Marian Brandau | Searcharxiv

arXiv2008

Exact solution of a stochastic SIR model

The susceptible-infectious-recovered (SIR) model describes the evolution of three species of individuals which are subject to an infection and recovery mechanism. A susceptible $S$ can become infectious with an infection rate $β$ by an infectious $I$- type provided that both are in contact. The $I$- type may recover with a rate $γ$ and from then on stay immune. Due to the coupling between the different individuals, the model is nonlinear and out of equilibrium. We adopt a stochastic individual-based description where individuals are represented by nodes of a graph and contact is defined by the links of the graph. Mapping the underlying Master equation into a quantum formulation in terms of spin operators, the hierarchy of evolution equations can be solved exactly for arbitrary initial conditions on a linear chain. In case of uncorrelated random initial conditions the exact time evolution for all three individuals of the SIR model is given analytically. Depending on the initial conditions and reaction rates $β$ and $γ$, the $I$-population may increase initially before decaying to zero. Due to fluctuations, isolated regions of susceptible individuals evolve and unlike in the standard mean-field SIR model one observes a finite stationary distribution of the $S$-type even for large population size. The exact results for the ensemble averaged population size are compared with simulations for single realizations of the process and also with standard mean field theory which is expected to be valid on large fully-connected graphs.

cond-mat.stat-mech↗

arXiv2005

Network of social groups or Let's have a party

We present a simple model for growing up and depletion of parties due to the permanent communication between the participants of the events. Because of the rapid exchange of information, everybody is able to evaluate its own and and all other parties by means of the list of its friends. Therefore the number of participants at different parties can be changed incessantly. Depending on the deepness of the social contacts, which will be characterized by a parameter $α$, a stable distribution of party members emerges. At a critical $α_c$ an abrupt depletion of almost all parties is observed and as the consequence all the peoples are assembled at a single party. The model is based on a hierarchical social network. The probability that a certain person is contacted to another one depends on the social distance introduced within the network and homophily parameter $α$.

physics.soc-ph↗