Acta Biotheoretica

We prove almost sure exponential stability for the disease-free equilibrium of a stochastic differential equations model of an SIR epidemic with vaccination. The model allows for vertical transmission. The stochastic perturbation is associated with the force of infection and is such that the total population size remains constant in time. We prove almost sure positivity of solutions. The main result concerns especially the smaller values of the diffusion parameter, and describes the stability in terms of an analogue $$\mathcal{R}_\sigma$$ of the basic reproduction number $$\mathcal{R}_0$$ of the underlying deterministic model, with $$\mathcal{R}_\sigma \le \mathcal{R}_0$$ . We prove that the disease-free equilibrium is almost sure exponentially stable if $$\mathcal{R}_\sigma <1$$ .

We extend recent information-theoretic phase transition approaches to evolutionary and cognitive process via the Rate Distortion and Joint Asymptotic Equipartition Theorems, in the circumstance of interaction with a highly structured environment. This suggests that learning plateaus in cognitive systems and punctuated equilibria in evolutionary process are formally analogous, even though evolution is not cognitive. Extending arguments by Adami et al. (2000), we argue that 'adaptation' is the process by which a distorted genetic image of a coherently structured environment is imposed upon a species.

Morphogenesis is a general concept in biology including all the processes which generate tissue shapes and cellular organizations in a living organism. Many hybrid formalizations (i.e., with both discrete and continuous parts) have been proposed for modelling morphogenesis in embryonic or adult animals, like gastrulation. We propose first to study the ventral furrow invagination as the initial step of gastrulation, early stage of embryogenesis. We focus on the study of the connection between the apical constriction of the ventral cells and the initiation of the invagination. For that, we have created a 3D biomechanical model of the embryo of the Drosophila melanogaster based on the finite element method. Each cell is modelled by an elastic hexahedron contour and is firmly attached to its neighbouring cells. A uniform initial distribution of elastic and contractile forces is applied to cells along the model. Numerical simulations show that invagination starts at ventral curved extremities of the embryo and then propagates to the ventral medial layer. Then, this observation already made in some experiments can be attributed uniquely to the specific shape of the embryo and we provide mechanical evidence to support it. Results of the simulations of the “pill-shaped” geometry of the Drosophila melanogaster embryo are compared with those of a spherical geometry corresponding to the Xenopus lævis embryo. Eventually, we propose to study the influence of cell proliferation on the end of the process of invagination represented by the closure of the ventral furrow.

Les emplois du mot fonction peuvent se grouper en: Tous dérivent d'un emploi unique: Fonction d'un instrument vu comme le controle d'une bifurcation.

Certain sections ofJosiah Willard Gibbs's thermodynamics papers might be applicable to biological equilibrium and growth, normal or abnormal.Gibbs added terms⌆ Μ i dm i to the differential of the internal energy dε=tdη−pdΝ, (t=temperature,p=pressure,η=entropy,Ν=volume) where $$\mu _i = \frac{{\delta \varepsilon }}{{\delta m_i }}$$ is the potential of substancem i , to provide for chemical as well as thermal and mechanical equilibrium. In this article a further generalization is suggested, to include biological equilibrium by adding to de terms of the form GdN, the variableN being the number of cells, where $$G = \frac{{\delta \varepsilon }}{{\delta N}}$$ is a “growth potential” that measures exactly the resistance toward spontaneous growth. The functionG, likeΜ i is intensive in nature (i.e. depends on intensive variables only) except for a conversion factor $$\frac{{dM}}{{dN}}$$ ,M=⌆m i , affording possible insight into why incipient abnormal growth is often independent of the number of cells. Useful applications might follow from identities between $$\frac{{\delta G}}{{\delta \eta }},\frac{{\delta G}}{{\delta v}}$$ , or $$\frac{{\delta G}}{{\delta m_i }}$$ and $$\frac{{\delta t}}{{\delta N}}, - \frac{{\delta p}}{{\delta N}}$$ or $$\frac{{\delta \mu _i }}{{\delta N}}$$ respectively. The following new function is studied, $$\bar \zeta = \zeta - GN$$ , a natural generalization of theGibbs free energy function ζ, the possibility of measuring it electrically, and comparison of its role with that of ζ for the possible experimental determination ofG. Gibbs's necessary and sufficient conditions for heterogeneous equilibrium ofn components inm phases are generalized and also modified to include broader restraining conditions like $$\mathop \sum \limits_{i = 1}^m \delta N_j (i) \geqslant o$$ ,j=1,f,n, the > being characteristic of only living cellular phases. Careful appraisal of the term “biological stability” is followed by new criteria for stability, instability, and limits of stability, (neutral equilibrium) in terms of derivatives ofG, with possible medical applications. Three different sections of Gibbs's works tend to indicate that, for a biological phase, lower pressure usually increases its stability. The equation $$p'' - p' = \sigma \left( {\frac{I}{r} + \frac{I}{{r'}}} \right)$$ , where σ=surface tension,p′, p′ = pressures,r, r′=radii of curvature, is applied to possible control of tissue growth at interfaces. Methods of altering the equilibrum between three phasesA, B, C by varying the interfacial tensionsσ AB ,σ BC ,σ AC , using relations like AB <σ AC + BC for stability of theA, B interface, suggest different means for shifting biological equilibrium between normal and abnormal cells through the introduction of new third phases at the interface. Various devices are mentioned for possible control of growth through proper channeling of surface or other equivalent forms of energy.

