Generalized Diffusive Epidemic Process with Permanent Immunity in Two Dimensions

TL;DR

This paper introduces a generalized epidemic model with permanent immunity on a lattice, revealing a phase transition in the percolation universality class and subexponential epidemic growth.

Contribution

The study extends epidemic modeling by incorporating diffusion, permanent immunity, and non-conservation of active particles, mapping the process to a growing infection source network.

Findings

Identifies a phase transition in the dynamic percolation universality class.

Shows epidemic growth is subexponential at the percolation threshold.

Demonstrates the model's independence from diffusion rates D_S and D_I.

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 rates $D_S$ or $D_I$ if they are susceptible or infected, respectively, and recovered individuals possess permanent immunity. Individuals can be contaminated with rate $\mu_c$ if they share the same lattice node with an infected individual and recover with rate $\mu_r$, 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 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 dynamic percolation universality class controlled by the population size, irrespective of diffusion rates $D_S$ and $D_I$ and a subexponential growth of the epidemics in the percolation threshold.