Nonlinear Analysis of Guillain Barré Time Series to Elucidate Its Epidemiology

The etiology of Guillain Barre Syndrome (GBS) is not fully clarified, and there is a lack of agreement concerning its putative epidemic character. The low incidence rate of this disease is a disadvantage for employing the traditional statistical methods used in the analysis of epidemics. The objective of this paper is to clarify the GBS epidemic behavior applying a nonlinear time series identification approach. The authors obtained one time series of GBS and nine series of classical infectious epidemics (5 national and 4 international). These data were processed with advanced techniques of statistical time series analysis. This paper shows that GBS behaves similar to the other time series of classical epidemic studied. It corresponds to a nonlinear dynamics, with a point attractor. The spectral analysis pointed to an annual periodicity, and preference for the warmest month of the year was found. These results might suggest that Guillain Barre Syndrome has an epidemic behavior. The adequacy of nonlinear methods for analyzing the dynamics of epidemics, particularly those with low incidence rate, such as GBS was revealed.


Introduction
e Guillain Barré Syndrome is an acute autoimmune neuropathy. It's epidemiology is a broadly approached topic in the literature, but there are still aspects where opinions diverge or are even diametrically opposed [1]. e epidemiological behavior of GBS is one of the most controversial issues. Some authors report outbreaks or "striking �uctuations" in the incidence and con�rm that it behaves periodically and shows seasonal variation [2][3][4][5][6][7][8][9][10][11]. e occurrence of outbreaks and the presence of periodicity and seasonal preference of the incidence are aspects that suggest an epidemic behavior. Nevertheless, some authors have found that incidence of this syndrome is stable throughout the year and even for longer periods of observation and deny any seasonal variation or periodicity [12][13][14][15][16][17]. e endemic channel and other statistical tools for epidemiological surveillance show limitations when the epidemic analyzed has a low level of endemicity or when �uctuations of the incidence, which could be considered as outbreaks are not numerically large, as occuring in GBS. Some models have been proposed to study the dynamics of epidemics, but these techniques cannot adequately describe behavior of a series with more complex dynamics [18][19][20][21]. e modeling of "biological systems" through nonlinear mathematics ("Chaos theory") has proven to be useful in the understanding of complexity and especially in epidemics [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. However, we have not found in the literature reports which study the behavior of GBS applying a nonlinear identi�cation approach. In this paper nonlinear mathematic tools were applied, for the �rst time, to analyze GBS time series data. In particular, we have addressed the question whether GBS behaves like an epidemic. e authors also analyzed other nine time series of classical infectious epidemics to compare and determine possible causal links between these ones and GBS. . Five time series of national epidemics were also constructed: acute diarrheal diseases (ADD), acute respiratory diseases (ARD), typhoid fever, and mononucleosis and scarlet fever (series available in a sufficient number of observations). is information was provided by the Statistics Department of the Ministry of Public Health. Additionally four time series of international epidemics were also included. e series of measles in New York and Baltimore and mumps in USA were provided by omas Schriber MD from Whupertal University (Germany). e series of cholera in the city of Calcutta was taken from the classical book "Health Statistics" [37].

Data
Analysis. e databases were created as open office spreadsheets and saved as ASCII �les. Advanced statistical techniques of time series analysis were applied and also systems for nonlinear time series analysis developed at the Cybernetics Centre Applied to Medicine.

Method Used to Determine Whether GBS Was a Random
Independent Process. Data were represented as an intervals' histogram between successive events and were plotted in semilogarithmic coordinates. In this analysis, if dependence is linear (exponential distribution), a random mechanism is suggested.

2.2.�. �inear Identi�cation Method.
Linear correlation between the present and the past values, based on the estimation of the lineal correlation coefficient in a multiple regression, model was employed.

2.2.�. �onlinear Identi�cation Method.
e time series is considered as dynamically emerging from a system of the type is a smooth map which has been designated as the skeleton of the underlying dynamical system [35]; { , = , , … , is a sequence of dynamical noise such that is independent of for ; and m is a positive integer, so-called lag. In our case, "x" refers to time intervals elapsed between two successive appearances of cases.
In the most general case, when very few assumptions are made about f (smooth, continuous, stationary) it is convenient to estimate the nonlinear autoregressive function f via a Naradaya-Watson nonparametric kernel [35]. In kernel nonparametric estimation, the estimate of f in (1) at a point ( , , … , ) of the state space is obtained as a weighted average of all the observed data ( , , … , ). Speci�cally: , , … , = ∑ = + ∏ = /ℎ Here, ( /ℎ) is the kernel function and the parameter ℎ is the bandwidth parameter. In our implementation, ℎ is selected via cross validation error minimization.
In this study the following exponential kernel was selected: ( ) = ( / ) . To �nd the bandwidth parameter ℎ, a cross validation one-step-ahead criterion was applied.

Viewing Attractors.
Once the function "f " was estimated, a noise-free realization was obtained by recursively calculating the next value of "f " from the past. As initial values a real segment of the signal was taken. Aer several hundreds of recursive estimations a series was generated that, by construction, contained no stochastic in�uences and allowed to detect the presence of a nonpoint attractor. In such cases, the series was considered as nonlinear, but in the cases in which there was a point attractor (such as our series of GBS), a complementary approach to demonstrate nonlinearity was necessary. In these situations, we estimated correlation dimension.

