Question

4. Scientists were conducting genetic experiments in the small town of Romero when a freak accident occurred, mutating one of the scientists into an unkillable, evil zombie. As a scientist working for the Centers for Disease Control, you must create a model of this situation to give to your response team using what you know about zombies and the town of Romero. • Your files show that Romero has a population of 1200, and the 911 call from the scientists indicated that only one of them was initially turned into a zombie. You know that zombies will turn normal humans into zombies if they can catch them and that, from watching various movies, the rate at which humans are turned into zombies is proportional to the product of the current number of humans and the current number of zombies in the town, with a constant of proportionality of a = 5.7. 104 people/hour. Therefore, the population of zombies will also increase according to this rate. You also know that the town has an empty military bunker to which citizens can retreat in case of emergency, which is impenetrable to zombies. You suspect that humans will retreat to this bunker as more zombies appear, so that humans from the town will go to the bunker at a rate proportional to the product of the current number of humans and the current number of zombies in the town, with a constant of proportionality of ay = 8.0.10“ people/hour. Therefore, the population of people in the bunker will increase at this rate. Schematically, the three populations will flow to different states according to the figure on the right. Evil Zombies In Town Your model should be a system of three coupled first order differential equations, with each equation tracking one population. Humans In Town Humans In Bunker a. (10%) Write down your model of the zombie outbreak as a system of three differential equations. Hint: Because zombies cannot die and cannot be cured, their population will never decrease; and because no one leaves the bunker, the bunker population will never decrease. Hint: You can verify your model system is coherent by integrating the system and showing that the total number of beings (town humans plus zombies plus bunker humans) stays constant.

Answer 1

Art given that Pt = popelation of human in town P = Population of Zomies the town Po= Population of human in bruker At any part of time Ptt Pzt Pt = 1200 (total populatiad Boon At time to Pt = 1199 P2=1 P620 Rate at which hinaus are turned into Zombies - Rate of increase & Zombies dpz = CPLPz people /hour 21=5.7 x104 people / hour Rate cat which people go into benker's - Rate Breuker of Increase of population in Csa!" CamScanne

dPb = x₂ Pt. P2 people/hour dt L2 = 8.0x105 people /hour. Rate at which people in town decrease Pe + P₂ + Pb = 1200 differentiate wort te dpt de les dt dt dt = – L. PEP2 – 22 Pt P2 –– (4 + 22 ) Pt Pz dpt dt a) Model of zombie outbreak dpt - - (2, +22) PEPz dt where Pt = humans in Pozomies town dlz = + 4 Pt P2 dt ale = & PER I Pb-humans in bunker, dt zomies and & we see, dpa, dpe is the, therefore. humans in bunker increases only dpt is re, therefore humans in town decreases only

- 1 dp - 155 at I (68408 - 57P). Pt at t=0 Pt = 1199 (68408-578)+5784 68408 I (68408-578). It 194 ost 099 . -10 t tt In ( 68408-57Pt - 68408 enor Sense - sh. 40103-54)". -16** ** to come on) * - -,•** 2 68608 = -) 0 x t x 68408 68408-57Pt/ ) 0.68408 t 65 1 68408-57 Pt -0.68408t 65 Pt 1199 (60408-57P4) -0.689084 -0.684087 e 65P = 1199 x 68408x - 0684087 1199 x 68 408 xe t = -0.68408t (65+ 1199 x 57 x en 65P2 = 68408 -57P 6512 2 -0.68 Godt = 68408 - 5X110 3 X 2 65 P2 = 6840865+1199x57x0684087 - 6800 - 57x1199xe 65 + 1199x57x pos

dt dpt - (est de) Pole dle & Pit, pz From these two equations dPt d P2 - (Qtd2) P2P2 di P P2 adpe - flatdz diz 2, P+ = (2, + de) Pz + c P2=1 P+= 1199 at t=0 B – 2,(1199) = x + 2a + c C = - 02 - 1200 m, = (dit de) Pz - da - 12002, a) – L. Pt di 57 X10-5 22= 8 x 10-5 – 57 Pt = 65 P2 - 8 - 68400 165 P2 = 68408 - 57 Pt J Now dpt & - (2, +22) Pt Pz dit dpt - 65 x 104 Pt (68408-57 P2) dt 05

68408 x 65 65 (65 + 1199 x 57 x eo P2 - 68408 65+ 11 99857x e obor P= 1200 – Pt - D2 - 6.68 4ody 2 e Pb = 1200 1408 1 68408 + 1199 x 68408 65 + 1199 x 57 x e P. Pb = 15, 95 9 2 65 - 0684087 - 1199 x 8 e & 1199 85 7 & 8.6840 -0.684094 . Pt & P2 + Po= 1200 for any time t

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