Parameters of a Bertalanffy type of temperature dependent growth model are fitted using data from a population of stone loach (Barbatula barbatula). Over two periods respectively in 1990 and 2010 length data of this population has been collected at a lowland stream in the central part of the Netherlands. The estimation of the maximum length of a fully grown individual is given special attention because it is in fact found as the result of an extrapolation over a large interval of the entire lifetime. It is concluded that this parameter should not at forehand be set at one fixed value for the population at that location due to varying conditions over the years.

Author continues the publication which appeared in the Acta Biotheoretica I, p. 113–132, regarding his results obtained in course of research work on superior plants:Picea excelsa trees, and furthermore on unicellular living beings, namely yeast cells (Saccharomyces spec). Author made a pure culture with the unicellular culture method, and by occasional inoculation produced successors therefrom. He established the progress in development by measuring, according to weight, the CO2 which arose in course of life. The ontogenetic course of development of the original culture as well as that of the successors took the form ofS but theseS curves were not equally precipitous (Fig. 1). When he drew theS-formed development curves in the measure of their time of inoculation in a rectangular co-ordinate system, he received a wave-surface (Fig. 2). When he intersected the wave-surface with the abscissa and the plane parallel with the vertical axis, wave-like lines were the result, which resembled vibratory motion evolving around an axis producing a regular picture (Fig. 3). Research has ascertained that the axis follows the laws of aperiodic vibratory motion, the undulating curve corresponds with the phenomenon of periodic vibratory motion, both of which are derived from the common differential equationd 2 s/dt 2 + 2r ds/dt +w 2 s=o. Any point of thes h axis following the aperiodic vibratory motion is given by the following equation: (8a) $$s_h = a_1 e^{ - r_1 t} \cdot \frac{{e^{ + t\sqrt {r_1 ^2 - w_1 ^2 } } - e^{ - t\sqrt {r_1 ^2 - w_1 ^2 } } }}{2} \cdot \cdot \cdot \cdot \cdot $$ while the wave-curve showing thes h periodic movement is given by the following equation: (8b) $$s_p = a_2 e^{ - r_2 t} \cos (t\sqrt {w_2 ^2 - r_2 ^2 } ) \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot $$ The completey movement is made up of the total of the two vibratory movementsy=s h +s p . The research for the time being refers merely to the aperiodic axis. Calculations show that with thes h equation, the aperiodic axes belonging to all the sections (Fig. 4) can be followed exactly, as the time function oft, and moreover, the change according to time in the size of the w1, r1, a1 coefficients also shows definite regularity (Figs 5, 6, 7). Author deals separately with the calculation of the axis maximal time =t max and the maximal value = s h max of the wave-curve, and establishes that the development speed of every cell and every organism built up from cells has a maximal point of time and a maximal value. Thet maximum point of time: (24) $$t_{max} = \frac{I}{{\sqrt {r_{1^2 } - w_{1^2 } } \log e}}\log \sqrt {\frac{{r + \sqrt {r_{1^2 } - w_{1^2 } } }}{{r - \sqrt {r_{1^2 } - w_{1^2 } } }}} \cdot \cdot \cdot \cdot $$ If we put thet max value in the place of the original (8a) equation of the axis we get the numeric value of thet max . Author finally establishes that the recognition of the fact that a harmonic vibratory motion plays a part in the evolution of living beings and that the axis of this complicated vibration follows the equation of aperiodic vibratory motion is a fundamental result in the theoretic study of the phenomena of life, which, besides, has also a great practical importance in the sphere of the study of improvement, inheritance and the biology of the species. L'auteur continue la publication des résultats de ses recherches faites sur des êtres vivants unicellulaires, notamment desSaccharomyces. Cette publication fut communiquée dans les Acta Biotheoretica I, p. 113–132. Par la méthode de la culture unicellulaire, l'auteur a créé une culture pure et, de celli-ci, au moyen d'inoculations périodiques, il a créé des descendants. Il a mesuré la marche du développement en pesant le CO2 produit au cours de la vie. La courbe d'évolution ontogénétique, tant de la culture-mère que des descendants, a présenté une ligne enS, mais chacun de cesS ne montait pas de la même façon (Fig. 1). En représentant ces courbes d'évolution enS en fonction du temps et à l'échelle des périodes d'inoculation sur des coordonnées rectilignes, il obtenait une surface d'ondes (Fig. 2). En coupant cette surface par un plan parallèle à l'abscisse et à l'axe vertical, il obtenait des lignes ondoyantes qui ressemblaient à un mouvement oscillatoire se formant autour d'un axe à l'image régulière (Fig. 3). Les examens ont révélé que l'axe suit les lois du mouvement vibratoire harmonique apériodique et que la courbe ondoyante répond au phénomène du mouvement vibratoire harmonique périodique. L'un et l'autre provenant de l'équation différentielle communed 2 s/dt 2 + 2r ds/dt +w 2 s=o. N'importe quel point de l'axes h qui suit un mouvement vibratoire harmonique apériodique est déterminé par l'équation suivante: (8a) $$s_h = a_1 e^{ - r_1 t} \cdot \frac{{e^{ + t\sqrt {r_1 ^2 - w_1 ^2 } } - e^{ - t\sqrt {r_1 ^2 - w_1 ^2 } } }}{2} \cdot \cdot \cdot \cdot \cdot $$ cependant que la courbe ondoyante s p qui montre un mouvement vibratoire harmonique périodique est exprimée par l'équation suivante (8b) $$s_p = a_2 e^{ - r_2 t} \cos (t\sqrt {w_2 ^2 - r_2 ^2 } ) \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot $$ Le mouvement complety se compose de la somme des deux mouvements oscillatoires,y =s h +s p . Pour le moment, les examens n'ont trait qu'à l'axe apériodique. Les calculs prouvent que par l'équation sh (8a) les axes apériodiques appartenant à toutes les coupes (Fig. 4) peuvent être suivis avec exactitude comme fonctions du tempst, et que même le changement de la valeur des coefficientsw 1 ,r 1 ,a 1, présente aussi une régularité nette (Fig. 5, 6, 7). L'auteur examine ensuite le calcul du temps maximum=t max et, en outre, la valeur maximum =s hmax de l'axe de la courbe vibratoire et il énonce que toute cellule et tout organisme composé de cellules ont dans le cours de leur développement un temps maximum et une valeur maximum. Le tempst maximum sera: (24) $$t_{max} = \frac{I}{{\sqrt {r_{1^2 } - w_{1^2 } } \log e}}\log \sqrt {\frac{{r + \sqrt {r_{1^2 } - w_{1^2 } } }}{{r - \sqrt {r_{1^2 } - w_{1^2 } } }}} \cdot \cdot \cdot \cdot $$ En substituant la valeur dut max dans l'équation originale de l'axe (8a), on obtient la valeur numérique dut max . Finalement, après avoir déterminé le fait que dans le développement des êtres vivants, on est en présence d'un mouvement vibratoire harmonique périodique et que l'axe de cette oscillation complexe suit l'équation du mouvement vibratoire harmonique apériodique, l'auteur constate que cette détermination constitue un résultat fondamental pour l'étude théoritique des phénomènes vitaux et ce résultat possède en même temps une grande importance pratique pour l'étude de l'amélioration, du sélectionnement, de l'hérédité et de la biologie de la race.

