Page - 226 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

226 8 SolvingOrdinaryDifferentialEquations Let the mathematical functionS(t) count how many individuals, at time t, that have the possibility to get infected. Here, t may count hours or days, for instance. These individuals make up a category called susceptibles, labeled as S. Another category, I, consists of the individuals that are infected. Let I(t) count how many thereare in categoryI at time t. An individualhaving recoveredfromthe disease is assumedtogainimmunity.Thereisalsoasmallpossibility thatan infectedwilldie. Ineither case, the individual is movedfrom the I category to a categorywe call the removed category, labeled with R. We letR(t) count the number of individuals in theRcategoryat time t.Thosewhoenter theRcategory,cannotleavethiscategory. Tosummarize,thespreadingofthisdiseaseisessentiallythedynamicsofmoving individuals fromthe S to the Iand then to theR category: We can use mathematics to more precisely describe the exchange between the categories.The fundamental idea is todescribe the changes that takeplaceduringa small time interval,denotedbyΔt. Ourdiseasemodel isoftenreferred toasacompartmentmodel,wherequantities are shuffled between compartments (here a synonym for categories) according to some rules. The rules express changes in a small time intervalΔt, and from these changeswecanletΔtgotozeroandobtainderivatives.Theresultingequationsthen gofromdifferenceequations(withfiniteΔt) todifferentialequations(Δt→0). We introduce a uniform mesh in time, tn = nΔt, n = 0,.. .,Nt, and seek S at the mesh points. The numerical approximation toS at time tn is denoted bySn. Similarly, we seek the unknown values of I(t) and R(t) at the mesh points and introduce a similar notation In andRn for the approximations to the exact values I(tn)andR(tn). In the time interval Δt we know that some people will be infected, so S will decrease. We shall soon argue by mathematics that there will be βΔtSI new infectedindividualsin this timeinterval,whereβ isaparameterreflectinghoweasy people get infected during a time interval of unit length. If the loss inS isβΔtSI, wehave that thechange inS is Sn+1 −Sn=−βΔtSnIn . (8.9) DividingbyΔt and lettingΔt →0, makes the left-handside approachS′(tn) such thatweobtainadifferentialequation S′ =−βSI . (8.10) The reasoning in going from the difference equation (8.9) to the differential equation(8.10) followsexactly thestepsexplainedinSect. 8.2.1. BeforeproceedingwithhowI andRdevelopsin time, letusexplain theformula βΔtSI.WehaveS susceptiblesandI infectedpeople.ThesecanmakeupSI pairs. Now, suppose that during a time interval T we measure that m actual pairwise meetings do occur among n theoretically possible pairings of people from the S