Mathematical modelling of the zombie apocalypse

V magistrskem delu predstavimo in obravnavamo načine modeliranja epidemije zombijev. Apokalipso zombijev, ki so v populacijo prišli z izbruhom epidemije, modeliramo z uporabo predpostavk, osnovanih na znanih filmih o zombijih. Spoznamo osnovni model za modeliranje napada zombijev, ga analiziramo in s tem določimo ravnovesna stanja in njihovo stabilnost. Nato model dopolnimo z vključevanjem latentne dobe zombifikacije, v kateri so ljudje okuženi, ne pa tudi kužni. V nadaljevanju model prilagodimo še z dodatkom karantene in pa vplivom cepiv na potek epidemije. Za modeliranje apokalipse zombijev v magistrskem delu spoznamo še nekaj drugačnih pristopov. Najprej model osnujemo na podlagi Lotka-Volterra plen-plenilec modela in predstavimo ciklična nihanja števila posameznikov v populaciji ljudi in zombijev. Predstavimo še model, v katerem privzamemo agresiven pristop človeške populacije v spopadu z epidemijo zombijev. Z analizo modelov poskušamo odgovoriti na vprašanje kako se zombi epidemija lahko razplete in ali človeštvo v primeru takšne epidemije lahko preživi. Srečanja ljudi in zombijev v populaciji zapišemo tudi v analogiji s kemijskimi reakcijami in iz tega oblikujemo nov model. Ob tem nas zanima, ali začetna velikost populacije vpliva na razplet epidemije in kakšnim pogojem mora zadoščati. Za konec predstavimo možno strategijo preživetja zombi epidemije, v kateri predlagamo model skupnosti. Srečanja ljudi in zombijev za konec obravnavamo s pomočjo difuzije in poskušamo ugotoviti, koliko časa imamo na voljo, preden nas ujame zombi.In the thesis, we introduce and study different approaches of modelling zombie epidemic. We model the apocalypse of zombies, who came into the population through an outbreak of an epidemic, with the use of assumptions based on popular zombie movies. We introduce and analyse the basic zombie model, determine steady states of the model and their stability. We expand our model by adding different compartments to the model, such as state of latency, quarantine and vaccination. In the thesis we introduce some other approaches for modelling the apocalypse of zombies. We begin with a model based on Lotka-Volterra predator-prey model and illustrate the periodic oscillations of the number of zombies and humans in population. Furthermore, we consider more aggressive approach in facing the zombie threat and try to answer the question whether the humanity survives. We describe the interactions between populations using an analogy with chemical reactions. We formulate a new model and consider whether the size of initial population affects the outcome of epidemic. As a strategy for surviving the zombie epidemic we propose the model in which we build a community. At the end we use diffusion to describe the movement of zombies and calculate how much time we have before getting caught by a zombie