5.1: Maxima and Minima MTH 124
Neuraminidase of influenza virus.
Source: rcsb.org
As we have now seen, we can use the derivative to understand
how a function changes over an interval or at a given point.
When modeling physical phenomenon, we’re often interested in
any maxima or minima the function modeling that behavior may
have. For example, the model given in figure 1 uses social media
to predict the number of new cases of influenza in an area of
Baltimore.
From a public health and financial standpoint, a sophisticated
understanding of a disease like influenza is extremely impor-
tant.
“Each year, on average, 5 percent to 20 percent of the
U.S. population gets the flu, tens of thousands are hos-
pitalized and thousands die from flu-related illness. This
costs an estimated $10.4 billion a year in direct medical ex-
penses and an additional $16.3 billion in lost earnings annu-
ally.”
Molinari NA et al. Vaccine 25 (2007)
According to Molinari et al this results in a total eco-
nomic burden of more than $87 billion dollars.
With this in mind we can see the impact that a good model could have regarding a context
like public health. Today we’ll investigate the connection between the derivative of a function
and any maxima or minima of the function.
Given the information above, it is easy to see why we would like to avoid maximizing such a
function, and similarly why we would want to take the appropriate steps to promote minimizing
such a function. Mathematical models like this, give public health officials and policy makers
tools to make the best use of the limited resources they have available.
Figure 1: Figure taken from “Using Social Media to Perform Local Influenza Surveillance in an Inner-City
Hospital: A Retrospective Observational Study” by Broniatowski DA et al. JMIR Public Health Serveill (2015)
75
