Seite - 154 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

154 4 SolvingOrdinaryDifferentialEquations c) For the case in b), find through experimentation the largest value of twhere the exact solution and the numerical solution byHeun’smethod cannotbedis- tinguished visually. It is of interest to see how far off the curve the Forward Euler method is whenHeun’s method can be regarded as “exact” (for visual purposes). Filename:ode_Heun.py. Exercise4.4: Findanappropriate timestep; logisticmodel Compute the numerical solution of the logistic equation for a set of repeatedly halved time steps: tk D 2 k t, k D 0;1;:: :. Plot the solutions correspond- ing to the last two time steps tk and tk 1 in the sameplot. Continuedoing this untilyoucannotvisuallydistinguish the twocurves in theplot. Thenonehasfound asufficiently small timestep. Hint Extend the logistic.pyfile. Introduce a loop over k, write out tk, and ask theuser if the loop is tobecontinued. Filename:logistic_dt.py. Exercise4.5: Findanappropriate timestep;SIRmodel RepeatExercise4.4 for theSIRmodel. Hint Import the ode_FE function from the ode_system_FEmodule and make amodified demo_SIR function that has a loop over repeatedly halved time steps. PlotS,I, andRversus time for the two last timestep sizes in the sameplot. Filename:SIR_dt.py. Exercise4.6:Modelanadaptivevaccinationcampaign In the SIRVmodelwith time-dependent vaccination fromSect. 4.2.9,wewant to test the effect of an adaptivevaccination campaignwherevaccination is offeredas longashalf of thepopulation is notvaccinated. The campaign starts after days. That is,pDp0 ifV < 12.S0CI0/and t > days,otherwisepD0. Demonstrate the effect of this vaccination policy: choose ˇ, , and as in Sect. 4.2.9, setpD0:001, D10days, andsimulate for200days. Hint Thisdiscontinuousp.t/ functioniseasiest implementedasaPythonfunction containing the indicatedif test. Youmayuse thefileSIRV1.pyas starting point, butnote that it implementsa time-dependentp.t/viaanarray. Filename:SIRV_p_adapt.py. Exercise4.7:MakeaSIRVmodelwith time-limitedeffectofvaccination Weconsider theSIRVmodel fromSect. 4.2.8, but now the effect of vaccination is time-limited. After a characteristic period of time, , the vaccination is nomore effectiveand individuals areconsequentlymovedfromtheVto theScategoryand canbe infected again. Mathematically, this can bemodeled as an average leakage 1V from the V category to the S category (i.e., a gain 1V in the latter).