Inverse Power Law Function Research Articles

SCALING PROPERTIES OF EPIDEMIOLOGICAL TIME SERIES

The statistical analysis monthly Hungarian (1967-2001) and Austrian (1989-2003) rabies and Hungarian bovine tuberculosis (1950-1979) cases were statistically analyzed. There were eradication campaigns against both diseases, which gave us the unique opportunity to observe how the statistical properties of the time series changed due to the eradication program. The fluctuations around the trend were got by removing the 12 month moving averages from the time series. In order to characterize the structure of the contacts among the individuals exposed to the diseases and the underlying processes governing the behavior of the epidemics, the fluctuations were analyzed before and after eradication separately. It turned out that the tails of the complementary cumulative distribution functions differ in the rabies cases, and do not differ in the case of tuberculosis. In each case the tail of the distribution follows an Inverse Power Law (IPL) function and describes the distribution of extreme events. It is possible to make conclusions about the dimensions of the processes from the fitted exponents. These dimensions are not related necessarily to the spatial dimensions, but to the possible connections. The knowledge about the distributions give us the opportunity to asses the risk of epidemic outbreaks.

Analysis of the effect of immunization in rabies time series

The authors analysed time series of rabies cases diagnosed in Hungary between 1967 and 2001. In Transdanubia (West Hungary), an oral immunization program started in 1992 and in East Hungary in 2001. Both long term and seasonal trends were identified in the time series of rabies cases. In order to characterize the underlying processes governing the behaviour of the epidemic, the fluctuations around the trend were analysed before and after immunization separately. It turned out that the tail of the complementary cumulative distribution functions differ. The tail of the distribution follows an inverse power law (IPL) function and describes the distribution of extreme events. The significant difference in the IPL exponents before and after immunization can be explained by the theory of Highly Optimized Tolerance (HOT).

Modelling of metastatic cure after radionuclide therapy: influence of tumor distribution, cross-irradiation, and variable activity concentration.

The objective was to study the influence of tumor number and size, cross-irradiation from normal tissue, and of variable activity concentration on metastatic cure after radionuclide therapy. A model to calculate the metastatic cure probability (MCP) was developed, in which it was assumed that the tumor response was an exponential function of the absorbed dose. All calculations were performed for monoenergetic electron emitters with different energies (10-1000 keV). The influence of tumor size and number of tumors were investigated with different log uniform distributions; the basic tumor distribution consisted of tumors with 1, 10, ..., 10(11) cells. The influence of cross-irradiation was assessed by calculating MCP for various tumor-to-normal tissue activity concentration ratios (TNC). The influence of variable activity concentration between tumors was calculated by assuming that the activity concentration in tumors was an inverse power law function of tumor mass. The required activity concentration (C0.9) and absorbed dose (D0.9) to obtain MCP=0.9 was calculated in the different models. The C0.9 and D0.9 needed to obtain MCP were very high; more than 25 MBq/g and 80 Gy, respectively. The lowest C0.9 and D0.9 for equal activity concentration in the different tumor sizes were obtained for electron energies less than 80 keV. For higher energies the low absorbed energy fraction in small tumors will increase the required C0.9 and D0.9 markedly. Cross-irradiation from normal cells surrounding the tumor will cause sterilization of the smallest tumors and decrease the required C0.9 and D0.9 for higher electron energies. Assuming that the activity concentration decreased with increased tumor mass caused a marked increase in C0.9 and D0.9 in favor of higher electron energies. With the MCP model we demonstrated significant influence of the number of tumors, their size, TNC and variable activity concentration on MCP. The results are valuable when evaluating optimal choices for radionuclides for internal-emitter therapy.

