Page - 231 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.3 SpreadingofDisease:ASystemofFirstOrderODEs 231 Fig. 8.12 Smalloutbreak offlu at a boarding school (β ismuch smaller than in Fig.8.11) or simpler βS(0) γ >1 (8.29) toincreasethenumberofinfectedpeopleandacceleratethespreadingofthedisease. Youcan run theSIR1.pyprogramwith asmallerβ such that (8.29) isviolatedand observe that there isnooutbreak. The powerofmathematicalmodeling Thereadershouldnoticeourcarefuluse ofwords in thepreviousparagraphs. We started out with modeling a very specific case, namely the spreading of flu among pupils and staff at a boardingschool. With purposewe exchanged wordslikepupilsandfluwithmoreneutralandgeneralwordslike individuals anddisease, respectively.Phrasedequivalently,weraisedtheabstractionlevel by moving from a specific case (flu at a boarding school) to a more general case (disease in a closed society).Veryoften,when developingmathematical models, we start with a specific example and see, through the modeling, that what is going on of essence in this example also will take place in many similarproblemsettings.Wetrytoincorporatethisgeneralizationinthemodel so that themodelhasamuchwiderapplicationarea thanwhatweaimedat in the beginning. This is the very power of mathematical modeling: by solving onespecificcase wehaveoftendevelopedmoregeneric tools that can readily be applied to solve seeminglydifferentproblems.The next sectionswill give substance to this assertion.