Seite - 228 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Band Second Edition

228 8 SolvingOrdinaryDifferentialEquations Since there is no loss in the R category (people are either recovered and immune, ordead),we are donewith the modelingof this category. In fact, we do not strictly needEq. (8.14)forR, butextensionsof themodel laterwillneedanequationforR. Dividing by Δt in (8.13) and (8.14) and letting Δt → 0, results in the correspondingdifferentialequations I′ =βSI−γI, (8.15) and R′ =γI . (8.16) To summarize, we have derived three differenceequationsand three differential equations,whichwe list here foreasy reference.Thedifferenceequationsare: Sn+1 =Sn−βΔtSnIn, (8.17) In+1 = In+βΔtSnIn−γΔtIn, (8.18) Rn+1 =Rn+γΔtIn . (8.19) Note that we have isolated the new unknown quantitiesSn+1, In+1, andRn+1 on the left-hand side, such that these can readily be computed if Sn, In, and Rn are known.Togetsuchaprocedurestarted,weneedtoknowS0,I0,R0.Obviously,we alsoneed tohavevalues for theparametersβ andγ . The threedifferentialequationsare: S′ =−βSI, (8.20) I′ =βSI−γI, (8.21) R′ =γI . (8.22) This differential equation model (and also its discrete counterpart above) is known as an SIR model. The input data to the differential equation model consist of the parametervaluesforβ andγ , aswellas the initialconditionsS(0)=S0,I(0)= I0, andR(0)=R0. 8.3.2 AFEMethodfortheSystemofODEs Letusapplythesameprinciplesaswedid inSect.8.2.2 todiscretize thedifferential equation system by the Forward Euler method. We already have a time mesh and time-discretequantitiesSn, In,Rn,n=0,.. .,Nt. The threedifferentialequations areassumedto bevalidat themeshpoints.At thepoint tn wethenhave S′(tn)=−βS(tn)I(tn), (8.23) I′(tn)=βS(tn)I(tn)−γI(tn), (8.24) R′(tn)=γI(tn), (8.25)