Auf Grund der Vorstellung, dass die Erregungsleitung auf einer Wiedererregung der benachbarten Gewebebezirke durch lokale bioelektrische Ströme beruht, wurde vorher eine mathematische Theorie der Fortpflanzung der Erregung im Nerv entwickelt, welche einige Tatsachen befriedigend darstellt. In der vorliegenden Arbeit wird die Theorie auf den Fall angewandt, dass die Erregung von einem Gewebe auf ein anderes übertragen wird, wobei die beiden Gewebe verschiedene elektrische Eigenschaften haben. Es zeigt sich, dass dabei gewisse Bedingungen für die Möglichkeit der Übertragung der Erregung erfüllt sein müssen, welche an den vonL. Lapique geforderten Isochronismus erinnern. Dans deux mémoires précédents nous avons développé une théorie mathématique de la propagation de l'excitation nerveuse, basée sur l'hypothèse, que cette propagation est due à une réexcitation par les courants bioélectriques. Dans le mémoire présent nous étudions le cas de deux tissus adjacents, différents dans leurs constantes électriques, du point de vue du passage de l'excitation d'un tissu sur l'autre. Il se trouve, que pour que ce passage put avoir lieu, certaines relations doivent être satisfaites, des relations qui rappellent l'isochronisme postulé parL. Lapique.