An out-of-equilibrium model of the distributions of wealth

The distribution of wealth among the members of a society is herein assumed to result from two fundamental mechanisms: trade and investment. An empirical distribution of wealth shows an abrupt change between the low–medium range, that may be fitted by a non-monotonic function with an exponential-like tail such as a gamma distribution, and the high wealth range, that is well fitted by a Pareto or inverse power-law function. We demonstrate that an appropriate trade-investment model, depending on three adjustable parameters associated with the total wealth of a society, a social differentiation among agents, and economic volatility referred to as investment, can successfully reproduce the distribution of empirical wealth data in the low, medium and high ranges. Finally, we provide an economic interpretation of the mechanisms in the model and, in particular, we discuss the difference between classical and neoclassical theories regarding the concepts of value and price. We consider the importance that out-of-equilibrium trade transactions, where the prices differ from values, have in real economic societies.

EVIDENCE OF "ESSENTIAL UNCERTAINTY" IN EMERGENCY-WARD LENGTH OF STAY

Scaling in random time series is a novel way of gaining insight into mechanisms of complex systems. Scaling parameters characterize searches for the presence of "essential" uncertainty in outcomes of complex systems. The presence of this type of uncertainty in the emergency system (ES) can point to unusual ways of relieving the financial burden to the health system and improve the public health. The objective of this paper is to test for the presence of "essential" uncertainty in an outcome variable of the ES, i.e. emergency-ward length of stay (EWLS). An inverse power law (IPL) function fit to data can assess "essential" uncertainty. The study of EWLS has been undertaken in ten large Quebec hospitals. We find that all hospital EWLS are well fit by an IPL, as determined by an aggregated allometric relation. The presence of "essential" uncertainty is further confirmed by stratifying the EWLS according to proxies of disease complexity and severity at entry in emergency ward. Results point out that the various hospital dynamic systems embed "essential" uncertainty to various degrees. We conclude that intervention to reduce hospital health care costs must be centered on the interaction and feedback characterizing the ES processing of the patients.

Molecular evolution modeled as a fractal statistical process

Modeling the rate of nucleotide substitutions in DNA as a dichotomous stochastic process with an inverse power-law correlation function describes evolution by a fractal stochastic process (FSP). This FSP model agrees with recent findings on the relationship between the variance and mean number of synonymous and nonsynonymous substitutions in 49 different genes in mammals, that being a power-law increase in the ratio of the variance to the mean, the index of dispersion, with the number of substitutions in a protein. The probability of a given number of substitutions occuring in a time t is determined by a fractional diffusion equation whose solution is a truncated Lévy distribution implying that evolution is a Lévy process in time and yields the same functional behavior for the variance in the number of substitutions as does the FSP model. In addition to obtaining these relationships, the FSP model implies lognormal statistics for the index of dispersion as a function of the mean number of substitutions in a protein, which is confirmed in the regression of the FSP model to data. Lognormal statistics suggest that molecular evolution can be viewed as a multiplicative stochastic process, rather than the linear additive processes of Darwinian selection and drift.

Random texts exhibit Zipf's-law-like word frequency distribution

It is shown that the distribution of word frequencies for randomly generated texts is very similar to Zipf's law observed in natural languages such as English. The facts that the frequency of occurrence of a word is almost an inverse power law function of its rank and the exponent of this inverse power law is very close to 1 are largely due to the transformation from the word's length to its rank, which stretches an exponential function to a power law function.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Background modeling alternatives in EELS and their implications on microanalysis

With the advent of parallel detectors, electron energy-loss spectroscopy (EELS) is expected to be employed increasingly in the routine microanalysis of light elements. The quantitation formulae that are used are relatively simple and straightforward and require only a measurement of the integrated core-loss intensity over a particular energy window beyond the ionization edge (Δi) and most often, a calculated ionization cross-section. Even though the hydrogenic cross-sections that are used routinely for microanalysis can be a source of error, it is observed that the procedure that largely determines the accuracy of quantification is the removal of the contribution of the background below the ionization edge. Based on empirical observations, an inverse power law function of the form I = AE-r, where the exponent ‘r’ takes values from 2 to 6, is now commonly used for the background. A background fitting region preceding the ionization edge (Δb) is chosen, the constants A and r determined by least squares refinement and the background extrapolated beyond the ionization edge for the required energy window (Δi).