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Classic epidemic models of disease transmission are described in
Compartmental models in epidemiology Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious ...
. 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 (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 and Grassberger argued that this growth is similar to the growth of percolation clusters, which are governed by the "dynamical 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 ::* Removed (R), where the removal from further participation in the process is assumed to be permanent, due to immunization or death 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: ::* 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 removal 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.


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 (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 related to the directed percolation model.


See also

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Mathematical modelling of infectious disease Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics alo ...
*
Compartmental models in epidemiology Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious ...
*
Epidemic model Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, ...
*
Percolation Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applicatio ...
*
Percolation threshold The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a ...
*
Percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnecte ...
*
2D percolation cluster In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
*
Directed percolation In statistical physics, directed percolation (DP) refers to a class of models that mimic filtering of fluids through porous materials along a given direction, due to the effect of gravity. Varying the microscopic connectivity of the pores, these ...
*
Bootstrap percolation In statistical mechanics, bootstrap percolation is a percolation process in which a random initial configuration of active cells is selected from a lattice or other space, and then cells with few active neighbors are successively removed from the ...
* Biological lattice-gas cellular automaton


References


Further reading

* {{cite book , author = J. Marro and R. Dickman , year = 1999 , title = Nonequilibrium Phase Transition in Lattice Models , publisher = Cambridge University Press , location = Cambridge , isbn = Epidemiology