Keywords: cellular automata, epidemic spreading, hexagonal cellulal space, mathematical modelling, rectangular cellular space, SIR model.

The main goal of this work is to introduce a new mathematical model based on the use of cellular automaton to simulate epidemic spreading. Specifically, it is a SIR-type model, that is, the population is divided into three classes: The susceptible individuals (S) are those capable to contracting the disease; the infected individuals (I) are those capable of spreading the disease; and the recovered individuals () are those immune from the disease, either died from the disease, or, having recovered, are definitely immune to it. Moreover, some assumptions will be taking into account in our model: (1) The disease is transmitted by contact between an infected individual and a susceptible individual. (2) There is no latent period for the disease, hence the disease is transmitted instantaneously upon contact. (3) All susceptible individuals are equally susceptible and all infected individuals are equally infectious. (4) The population under consideration is fixed in size. This means that no births or migration occurs, and all deaths are taken into account.

Cellular automata (CA for short) are simple models of computation capable to simulate complex physical, biological or environmental phenomena. Roughly speaking, a two-dimensional CA is formed by a two-dimensional array of identical objects called cells, which are endowed with a state that change in discrete steps of time according to a specific rule. As the CA evolves, the updated function (whose variables are the states of the neighbors) determines how local interactions can influence the global behaviour of the system.

The main features of the model are the following:

• Each cell stands for a regular area of the land in which the epidemic is spreading; we will consider square cells for rectangular cellular spaces and hexagonal cells for hexagonal cellular spaces.
• It is assumed that the population distribution is homogeneous, i.e., all cells have the same population, at every step of time.
• The state of each cell (a,b), which is defined as , at each time step is obtained from the fraction of the number of individuals of the cell which are susceptible, infected and recovered from the epidemic, that is, it is a suitable discretization of the following vector:

  (65)

  
  
  where N is the total population of the cell (a,b). Consequently,
  
  where is the portion of susceptible individuals, is the portion of infected individuals, and is the portion of recovered individuals. As the three coordenates of the vector given in (65) are real numbers and the state set must be finite, we must discretize such values in order to obtain an element of , where is the following set with elements:
  

• The states of the CA change according to the following local transition function:
  
  
  where is the following discretization function:
  
  Moreover, the function stands for the recovering process of the infected cells. The parameter stands for the virulence of the epidemic. Finally, where is the connection factor, and is the movement factor.

Rectangular and hexagonal cellular spaces are considered with different types of neighborhoods: Von Neumann neighborhood, Moore neighborhood, hexagonal Moore neighborhood and extended Moore neighbordhood.

It is shown that the simulations obtained are more accute when hexagonal cellular space is considered. Moreover, the laboratory simulations obtained seem to be in agreement with the expected behaviour of a real epidemic.

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