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Working in the multi-type Galton-Watson branching-process framework we analyse the spread of a pandemic via a most general random contact graph. Our model consists of several communities, and takes an input parameters that outline the contacts between individuals in different communities. Given these parameters, we determine whether there will be a pandemic outbreak and if yes, we calculate the size of the giant--connected-component of the graph, thereby, determining the fraction of the population of each type that would contract the disease before it ends. We show that the disease spread has a natural evolution direction given by the Perron-Frobenius eigenvector of a matrix whose entries encode the average number of individuals of one type expected to be infected by an individual of another type. The corresponding eigenvalue is the basic reproduction number of the pandemic. We perform numerical simulations that compare homogeneous and heterogeneous disease spread graphs and quantify the difference between the pandemics. We elaborate on the difference between herd immunity and the end of the pandemics and the effect of countermeasures on the fraction of infected population.
We study the spread of information on multi-type directed random graphs. In such graphs the vertices are partitioned into distinct types (communities) that have different transmission rates between themselves and with other types. We construct multiv
We study the kinetics for the search of an immobile target by randomly moving searchers that detect it only upon encounter. The searchers perform intermittent random walks on a one-dimensional lattice. Each searcher can step on a nearest neighbor sit
We study tilings of the square lattice by linear trimers. For a cylinder of circumference m, we construct a conserved functional of the base of the tilings, and use this to block-diagonalize the transfer matrix. The number of blocks increases exponen
A self-organized model with social percolation process is proposed to describe the propagations of information for different trading ways across a social system and the automatic formation of various groups within market traders. Based on the market
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $mathcal{P}_N(K,lambda)$ that a large $N times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $l