Are Models Useful? Reflections on Simple Epidemic Projection Models and the Covid-19 Pandemic
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rediction is very difficult, especially if it’s about the future’’ is a quotation one might expect from Groucho Marx or Yogi Berra. Yet it is attributed to the twentieth-century Nobel laureate Danish physicist Niels Bohr. The quotation may be apocryphal, but it makes a valid point, particularly when it comes to epidemiological predictions. Fast forward to April 2020 in the midst of the Covid-19 pandemic. Dr. Anthony Fauci, director of the USA’s National Institute of Allergy and Infectious Diseases and a leading member of the White House Coronavirus Task Force, is quoted on April 2 as saying, ‘‘I’ve looked at all the models. I’ve spent a lot of time on the models. They don’t tell you anything. You can’t really rely upon models’’ [4]. This quotation is too recent to be called apocryphal, but it is surprising and possibly out of context. Indeed, to end with a final aphorism, Dr. Fauci surely knows that ‘‘all models are wrong, but some are useful’’—a pronouncement attributed to the statistician George Box. This is another principle that is particularly true in the case of epidemiological models. The first goal of this paper is to introduce mathematically inclined readers to a few simple epidemic projection models. We will start with the basic notion of an epidemic
growing exponentially. We will then move on to a brief review of ‘‘compartmental’’ models, which capture the demographic dynamics of an infected population whose growth is limited endogenously by the size of the underlying population. We next introduce a simple yet novel variant of these models that is driven by an exogenously determined growth rate of the infected population. We call it the ‘‘Exo-r’’ model and fit it to data on infections in China and deaths in the United States. Informed by these projections, we will close by reflecting on the questions raised above: why are epidemiological predictions so difficult, and how do we reconcile an understandable dose of skepticism with the fact that projection models may be useful despite being wrong?
Basic Epidemic Projection Models The Exponential Model Epidemic models require an initial pool of infected individuals. A single person, known as ‘‘Patient 0,’’ is enough to trigger an outbreak. We assume that every day, an infected person is in contact with c randomly chosen uninfected individuals (the ‘‘susceptibles’’). A contact in itself does not guarantee transmission. There is a probability p, which
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depends on the virulence of the disease, that a contact leads to the susceptible becoming infected. If we define r ¼ cp, then during a time interval dt, each infected person generates, on average, r dt new infections.1 The total (cumulative) number of infected individuals I(t) up to day t then satisfies the differential equation dI =dt I_ ð1Þ ¼ ¼ r; I I where we use the physicist’s dot notation over a variable to indicate its derivative with respect to time. Equation (1) is the world’s simplest diffe
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