Sergio Invernizzi

On lags and chaos in economic dynamic models

Abstract This paper discusses the role of lags in dynamic economic models. Applications of discrete dynamical systems to economics are considered and the shortcomings of their (implicit) treatment of lags are criticized. An abstract, probabilistic view of lags is then provided, within which fixed delay lags are shown to be a special case. A basic equivalence is proved between: (i) a system with an indefinitely large number of agents, reacting to inputs with randomly gamma distributed, discrete lags, and (ii) a system with one single ‘representative‘ agent reacting to inputs with a continuous, multiple exponential lag. Finally, the paper analyses a single-loop feedback system cuopling a multiple exponential lag and a non-linearity of the one-hump type. The dynamic behavior of the system is studied by means of analytical and numerical methods, and the conditions for periodic and chaotic solutions are investigated.

Pharmacokinetics and immunomodulatory effects of phytotherapeutic lozenges (bonbons) with Echinacea purpurea extract.

The relative bioavailability of the major alkamides, dodeca-2E,4E,8Z,10E/Z-tetraenoic acid isobutylamides, from Echinacea purpurea phytotherapeutic lozenges at three different dose levels (0.07, 0.21 and 0.9 mg) was evaluated in a pharmacokinetic study in humans and the possible effects on the immunological system were measured. Alkamides were found to be rapidly absorbed and measurable in plasma 10 min after administration of 0.21 and 0.9 mg lozenges and remained detectable for 3h for the 0.21 mg lozenges and for more then 3h for the 0.9 mg lozenges; 0.07 mg lozenges were measurable 20 min after administration and remained detectable for only 2h after the administration. A significant dose-independent down-regulation of the pro-inflammatory cytokines IL-12p70, IL-8, IL-6, IL-10 and TNF was observed 24h after oral administration. The results of non-compartmental pharmacokinetic analysis revealed that a C(max) of (0.65+/-0.41 ng/ml) was reached at 32 min with the 0.07 mg lozenges, (1.00+/-0.21ng/ml) at 25 min with the 0.21 mg lozenges and (8.88+/-5.89 ng/ml) at 19 with the 0.9mg lozenges. As evidenced by the dose-exposure relationship, no significant departure from dose proportionality was observed, indicating linearity in pharmacokinetics. To get a further insight in pharmacokinetics of dodeca-2E,4E,8Z,10E/Z-tetraenoic isobutylamides a compartmental population pharmacokinetic model was developed applying mixed effect modelling procedure. The results demonstrate that within the dose range studied pharmacokinetics of dodeca-2E,4E,8Z,10E/Z-tetraenoic isobutylamides are linear and that absorption is very rapid (t(1/2)=6 min) with apparently no lag time, thus indicating the possibility that a fraction of the drug is absorbed through the oral mucosa.

Enhanced oral bioavailability of vinpocetine through mechanochemical salt formation: physico-chemical characterization and in vivo studies.

ABSTRACTPurposeEnhancing oral bioavailability of vinpocetine by forming its amorphous citrate salt through a solvent-free mechanochemical process, in presence of micronised crospovidone and citric acid.MethodsThe impact of formulation and process variables (amount of polymer and citric acid, and milling time) on vinpocetine solubilization kinetics from the coground was studied through an experimental design. The best performing samples were characterized by employing a multidisciplinary approach, involving Differential scanning calorimetry, X-ray diffraction, Raman imaging/spectroscopy, X-ray photoelectron spectroscopy, solid-state NMR spectroscopy, porosimetry and in vivo studies on rats to ascertain the salt formation, their solid-state characteristics and oral bioavailability in comparison to vinpocetine citrate salt (Oxopocetine®).ResultsThe analyses attested that the mechanochemical process is a viable way to produce in absence of solvents vinpocetine citrate salt in an amorphous state.ConclusionFrom the in vivo studies on rats the obtained salt was four times more bioavailable than its physical mixture and bioequivalent to the commercial salt produced by conventional synthetic process implying the use of solvent.

Multidisciplinary Approach on Characterizing a Mechanochemically Activated Composite of Vinpocetine and Crospovidone

Significant improvement of solubilization kinetics of poorly soluble vinpocetine was obtained through a mechanochemical activation process in presence of micronized crospovidone. Drug-to-polymer weight ratio and milling time variables resulted to have statistically significant impacts on the activation of the product. The complete amorphization was obtained in the coground with the highest crospovidone contents (>80% wt), milled for the longest time (180 min). An ad hoc software was then used to calculate the dimensions of the drug crystallites in the samples on the basis of the calorimetric data. The thermal analyses were then accompanied and confirmed by an extensive solid-state characterization, performing X-ray diffraction, Raman imaging/spectroscopy, DRIFT, and SS-NMR spectroscopy, followed by laser diffraction and solubilization kinetics tests. All the analyses agreed on attesting the progressive loosing of crystalline structure of the drug when increasing milling time and amount of polymer in the formulations. This activated status of the drug, which resulted to be homogeneously distributed on the coground sample and stable for at least 1 year, was reflected on favorable solubilization kinetics. The in vivo studies on rats revealed that coground systems promoted a fivefold higher oral bioavailability enhancement in comparison to a commercial formulation (Vimpocetin 5mg Capsules, Pharma).

