Page - 116 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

116 4 SolvingOrdinaryDifferentialEquations 4.2.4 OutbreakorNot Lookingat theequation forI, it is clear thatwemusthaveˇSI I >0 forI to increase.Whenwestart thesimulation itmeans that ˇS.0/I.0/ I.0/>0; or simpler ˇS.0/ >1 (4.29) to increase the number of infected people and accelerate the spreading of the dis- ease.YoucanruntheSIR1.pyprogramwithasmallerˇ suchthat(4.29)isviolated andobserve that there isnooutbreak. Thepowerofmathematicalmodeling The reader should notice our careful use ofwords in the previous paragraphs. Westartedoutwithmodelingaveryspecificcase, namely the spreadingofaflu amongpupilsandstaff at aboardingschool.Withpurposeweexchangedwords like pupils and flu with more neutral and general words like individuals and disease, respectively. Phrased equivalently, we raised the abstraction level by moving from a specific case (flu at a boarding school) to amore general case (disease in a closed society). Very often,when developingmathematicalmod- els,westartwith a specificexampleandsee, through themodeling, thatwhat is going on of essence in this example alsowill take place inmany similar prob- lem settings. We try to incorporate this generalization in themodel so that the model has amuchwider application area thanwhatwe aimed at in the begin- ning. This is theverypowerofmathematicalmodeling: bysolvingonespecific casewehaveoften developedmoregeneric tools that can readily be applied to solveseeminglydifferentproblems. Thenextsectionswillgivesubstance to this assertion. 4.2.5 AbstractProblemandNotation Whenwehadaspecificdifferentialequationwithoneunknown,wequickly turned toanabstractdifferential equationwritten in thegeneric formu0 Df.u;t/.Were- fer tosuchaproblemasascalarODE.Aspecificequationcorrespondstoaspecific choiceof the formulaf.u;t/ involvinguand(optionally) t. It is advantageous to also write a system of differential equations in the same abstractnotation, u0 Df.u;t/; but this time it isunderstoodthatu is avectorof functionsandf is alsovector.We say thatu0 Df.u;t/ is avector ODEor system of ODEs in this case. For theSIR modelwe introduce the two3-vectors,one for theunknowns, uD .S.t/;I.t/;R.t//;