solving ordinary differential equations cfor the case in )find through experimentation the largest value of where the exact solution and the numerical solution by heun' method cannot be distinguished visually it is of interest to see how far off the curve the forward euler method is when heun' method can be regarded as "exact(for visual purposesfilenameode_heun py exercise find an appropriate time steplogistic model compute the numerical solution of the logistic equation for set of repeatedly halved time stepstk tk plot the solutions corresponding to the last two time steps tk and tk in the same plot continue doing this until you cannot visually distinguish the two curves in the plot then one has found sufficiently small time step hint extend the logistic py file introduce loop over kwrite out tk and ask the user if the loop is to be continued filenamelogistic_dt py exercise find an appropriate time stepsir model repeat exercise for the sir model hint import the ode_fe function from the ode_system_fe module and make modified demo_sir function that has loop over repeatedly halved time steps plot si and versus time for the two last time step sizes in the same plot filenamesir_dt py exercise model an adaptive vaccination campaign in the sirv model with time-dependent vaccination from sect we want to test the effect of an adaptive vaccination campaign where vaccination is offered as long as half of the population is not vaccinated the campaign starts after days that isp if daysotherwise demonstrate the effect of this vaccination policychoose vand as in sect set : daysand simulate for days hint this discontinuous tfunction is easiest implemented as python function containing the indicated if test you may use the file sirv py as starting pointbut note that it implements time-dependent tvia an array filenamesirv_p_adapt py exercise make sirv model with time-limited effect of vaccination we consider the sirv model from sect but now the effect of vaccination is time-limited after characteristic period of timethe vaccination is no more effective and individuals are consequently moved from the to the category and can be infected again mathematicallythis can be modeled as an average leakage from the category to the category ( gain in the latter