Periodic Solutions for Systems of Forced Coupled Pendulum-Like Equations

Abstract The existence of periodic solutions for systems of forced pendulum-like equations was studied in the papers by J. A. Marlin ( Internat. J. Nonlinear Mech. 3 (1968), 439–447) and J. Mawhin ( Internat. J. Nonlinear Mech. 5 (1970), 335–339). In both works some symmetry hypotheses on the forcing terms were considered. This paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms. Note that as a model example it is possible to consider the motion of N coupled pendulums (see the already mentioned paper by J. A. Marlin) or the oscillations of an N -coupled point Josephson junction with external time-dependent disturbances studied in the autonomous case by M. Levi, F. C. Hoppensteadt, and W. L. Miranker ( Quart. Appl. Math. 36 (1978), 167–198).

Mechanochemically induced disordered structures of vincamine: The different mediation of two cross-linked polymers

The aims of this research were to prepare highly bioavailable binary cogrounds (vincamine-AcDiSol(®) or PVP-Cl) by means of a mechanochemical process and to study the mediation of each polymer in the induction of physical transformations of the drug. From a set of fifteen cogrounds for each crosslinked polymer, two samples were selected in each group on the basis of the AUC of in vitro dissolution profiles with the help of a statistical comparison. The chosen samples were analysed by means of TEM, XRPD, Raman-spectroscopy/imaging, SSNMR, also including the study of (1)H spin-lattice relaxation times. The research encompassed in vivo oral absorption studies in rats, pharmacokinetic analysis and physical stability studies during 1 year. An intimate drug-polymer mixing was found in the coground samples with domain average dimensions smaller than 100 Å; this reflected in a remarkable enhancement of the in vitro and in vivo bioavailability. Different disordered states were detected in the coground samples as a function of cogrinding time and the type and amount of polymer used. Though both crosslinked polymers produced a remarkable enhancement of the oral bioavailability, coground systems based on AcDiSol(®) are preferable in terms of pharmacokinetic performance and physical stability.

A generalized logistic model for photosynthetic growth

Abstract This paper presents a model for the photosynthetic growth of C 3 green plants, consisting of a 1-periodic scalar first order nonautonomous ordinary differential equation, of the generalized logistic type. The model is tested both mathematically and experimentally. For a very large biologically significant range of the parameter values, the equation has a repelling trivial solution and a unique positive one-year-periodic solution, which is globally attractive from all positive initial data. Biomass production predicted by the model matches field data.

Immunomodulation mediated by a herbal syrup containing a standardized Echinacea root extract: A pilot study in healthy human subjects on cytokine gene expression

In this study, the immunomodulatory effect of a triply standardized Echinacea angustifolia root extract (Polinacea(®)) was evaluated in 10 healthy subjects. Ten ml of syrup containing one hundred mg of extract (corresponding to 4.7 mg of Echinacoside and 8.0mg of a high molecular weight-20,000 Da- polysaccharide) were administered as a herbal syrup once a day for one month. The immunomodulatory effect was evaluated before and after herbal syrup administration evaluating the expression levels of the cytokines IL-2, IL-8, IL-6 and TNF-α. Cytokine expression was studied in lympho-monocytes and in plasma samples measuring the mRNA and protein levels, respectively. The results were analysed by ANOVA and non-parametric Friedman rank sum tests; when possible it was adopted a pair-wise comparisons at different post-treatment times, using the paired t-tests with Holm correction. The correlation between the variations of cytokine plasma levels and the respective mRNA was carried out using a linear regression model. In lympho-monocytes our data indicate the up-regulation of the mRNA levels of IL-2 and IL-8 and the down regulation of the mRNA levels of the pro-inflammatory cytokines TNF-α and IL6. The differential regulation was maximal after 14 days of treatment. IL-2 up-regulation and IL-6 down-regulation were also confirmed at the protein level in plasma. Finally, the up-regulation of the mRNA of IL-2/IL-8 and the down-regulation of IL-6 positively correlated with the protein levels detected in the plasma. In conclusion, this pilot study suggests a relevant role for the standardized Echinacea angustifolia root extract in the control of cytokine expression. This first demonstration of the immuno-modulating activity of Echinacea angustifolia root extract in the healthy subject, supports at least in part the common use of such products as health promoting supplement.

Quantitative analysis of the Hopf bifurcation in the Goodwin n-dimensional metabolic control system

We study, from a quantitative point of view, the Hopf bifurcation in an ODE model of feedback control type introduced by Goodwin (1963) to describe the dynamics of end-product inhibition of gene activity. We formally prove that the exchange of linear stability of the positive equilibrium in the n-dimensional Goodwin system with equal reaction constants coexists with a Hopf bifurcation of nontrivial periodic solutions emanating from this equilibrium, without any further restriction on the dimension n ⩾ 3 or on the Hill coefficient ϱ. The direction of the bifurcation, and the stability and the period of the bifurcating orbits are estimated by means of the algorithm proposed by Hassard et al. (1981).

Periodic Solutions of Functional-Differential Systems with Sublinear Nonlinearities

(a) if H and K are defined by (Hx)(t) = x(t t), (Kx’)(t) = x’(t a), our theorem gives the existence of a T-periodic solution of a differential delay equation of Lienard type. As corollaries, we obtain full extensions of the results of Martelli and Schuur [8] and Mawhin [9], not only because of considering delays, but also because of enlarging the class of the maps g which can be considered. (b) suppose that D is a (possibly nonlinear) differential operator which admits an inverse H. Then, the system