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A susceptible-infective-recovered (SIR) epidemiological model based on probabilistic cellular automaton (PCA) is employed for simulating the temporal evolution of the registered cases of chickenpox in Arizona, USA, between 1994 and 2004. At each time step, every individual is in one of the states S, I, or R. The parameters of this model are the probabilities of each individual (each cell forming the PCA lattice) passing from a state to another state. Here, the values of these probabilities are identified by using a genetic algorithm. If nonrealistic values are allowed to the parameters, the predictions present better agreement with the historical series than if they are forced to present realistic values. A discussion about how the size of the PCA lattice affects the quality of the model predictions is presented.

In control engineering, it is fundamental to identify the system to be controlled in order
to determine the best strategy for achieving the intended goals. The task of
identifying a system is to determine the best model able of describing the
dynamical behavior of such a system from measured data (e.g., [

Scaling problems are ubiquitous in nature and
prominent in biology. However, such problems are not easily treated, or even
recognized, in a plethora of cases. As put by Haldane in the beginning of the
20th century, “The most obvious differences between different animals are
differences in size, but for some reason the zoologists have paid singularly
little attention to them” (cited in [

There are several approaches to scaling problems, from
dimensional analysis to fractal dimensions (e.g., [

Here, we report a numerical study revealing the
influence of scale on parametric identification of an epidemiological model
used for predicting the temporal evolution of the number of chickenpox cases
registered by the Arizona Department of Health Services [

In Section

In our cellular automaton (CA) model [

The coupling topology influences the dynamical
behavior of biological, electronic, or social networks (e.g., [

With this model, we intend to fit the data shown in
Table

Registered cases of chickenpox in Arizona between 1994 and 2004.

Year | Number of cases (scale 1:1) | Number of cases (scale 1:20) |
---|---|---|

1994 | 6783 | 339 |

1995 | 2658 | 133 |

1996 | 3319 | 166 |

1997 | 1987 | 99 |

1998 | 1673 | 83 |

1999 | 960 | 48 |

2000 | 1522 | 76 |

2001 | 951 | 47 |

2002 | 606 | 30 |

2003 | 1620 | 81 |

2004 | 1091 | 54 |

There have been published several works on
identification of transition rules of CA by using GA (e.g., [

Here each generation of our GA is composed by 15
chromosomes (15-candidate solutions), where the length of the chromosome
depends on the kind and on the radius of the neighborhood. For instance, when
the von Neumann neighborhood of

The fitness

The function

The function

An initial population of chromosomes is randomly
generated, and the value

Crossover is performed by selecting a single point in
two randomly picked chromosomes (the parents) and swapping

Every parent chromosome has

After applying these genetic operations for producing
new chromosomes, the values of

To choose eight chromosomes for the next GA
generation, the pool of parents and children is organized in a crescent order
in terms of

Notice that the value of

Because

About 5 million people live in Arizona. For a lattice
of size

Each time step of the PCA is equivalent to 2 months. Thus, to predict the number of infective individuals from one year to another, the PCA must be iterated by 6 time steps.

At first, we performed simulations considering von
Neumann and Moore neighborhoods of

Historical series (solid line) and
simulations in the spatial scale 1:20 with

In order to obtain realistic values for the state
transition probabilities, the limit

Then, more modifications on the GA were accomplished.
Now each time step of the PCA can correspond to 1 or 2 months (thus, 12 or 6
time steps are necessary to complete 1 year, resp.). And additional limits were
imposed to the following genes:

In the simulations reported above,

Historical series (solid line) and
simulations in the spatial scale 1:5 with

In the scale 1:5, if those additional limits on

We found that the size of the PCA lattice affects the
quality of the predictions of our epidemiological model, after being identified
by using a GA. The best set of state transition probabilities, with

Here, we had to deal with the problem of scaling the lattice of the CA in relation to the number of individuals, healthy and infective ones. The most obvious first approach is to maintain the ratios from the original data, that is, a linear relation of size. In this sense, space is completely embedded in the number of individuals, which, ultimately, becomes the size reference frame. One potential problem in doing this is that the neighborhood (either Moore or von Neumann) maintains, therefore, its absolute size. Let us exemplify this.

Epidemiological models based on differential equations
can be used for describing the spreading of infectious diseases (e.g.,
[

Turner et al. [

DeAngelis and Petersen [

The discussion of system identification in ecological/epidemiological studies and scaling issues is still an open question. Our investigation goes in the same direction as many previous ones (see above), that is, that size matters. We add to the query the problem of interchanging space with individuals (and their neighborhoods). Therefore, if modeling is concerned with real predictions much more than simple data fitting, then a judicious evaluation of the underlying scale should be performed as a first step. It seems that these ecological/epidemiological problems need their particular definitions of size, whether this means space itself or number of individuals or a combination of both. In other words, the geometry for scaling these systems is still awaiting the due attention.

L. H. A. Monteiro is partially supported by CNPq.