Correlation
Dimension. e correlation dimension corresponding to the original series was estimated as well as to one generated from a linear autoregressive model estimated from the same original data. A nonlinearity criterion was considered if the value of the dimension of the original series was less than the surrogate one.

Programs.
For the nonparametric estimation a program designed in SciLab by the bioinformatics group of CECAM was developed. e SciLab system was also used for plotting of original and generated series. To calculate the correlation dimension the MTRCHAOS program, developed by Michael T. Rosenstein in 1993, was used.

To De�ne If GBS Beha�ed as a Random or Deterministic
System. Graphic Representation of the GBS Time's Series. e time series of monthly incidence of the disease is represented in Figure 1. Simple observation of the data might suggest that GBS has an epidemic behavior showing spikes in certain months, where the incidence is higher (It could correspond to outbreaks). On the other hand, in another month there were no cases. �eri�cation of peaks in time's series of typical epidemics is a graphic evidence of the irrefutable existence of an outbreak, but in the case of GBS, where the daily and outbreak incidence is very low, we found this criterion unreliable to assert the epidemic behavior of any disease.  Figure 2 that shows the graphical representation of an arti�cial series obtained randomizing the original one. is technique makes random and original series comparable. We can see that in the random series there are also peaks that could suggest the presence of outbreaks; even in the spectrum of both series there were no great differences, but a closer look at both �gures shows that in the surrogate series the values are regularly distributed throughout each month, while in the original one there are periods of several months where there are no cases (arrows) and others where the incidence is more constant. is aspect might suggest that this is not a random process and has a structure or laws that govern the behavior, but this small detail is not enough.
Testing the GBS Time Series for Randomness. A histogram was constructed for the distribution of intervals between two successive admissions. e x-axis contains the logarithms of the intervals and the y-axis contains the frequency of appearance of an interval with this size or higher ( Figure  3). e graphic reveals that the data are not distributed exponentially, the dependence is not linear, and therefore it is not a random process. is is the �rst evidence of structure in this data; it behaves (at least partly) as a deterministic system. e next step was then to determine whether the system is linear or nonlinear.

Testing the GBS Time Series for Linearity. Linear Correlation in GBS.
e correlation of GBS with their previous values was estimated. e linear autoregressive model had a correlation of 0.45, which was signi�cant for the number of cases studied ( ). us, we concluded that GBS presents correlations with the past. 8.087. It supports the presence of a nonlinear structure in the GBSs' time series. Correlation dimension is one of the fundamental concepts of nonlinear dynamics and "chaos theory". In a space reconstruction following the Takens algorithm, correlation dimension appears as one of the dynamical systems invariants. In general it is regarded as a measure of complexity, since it is related to the minimal number of variables required to describe a given system [38].
Estimation of the Attractor. e nonlinear identi�cation method showed the presence of a point attractor. So far, we could identify that GBS behaves as a deterministic, nonlinear system with a point attractor (Figure 4).
According to Philippe, the estimation of the attractor and the spectral analysis are useful tools to detect chaos in time series data [26]. In this paper we tested the usefulness of this method to determine the existence of a deterministic component.
is analysis supports that GBS behaves mathematically as a nonlinear deterministic system. is kind of behavior was also found in other typical infectious epidemics studied

Detection of Periodicity in the Time Series of GBS.
e power spectrum of the signal reveals two peaks corresponding to 11 and 14 months. We assume that these two local peaks are indeed part of a common peak at 12 months period (annual periodicity). e fact is that many epidemics have a year cycles and curiously GBS seems to share also this property of classical epidemics ( Figure 5).
Possible Dependence of the Periodicity with the Seasons. Data were averaged by month (for 30 years), and it was observed that in August incidence was almost doubled ( Figure 6). e incidence in our study varies throughout the year but with greater preference for the month of August (one of the warmest months of the year). is is possibly related to environmental factors. e seasonal variation in the incidence in GBS has been analyzed by several researchers [1,2,12,13,16,[39][40][41].

Dynamics of Typical
Epidemics. e series of mumps in the USA can be modeled as a nonlinear stochastic system with a chaotic attractor. e measles series in Baltimore and New York have a chaotic attractor too. e series of cholera in Calcutta can be modeled as a limit cycle perturbed by noise. e national Cuban series of acute diarrheal diseases (ADD) has a change of dynamics in the vicinity of the month 150. e �rst part of the series shows a chaotic dynamics and the second part of the series, a limit cycle. e national series of acute respiratory disease (ARD) corresponds to a stochastic system with a point attractor. National series of typhoid fever shows a point attractor. National series of mononucleosis and scarlet fever were associated with a point attractor perturbed by noise.

Conclusion
e random nature of GBS was discarded. It can be modeled as a nonlinear deterministic system with a point attractor. is behavior is frequently observed in other typical epidemics. ere is an annual periodicity in the GBS series with greater incidence in August.
In this work we applied a nontraditional mathematical method, to clarify some aspects of GBS in which some researchers disagree. Apparently, this is the �rst attempt in the literature to apply a nonlinear identi�cation technique to a broad group of epidemics datasets and in particular the GBS. We consider that this approach adds additional evidence in favor of the epidemic behavior of this polemic syndrome.