Matematičko modeliranje nekih društvenih fenomena

U ovom radu predstavili smo matematičke modele koji opisuju neke društvene fenomene kao što su dinamika ekonomije, kriminologija te posebno, utaja poreza. U prvom poglavlju smo opisali evoluciju raspodjele bogatstva u društvu. Prvo, kompartmentalnim modelom gdje smo društvo podijelili u \(n\) klasa ovisno o njihovom socijalnom indeksu, a kasnije smo predstavili poopćenje modela neprekidnim modelom gdje na dinamiku društva utječe faktor ekonomske mobilnosti, tj. promjena socijalnog indeksa. Zatim, u drugom poglavlju smo imali za cilj dodatno razumijevanje teorijskih mehanizama u kriminologiji. Koristili smo pristup koji nam pomaže u boljem razumijevanju činjenica koje karakteriziraju modeliranje u kriminologiji te u planiranju strategija u suprotstavljanju kriminala. Predstavili smo klasu modela baziranu na Lotka-Volterra teoriji „predator-plijen'' populacije gdje smo iznijeli model trokuta podpopulacija kriminalaca, čuvara i meta, te model proizvodnje i prodaje droge gdje smo imali podpopulacije dilera i proizvođača droga. Zatim, predstavili smo klasu epidemioloških modela gdje se regrutacija među podpopulacijama događa nekom vrstom „zaraze''. Tu smo imali model veze siromaštva i kriminala te model četiri podpopulacije gdje smo imali podpopulaciju kriminalaca, podložnih i nepodložnih „zarazi'' te zatvorenika. Na kraju smo iznijeli složeniji pristup problemu koji uzima u obzir društveno-ekonomsku dinamiku iz prvog poglavlja. Konačno, u posljednjem poglavlju predstavili smo modele bazirane na staničnom automatu koji su se bavili posebnom vrstom kriminala, a to je utaja poreza. Koristeći dvodimenzionalni stanični automat predstavili smo model gdje se pojedinac može opisati kao pridonosioc društvu (tj. onaj koji plaća porez) ili onaj koji ne pridonosi društvu (tj. utajivač poreza). Na kraju poglavlja smo proširili model uvođenjem fiskalne politike.In this thesis we present mathematical models that describe social phenomena such as economic dynamics, criminology and tax evasion. In the first chapter we describe the evolution of the distribution of wealth in society. Firstly, we use compartmental model where we divide the society into the n classes depending on their social index, and later we introduce the generalization of the model with continuous model where the dynamics of society are influenced by economic mobility, i.e. the change in the social index. Then, in the second chapter, we aim at further understanding the theoretical mechanisms in criminology. We have an approach that helps us to better understand the facts that characterize modeling in criminology and plan crime prevention strategies. We introduce a class of models based on Lotka-Volterra’s theory of ”predator-prey” populations where we present a triangle model consisting of sub-populations of criminals, guards and targets, and a model of drug production and sales where we have sub-populations of dealers and drug manufacturers. Next, we present the class of epidemiological models where recruitment among sub-populations occurs with some type of ”infection”. Here we describe a model of poverty and crime, and a model of four sub-populations where we have sub-populations of criminals, those who are not susceptible to commit a crime, those who are susceptible to commit a crime and prisoners. Finally, we introduce a more complex approach to the problem that takes into account the socio-economic dynamics introduced in the first chapter. In the last chapter, we describe models based on cellular automata where we consider a particular kind of crime, i.e. tax evasion. Using a two-dimensional cellular automata approach, we present a model where an individual can either be contributor (tax payer) or no contributor (tax evader or cheater). At the end of the chapter we expand the model by introducing the fiscal policy