Mathematics is a useful tool in studying the growth of infections in a population, such as what occurs in epidemics.  A simple model is given by a first-order differential equation, the logistic equation, $\frac{dx}{dy}=\beta x(1-x)$ which is discussed in almost any textbook on differential equations.  It can be found, for instance, in Chapter 2 of Boyce and DiPrima’s book Elementary Differential Equations and Boundary Value Problems.  (See MR0179403 for a short review of the first edition, from 1965.)  This is a rudimentary model, but mathematicians have built on it to create more realistic, hence more useful models.  There is an informative explanation of how to use a mathematical model for epidemics, including the importance of determining the reproductive number $R_0$ of an infectious disease, in this video made by Tom Britton, a professor of mathematical statistics at Stockholm University.  Britton is one of the authors of

MR3015083
Diekmann, Odo(NL-UTRE-NDM); Heesterbeek, Hans(NL-UTRE-NDM); Britton, Tom(S-STOC-NDM)
Mathematical tools for understanding infectious disease dynamics.
Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2013. xiv+502 pp. ISBN: 978-0-691-15539-5
92-01 (62P10 92D30)

the review of which is copied below.

If you are interested in exploring some of the mathematics used in modeling epidemics, you can search MathSciNet using the MSC 92D30, which is the five-digit class for epidemiology, in particular, in the context of population dynamics.  Besides the review of Britton’s book, some other reviews are also copied below, to help give a sense of the mathematics used in epidemiology. Continue reading →