On Ecological Models and the Coronavirus

We are caught up now in a coronavirus pandemic with an unknown end point. There is a great deal now available about COVID-19, and I want to concentrate on the models of this pandemic that currently fill our media channels. In particular I want to use the current situation to reflect on the role of mathematical models in helping to solve ecological problems and make predictions of future trends. To oversimplify greatly, the scientific world is aligned along an axis from those supporting simple models to those tied up in complex multifactor models. To make this specific, the simple epidemic model approach provides us with a coronavirus model that has three classes of actors – susceptible, infected, and recovered individuals, and one key parameter, the relative infection rate of one person to another. If you as an infected person pass on the disease to more than one additional person, the pandemic will grow. If you pass the disease on to less than one person (on average), the pandemic will collapse. Social distancing will flip us into the favourable state of declining infections. There is a similar sort of model in ecology for predator-prey interactions, called the Lotka-Volterra model, in which one predator eating one prey species will change the population size of both depending on the rate of killing of the predator and the rate of reproduction of the prey.

So far so good. We can all have an intuitive understanding of such simple models, but of course the critics rise up in horror with the cry that “the devil is in the details”. And indeed this is also a universal truth. All humans are not equally affected by COVID-19. Older people do poorly, young children appear to be little bothered by the virus. All prey individuals in nature are also not equally susceptible to being caught by a predator. Young prey may not run as fast as adults, poorly fed prey in winter may run more slowly than well fed animals. The consequences of this ‘inequality’ is what leads to the need for an increasing investment in scientific research. We can pretend the world is simple and the virus will just “go away”, and a simple view of predation that “larger animals eat smaller animals” could fail to recognize that a small predator might drive a dinosaur species extinct if the small predator eats only the eggs of the prey and avoids the big adults. The world is complicated, and that is what makes it both interesting to many and infuriating to some who demand simplicity.

One of the purposes of a mathematical model is to allow predictions of coming events, and we hear much of this with the COVID-19 models currently in circulation. A simple principle is “all models are wrong’ but this must be matched with the corollary that in general “the simpler the model the more likely it is to provide poor forecasts. But there is a corollary that might be called the “Carl Walters’ Law” that there is some optimal level of complexity for a good result, and too much complexity is also a recipe for poor projections. The difficulty is that we can often only find this optimal point after the fact, so that we learn by doing. This does not sit well with politicians and business-people who demand “PRECISE PRECISION PROMPTLY!” 

These uncertainties reflect on to our current decision making in the coronavirus pandemic, in issues to fight climate change, and in the conservation of threatened species and ecosystems. Our models, our scientific understanding, and our decisions are never perfect or complete, and as we see so clearly with COVID-19 the science in particular can be pushed but cannot be rushed, even when money is not limiting. The combination of planning, judgement and knowledge that we call wisdom may come more slowly than we wish. Meanwhile there are many details that need investigation.  

Adam, D. (2020) Modelling the Pandemic: The simulations driving the world’s response to COVID-19. Nature, 580, 316-318. Doi: 10.1038/d41586-020-01003-6 

Neher, R.A., Dyrdak, R., Druelle, V., Hodcroft, E.B. & Albert, J. (2020) Potential impact of seasonal forcing on a SARS-CoV-2 pandemic. Swiss Medical Weekly 150, w20224. Doi: 10.4414/smw.2020.20224.

Xu, B., Cai, J., He, D., Chowell, G. & Xu, B. (2020) Mechanistic modelling of multiple waves in an influenza epidemic or pandemic. Journal of Theoretical Biology, 486, 110070. Doi: 10.1016/j.jtbi.2019.110070.

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