Numerical and theoretical analysis of epidemiological SIR models and variants

U ovom radu predstavljen je epidemiološki SIR model te nekoliko njegovih inačica, kao što su SIRS, SIS i SEIR modeli. Osim teorijskom analizom sustava običnih diferencijalnih jednadžbi modela, rad se bavi i metodama za procjenu parametara modela iz dostupnih podataka, uz primjenu na pandemiju koronavirusa. Opisane su neke metode za numeričko rješavanje jednadžbi. Napravljena je analiza stacionarnih točaka sustava u čiju je svrhu korišten Routh-Hurwitz kriterij stabilnosti. Osnovni reprodukcijski broj \(R_0\) predstavljen je kao bitan parametar koji odreduje hoće li doći do epidemije ili ne. Osim klasičnog SIR modela, obrađene su njegove inačice s nekonstantnim parametrima te uz vitalnu dinamiku.This thesis presents the epidemiological SIR model and several variants of it, such as the SIRS, SIS and SEIR models. In addition to the theoretical analysis of the system of ordinary differential equations of the model, the thesis also deals with methods for estimating the model parameters from the available data, with application to the coronavirus pandemic. A few methods for numerical solution of the equations are described. An analysis of the equilibrium points of the system was made, using the Routh-Hurwitz stability criterion. The basic reproduction number \(R_0\) is presented as an important parameter that determines whether an epidemic will occur or not. In addition to the classic SIR model, its versions with non-constant parameters and with vital dynamics are described