We present a modified diffusive epidemic process (DEP) that has a finite threshold on scale-free graphs, motivated by the COVID-19 pandemic. The DEP describes the epidemic spreading of a disease in a non-sedentary population, which can describe the spreading of a real disease. Our main modification is to use the Gillespie algorithm with a reaction time t max, exponentially distributed with mean inversely proportional to the node population in order to model the individuals’ interactions. Our simulation results of the modified model on Barabasi–Albert networks are compatible with a continuous absorbing-active phase transition when increasing the average concentration. The transition obeys the mean-field critical exponents β = 1, γ′ = 0 and ν ⊥ = 1/2. In addition, the system presents logarithmic corrections with pseudo-exponents β ̂ = γ ̂ ′ = − 3 / 2 on the order parameter and its fluctuations, respectively. The most evident implication of our simulation results is if the individuals avoid social interactions in order to not spread a disease, this leads the system to have a finite threshold in scale-free graphs.

Abstract:On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabási-Albert (BA), small-worlds (SW), Voronoi-Delaunay (VD) and Erdös-Rényi (ER) random graphs. The review here is on phase transitions, critical points, exponents and universality classes that are compared to the results obtained for these models on regular square lattices (SL).

We present an analysis of an epidemic spreading process on an Apollonian network that can describe an epidemic spreading in a non-sedentary population. We studied the modified diffusive epidemic process using the Monte Carlo method by computational analysis. Our model may be helpful for modeling systems closer to reality consisting of two classes of individuals: susceptible (A) and infected (B). The individuals can diffuse in a network according to constant diffusion rates and , for the classes A and B, respectively, and obeying three diffusive regimes, i.e., , , and . Into the same site i , the reaction occurs according to the dynamical rule based on Gillespie’s algorithm. Finite-size scaling analysis has shown that our model exhibits continuous phase transition to an absorbing state with a set of critical exponents given by , , and familiar to every investigated regime. In summary, the continuous phase transition, characterized by this set of critical exponents, does not have the same exponents of the mean-field universality class in both regular lattices and complex networks.

Through Monte Carlo simulations, we studied the critical properties of kinetic models of continuous opinion dynamics on (3,4,6,4) and (3 4 , 6) Archimedean lattices. We obtain p c and the critical exponents' ratio from extensive Monte Carlo studies and finite size scaling. The calculated values of the critical points and Binder cumulant are p c = 0.085(6) and O * 4 = 0.605(9); and p c = 0.146(5) and O * 4 = 0.606(3) for (3, 4, 6, 4) and (3 4 , 6) lattices, respectively, while the exponent ratios β/ν, γ/ν and 1/ν are, respectively: 0.126(1), 1.50 (7), and 0.90(5) for (3, 4, 6, 4); and 0.125(3), 1.54(6), and 0.99(3) for (3 4 , 6) lattices. Our new results agree with majority-vote model on previously studied regular lattices and disagree with the Ising model on square-lattice.

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