Page - 237 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.3 SpreadingofDisease:ASystemofFirstOrderODEs 237 Thenew,extendeddifferentialequationswith theV quantitybecome S′ =−βSI+νR−pS, (8.36) V ′ =pS, (8.37) I′ =βSI−γI, (8.38) R′ =γI−νR. (8.39) We shall refer to thismodelas theSIRV model. The new equation for V ′ poses no difficulties when it comes to the numerical method. Ina ForwardEuler schemewe simplyaddanupdate Vn+1 =Vn+pΔtSn . TheprogramneedstostoreV(t) inanadditionalarrayV, and theplottingcommand must be extended with more arguments to plot V versus t as well. The complete code is found in thefileSIRV1.py. Using p = 0.0005 and p = 0.0001 as values for the vaccine efficiency parameter, the effect of vaccination is seen in Figs. 8.15 and 8.16, respectively. (otherparametersareas inFig.8.13). 8.3.9 DiscontinuousCoefficients:AVaccinationCampaign What about modeling a vaccination campaign? Imagine that 6 days after the outbreak of the disease, the local health station launches a vaccination campaign. They reach out to many people, say 10 times as efficiently as in the previous (constantvaccination)case. If thecampaign lasts for10dayswecanwrite p(t)= { 0.005, 6 ·24≤ t≤15 ·24, 0, otherwise Note that we must multiply the t value by 24 because t is measured in hours, not days.Inthedifferentialequationsystem,pS(t)mustbereplacedbyp(t)S(t), andin thiscasewegetadifferentialequationsystemwithatermthat isdiscontinuous.This