A Modified Discrete SIR Model Jennifer Switkes

Epidemiology  Epidemiology studies the causes, distribution, and control of disease in populations.

What is an SIR Model?  An SIR model is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time. The name of this class of models derives from the fact that they involve coupled equations relating the number of susceptible people S(t), number of people infected I(t), and number of people who have recovered R(t). One of the simplest SIR models is the Kermack-McKendrick Model. 1

Discrete vs. Continuous Discrete  Distinct separate values.  Having consecutive values that are not infinitesimally close, so that its analysis requires summation rather than integration. 2 Continuous  Objects that vary smoothly.  uninterrupted connection or union

Kermack-McKendrick model  t = time (days)  S(t) = Susceptible  I(t) = Infected  R(t) = Recovered*  a = Infection rate  b = Recovery rate

Kermack-McKendrick model

Question ?

Cont.  We want to show the idea behind the natural discrete approximation to the continuous SIR model to help us answer this question.

Kermack-McKendrick model  t = time (days)  S(t) = Susceptible  I(t) = Infected  R(t) = Recovered*  a = Infection rate  b = Recovery rate

Euler’s Method

Applying Euler’s

Modified Model

Initial Values & Scenario  Population of  1 student with highly contagious disease

Output

Results  The modified SIR model indicates a much higher maximum number of infecteds but also a much faster recovery for the student body as a whole.  Yes it does.

S-I-R Model is so cool  Good Intro to calculus  Art of Math Modeling  Learn how to analyze numerically a system of ODE’s  See connection between discrete and continuous dynamical systems  Tie sophisticated math to spread of an epidemic

Citations 1. Weisstein, Eric W. "SIR Model." From MathWorld--A Wolfram Web Resource Collins English Dictionary – Complete and Unabridged 6th Edition © William Collins Sons & Co. Ltd 1979, 1986 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003 Collins English Dictionary 3. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.Cambridge, England: Cambridge University Press, p. 710, 1992.Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.

Reference Page  A Modified Discrete SIR Model, Jennifer Switkes, The College Mathematics Journal, Vol. 34, No. 5 (Nov., 2003), pp  W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A 115 (1927), ; 138 (1932), 55-83; 141 (1933),  J. D. Murray, Mathematical Biology, Springer,  Callahan, J. "The Spread of a Contagious Illness."