Epidemic models on lattices


Classic epidemic models of disease transmission are described in Compartmental models in epidemiology. Here we discuss the behavior when such models are simulated on a lattice.

Introduction

The mathematical modelling of epidemics was originally implemented in terms of differential equations, which effectively assumed that the various states of individuals were uniformly distributed throughout space. To take into account correlations and clustering, lattice-based models have been introduced. Grassberger
considered synchronous versions of models, and showed how the epidemic growth goes through a critical behavior such that transmission remains local when infection rates are below critical values, and spread throughout the system when they are above a critical value. Cardy and Grassberger
argued that this growth is similar to the growth of percolation clusters, which are governed by the "dynamical percolation" universality class. In asynchronous models, the individuals are considered one at a time, as in kinetic Monte Carlo or as a "Stochastic Lattice Gas."

SIR model

In the "SIR" model, there are three states:
It is to be distinguished from the "SIS" model, where sites recover without immunization, and are thus not "removed".
The asynchronous simulation of the model on a lattice is carried out as follows:
Making a list of I sites makes this run quickly.
The net rate of infecting one neighbor over the rate of removal is λ = /c.
For the synchronous model, all sites are updated simultaneously as in a cellular automaton.
Latticezccλc = /cc
2-d asynchronous SIR model triangular lattice60.199727,0.249574
2-d asynchronous SIR model square lattice40.1765, 0.17650054.66571
2-d asynchronous SIR model honeycomb lattice30.1393 6.179
2-d synchronous SIR model square lattice40.223.55
2-d asynchronous SIR model on Penrose lattice0.1713
2-d asynchronous SIR model on Ammann-Beenker lattice0.1732
2-d asynchronous SIR model on random Delaunay triangulations0.1963

Contact process (asynchronous SIS model)

I → S with unit rate;
S → I with rate λnI/z where nI is the number of nearest neighbor I sites, and z is the total number of nearest neighbors
.
The simulation of the asynchronous model on a lattice is carried out as follows, with c = 1 / :
Note that the synchronous version is the same as the directed percolation model.
Latticezλc
1-d23.2978, 3.29785
2-d square lattice41.6488, 1.64874, 1.64872, 1.64877
2-d triangular lattice61.54780
2-d Delaunay triangulation of Voronoi Diagram6 1.54266
3-d cubic lattice61.31685, 1.31683, 1.31686
4-d hypercubic lattice81.19511
5-d hypercubic lattice101.13847