Epidemic models on lattices - AbsoluteAstronomy.com

Classic epidemic models of disease transmission are described in Epidemic model

Epidemic model

An Epidemic model is a simplified means of describing the transmission of communicable disease through individuals.-Introduction:The outbreak and spread of disease has been questioned and studied for many years...

and Compartmental models in epidemiology

Compartmental models in epidemiology

In order to model the progress of an epidemic in a large population, comprising many different individuals in various fields, the population diversity must be reduced to a few key characteristics which are relevant to the infection under consideration...

. Here we discuss the behavior when such models are simulated on a lattice.

Introduction

The mathematical modeling 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 (cellular automaton) 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

John Cardy

John Lawrence Cardy FRS is a British theoretical physicist at the University of Oxford. He is best known for his work in theoretical condensed matter physics and statistical mechanics, and in particular for research on critical phenomena and conformal field theory.He was an undergraduate and...

and Grassberger
argued that this growth is similar to the growth of percolation clusters, which are governed by the "dynamics percolation" universality class (finished clusters are in the same class as static percolation, while growing clusters have additional dynamic exponents). 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:

Susceptible (S) -- has not yet been infected, and has no immunity
Infected (I)-- currently "sick" and contagious to Susceptible neighbors
Recovered (R), where the recovery is assumed to be permanent (immunized against further infection)

The asynchronous simulation of the model on a lattice is carried out as follows:

Pick a site. If it is I, then generate a random number x in (0,1).
If x < c then let I go to R.
Otherwise, pick one nearest neighbor randomly. If the neighboring site is S, then let it become I.
Repeat as long as there are S sites available.

Making a list of I sites makes this run quickly.

The net rate of infecting one neighbor over the rate of recovery is λ = (1-c)/c.

For the synchronous model, all sites are updated simultaneously (using two copies of the lattice) as in a cellular automaton.

Lattice	z	c_c	λ_c = (1 - c_c)/c_c
2-d asynchronous SIR model square lattice	4	> 4.66571(3)
2-d asynchronous SIR model honeycomb lattice	3	0.1393(1)	6.179(5)
2-d synchronous SIR model square lattice	4	0.22	3.55

Contact process (asynchronous SIS model)

I → S with unit rate;
S → I with rate λn_I/z where n_I is the number of nearest neighbor I sites, and z is the total number of nearest neighbors (equivalently, each I attempts to infect one neighboring site with rate λ)

(Note: S → I with rate λn in some definitions, implying that lambda has one-fourth the values given here).

The simulation of the asynchronous model on a lattice is carried out as follows, with c = 1 / (1 + λ):

Pick a site. If it is I, then generate a random number x in (0,1).
If x < c then let I go to S.
Otherwise, pick one nearest neighbor randomly. If the neighboring site is S, then let it become I.
Repeat

Note that the synchronous version is the same as the directed percolation model.

Lattice	z	λ_c
1-d	2	3.2978(2) , 3.29785(2)
2-d square lattice	4	1.6488(1) , 1.64874(2) , 1.64872(3) , 1.64877(3)
2-d triangular lattice	6	1.54780(5)
2-d Delaunay triangulation of Voronoi Diagram	6 (av)	1.54266(4)
3-d cubic lattice	6	1.31685(10) , 1.31683(2) , 1.31686(1)
4-d hypercubic lattice	8	1.19511(1)
5-d hypercubic lattice	10	1.13847(1)

Introduction

SIR model

Contact process (asynchronous SIS model)

See also