models are simple, but they provide a starting point for studying the course of disease with various measures.

Room for failures

When discussing mathematical modeling, it is important to remember George Box's famous quote "All models are wrong, but some are useful". The model is useful if it explains observed data. The model is a formulation of the mechanisms we believe govern the phenomenon, while data tells us if we are astray in relation to reality. It is therefore crucial that as much data as possible is collected and made available, and that the data is accurate.

In "The fifth Discipline Fieldbook" Peter Senge philosophizes on how we interpret observations based on the models we have about the world. Senge claims that we "see" data that fits what we are looking for. We add data to opinions we already have and make assumptions that confirm our opinions. Thus, there is room for errors in both the modeling, the observations and when we compare data with models.

More testing can give a better picture of the spread of infection

The official infection figures must therefore be interpreted with a good model that considers data uncertainty and is consistent with data describing how the infection came to Norway. The most secure data is the number of dead and hospitalized. The health authorities should have done even more extensive testing in order to get a better picture of the spread of infection in Norway.

In addition, we must ensure that we do not see anything in the data that is not there. In the TV  debate mentioned earlier, a simple model was projected on insufficient data. Like the demographer Malthus, the researcher Nyborg saw the exponential disaster, and social isolation was the conclusion. Interestingly, this was also Malthus's end, except that he advocated chastity.

FHI also uses a simplified model, but it allows more mechanisms than unrestrained growth. This model, which epidemiologists often refer to as the SEIR model, allows moderation of the spread of infection, and this provides a rational basis for the alternative that FHI has chosen as the long-term solution: Flock immunity with some control of the development of infection, i.e. the braking strategy.

The fact that the government now seems to choose a tougher strategy may indicate that the gut feeling still plays a role.