Keywords: epidemiology, susceptible-exposed-infected-recovered, mathematical modelling, cellular automata.

Infectious diseases have been a human enemy from time immemorial. Currently, they are events of concern and interest to many people worldwide: remember for example the 2002 outbreak of SARS, or more recently, the 2009 H1N1 flu. Epidemics and pandemics can place sudden and intense demands on health systems and they can disrupt economic activity and development.

As a consequence, the importance of understanding the dynamics and evolution of infectious diseases is steadily increasing in the contemporary world. In this sense, the study, design and analysis of mathematical models to simulate an epidemic spreading is a very important challenge.

Several mathematical models have been appeared in the literature. The majority of such models are based on the use of differential equations and the paradigm stated by Kermack and McKendrick in 1927 [1]. Unfortunately, these models exhibit some important drawbacks since they do not take into account spatial factors such as population density, they neglect the local character of the spreading process, they do not include variable susceptibility of individuals, they cannot comprehensively depict complex contagion patterns (which are mostly caused by the human interaction induced by modern transportation, etc.) More suitable mathematical tools used to simulate the spread of an epidemic disease are a particular type of finite state machine called cellular automata [2].

In this work a new mathematical epidemiological model based on cellular automata of graphs is introduced. Specifically, it is a SEIR (susceptible-exposed-infected-recovered) model where each node of the graph defining the topology of the cellular automata stands for an only one individual. Moreover, the edges of the graph represent the physical contact between two individuals/nodes. As these physical contacts can change with time, the graph also changes.

Some epidemiological periods are taken into account: incubation period, latent period and infection period. The basic point of the model lies in the transition function that governs the transition from the susceptible state to the exposed state. In the model proposed in this work, this function depends on some parameters of the individuals and the epidemic disease: immune state, preventive treatment, risk behaviours, virulence of the disease, the transmission capacity, etc.

1

O. Diekmann, J.A.P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation", Wiley, 2000.

2

S. Wolfram, "A New Kind of Science", Wolfram Media Inc., 2002.

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