Dynamics of contagious disease spreading on complex networks

Pojavom pandemije virusa COVID-19, tema širenja zaraze postala je vrlo aktualna te dio svakodnevice prosječnog laika. Modeliranje širenja zaraze već je i ranije izazivalo velik interes raznih znanstvenih zajednica te je postalo područje multidisciplinarnog istraživanja budući da je zaintrigiralo matematičare, fizičare, biologe, sociologe i druge znanstvenike. U ovome ćemo radu sažeti najvažnije doprinose epidemiološkom modeliranju, općenita svojstva kompleksnih mreža te najznačajnije doprinose u problemu detekciji izvora zaraze na kompleksnim mreža. Također, razmotrit ćemo karakteristike prosječne udaljenosti promatranog vrha do svih zaraženih vrhova i njenog gradijenta na primjeru SIS modelu širenja zaraze na mreži malog svijeta. Dokazat ćemo da je gradijent prosječne udaljenosti promatranog vrha do svih zaraženih vrhova najveći za izvor zaraze, što ćemo provjeriti i na skupu podataka koji opisuju neku stvarnu mrežu. SIS model uvelike svojim karakteristikama podsjeća na dinamiku širenja koronavirusa (oporavljeni pacijent nakon nekog vremena ponovno postaje podložan novoj zarazi). Modeli homogenog miješanja tek aproksimativno opisuju širenje zaraze, stoga se u ovom radu promatra širenje zaraze SIS modelom na kompleksnim mrežama, konkretno na mrežama malog svijeta. Posebnost kompleksnih mreža jest vjeran prikaz raspodjele ljudi te njihovih interakcija, gdje su ljudi vrhovi grafa, a interakcije su veze medu vrhovima. Ovaj je rad usmjeren na problem detekcije izvora zaraze pri širenju zaraze po SIS modelu na mreži malog svijeta.With the appearance of the COVID-19 virus pandemic, the topic of the spread of the infection became very usual and part of the everyday life of an average man. Modelling the spread of the infection has already aroused great interest in various scientific communities and became an area of multidisciplinary research since it has intrigued mathematicians, physicists, biologists, sociologists, and other scientists. In this thesis, we summarize the most important contributions to epidemiological modelling, the general properties of complex networks, and the most significant contributions to the problem of detecting sources of infection on complex networks. Also, we will consider the characteristics of the average distance of the observed node to all infected ones and its gradient on the example of the SIS model infection spreading on the small-world network. We will prove that the gradient of the average distance of the observed node to all infected ones is the largest for the source of infection, which we will also check on a set of data describing a real network. The SIS model is very similar in its characteristics to the dynamics of coronavirus spread (a recovered patient after some time becomes susceptible to a new infection again). Homogeneous mixing models only approximately describe the disease spread, so in this paper, we observe the SIS model spreading of infection on complex networks, specifically on small-world networks. The peculiarity of complex networks is a possibility of correct representation of the distribution of people and their interactions, where people are the vertices of the graph, and the interactions are the connections between the vertices. This paper deals with the source detection problem in SIS model disease spreading on small-world networks