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Simple Summary: Neuroblastoma is a rare type of cancer that usually affects children. The high-risk patients' expected survival rate is less than 50%. One reason is the lack of precision in the standard treatment protocol: a one-size-fits-all multi-modal therapy. The study presented in this paper was designed to address this deficit by optimising the use of two chemotherapeutic agents—vincristine and cyclophosphamide—during induction chemotherapy—the part of the protocol that shrinks the primary tumour before surgical removal. We combined a mathematical model and an optimisation algorithm to identify the best chemotherapy schedules for a cohort of virtual patients with different initial tumour compositions. Our results reveal novel strategies to exploit a pair of drugs with different levels of efficacy, provide a platform on which to individualise induction chemotherapy, and lay the foundation for a personalised therapy that leverages targeted therapies, multi-region sequencing, liquid biopsies, and modern computational methods to improve today's multi-modal therapy. Neuroblastoma is the most common extra-cranial solid tumour in children. Despite multi-modal therapy, over half of the high-risk patients will succumb. One contributing factor is the one-size-fits-all nature of multi-modal therapy. For example, during the first step (induction chemotherapy), the standard regimen (rapid COJEC) administers fixed doses of chemotherapeutic agents in eight two-week cycles. Perhaps because of differences in resistance, this standard regimen results in highly heterogeneous outcomes in different tumours. In this study, we formulated a mathematical model comprising ordinary differential equations. The equations describe the clonal evolution within a neuroblastoma tumour being treated with vincristine and cyclophosphamide, which are used in the rapid COJEC regimen, including genetically conferred and phenotypic drug resistance. The equations also describe the agents' pharmacokinetics. We devised an optimisation algorithm to find the best chemotherapy schedules for tumours with different pre-treatment clonal compositions. The optimised chemotherapy schedules exploit the cytotoxic difference between the two drugs and intra-tumoural clonal competition to shrink the tumours as much as possible during induction chemotherapy and before surgical removal. They indicate that induction chemotherapy can be improved by finding and using personalised schedules. More broadly, we propose that the overall multi-modal therapy can be enhanced by employing targeted therapies against the mutations and oncogenic pathways enriched and activated by the chemotherapeutic agents. To translate the proposed personalised multi-modal therapy into clinical use, patient-specific model calibration and treatment optimisation are necessary. This entails a decision support system informed by emerging medical technologies such as multi-region sequencing and liquid biopsies. The results and tools presented in this paper could be the foundation of this decision support system. [ABSTRACT FROM AUTHOR]

We study a mathematical model based on ordinary differential equations to describe the dynamic interaction in the market of two types of energy called standard and innovative. The model consists of an adaptation of the generalized Lotka-Volterra system in which the parameters are assumed to depend on a quantitative and continuous attribute characteristic of energy generation. Using the analysis of the model the fitness function for the innovative energy is determined, from which conditions of invasion can be established in a market dominated by the conventional power. The canonical equation of the adaptive dynamics is studied to know the long-term behavior of the characteristic attribute and its impact on the market. Then we establish conditions under which evolutionary ramifications occur, that is to say, the requirements of coexistence and divergence of the characteristic attributes, whose occurrence leads to the origin of diversity in the energy market. [ABSTRACT FROM AUTHOR]

Love stories are dynamic processes that begin, develop, and often stay for a relatively long time in a stationary or fluctuating regime, before possibly fading. Although they are, undoubtedly, the most important dynamic process in our life, they have only recently been cast in the formal frame of dynamical systems theory. In particular, why it is so difficult to predict the evolution of sentimental relationships continues to be largely unexplained. A common reason for this is that love stories reflect the turbulence of the surrounding social environment. But we can also imagine that the interplay of the characters involved contributes to make the story unpredictable--that is, chaotic. In other words, we conjecture that sentimental chaos can have a relevant endogenous origin. To support this intriguing conjecture, we mimic a real and well-documented love story with a mathematical model in which the environment is kept constant, and show that the model is chaotic. The case we analyze is the triangle described in Jules et Jim, an autobiographic novel by Henri-Pierre Roché that became famous worldwide after the success of the homonymous film directed by Francois Truffaut. [ABSTRACT FROM AUTHOR]

Through the analysis of a mathematical model we construct in this paper the full catalog of possible love stories among standard (i.e. secure and non-synergic) individuals. For this, we first distinguish between robust and fragile couples, i.e. couples which have one or two stable equilibria. We also distinguish between high and low quality relationships by looking at the sign of the feelings among the partners. Finally, we split fragile couples into two distinct classes, namely those that start from the state of indifference to evolve toward their most desirable equilibrium, as opposed to those that have an unfavorable evolution. The boundaries separating the various classes in any two-dimensional parameter space are either bifurcation curves or special curves characterized by the annihilation of a suitably defined indicator associated to the equilibria or to their basins of attraction. The analysis is performed in part analytically and in part numerically (through continuation methods), and is mainly focused on the role of the appeals of the individuals. The results are in agreement with common wisdom and observations and show, in particular, that love stories can evolve positively even if one of the two individuals has negative feelings in the initial phase of the relationship. [ABSTRACT FROM AUTHOR]

We show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits. For this, we consider the standard Rosenzweig-MacArthur prey-predator model [The American Naturalist, 97 (1963), pp. 209-223] and, following the so-called adaptive dynamics approach, we derive from it a second-order evolutionary model composed of two ODEs, one for the prey trait and one for the predator trait. Then, we perform a detailed bifurcation analysis of the evolutionary model with respect to various environmental and demographic parameters. Surprisingly, the evolutionary dynamics turn out to be much richer than the population dynamics. Up to three evolutionary attractors can be present, and the bifurcation diagrams contain numerous global bifurcations and codimension-2 bifurcation points. Interesting biological properties can be extracted from these bifurcation diagrams. In particular, one can conclude that evolution of the traits can be cyclic and easily promote prey species diversity. [ABSTRACT FROM AUTHOR]