Generalized diffusive epidemic process with permanent immunity in two dimensions

Abstract We introduce the generalized diffusive epidemic process, which is a metapopulation model for an epidemic outbreak where a non-sedentary population of walkers can jump along lattice edges with diffusion probabilities <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">I</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> if they are susceptible or infected, respectively, and recovered individuals possess permanent immunity. Individuals can be contaminated with probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> if they share the same lattice node with at least one infected individual and recover with probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , being removed from the dynamics. Therefore, the model does not have the conservation of the active particles composed of susceptible and infected individuals. The reaction-diffusion dynamics are separated into two stages: (i) Brownian diffusion, where the particles can jump to the neighboring nodes, and (ii) contamination and recovery reactions. The dynamics are mapped into a growing process by activating lattice nodes with successful contaminations where activated nodes are interpreted as infection sources. In all simulations, the epidemic starts with one infected individual in a lattice filled with susceptibles. Our results indicate a phase transition in the dynamical percolation universality class controlled by the population size, irrespective of diffusion probabilities <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">S</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">I</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and a subexponential (power-law) growth of the epidemics in the percolation threshold.