Geometric method for global stability and repulsion in Kolmogorov systems

A class of autonomous Kolmogorov systems that are dissipative and competitive with the origin as a repellor are considered when each nullcline surface is either concave or convex. Geometric method is developed by using the relative positions of the upper and lower planes of the nullcline surfaces for global asymptotic stability of an interior or a boundary equilibrium point. Criteria are also established for global repulsion of an interior or a boundary equilibrium point on the carrying simplex. This method and the theorems can be viewed as a natural extension of those results for Lotka-Volterra systems in the literature

Trivial dynamics in discrete-time systems : carrying simplex and translation arcs

In this paper we show that the dynamical behavior in R-+(3) (first octant) of the classical Kolmogorov systems T(x(1), x(2), x(3)) = (x(1)F(1)(x), x(2)F(2)(x), x(3)F(3)(x)) of competitive type admitting a carrying simplex can be sometimes determined completely by the number of fixed points on the boundary and the local behavior around them. Roughly speaking, T has trivial dynamics (i.e. the omega limit set of any orbit is a connected set contained in the set of fixed points) provided T has exactly four hyperbolic nontrivial fixed points {p(1), p(2), p(3), p(4)} in partial derivative R-+(3) with {p(1), p(3)} local attractors on the carrying simplex and {p(2), p(4)} local repellers on the carrying simplex; and there exists a unique hyperbolic fixed point in IntR(+)(3). Our results are applied to some classical models including the Leslie-Gower models, Atkinson-Allen systems and Ricker maps.Peer reviewe

Mathematical Modeling of Biological Systems

Mathematical modeling is a powerful approach supporting the investigation of open problems in natural sciences, in particular physics, biology and medicine. Applied mathematics allows to translate the available information about real-world phenomena into mathematical objects and concepts. Mathematical models are useful descriptive tools that allow to gather the salient aspects of complex biological systems along with their fundamental governing laws, by elucidating the system behavior in time and space, also evidencing symmetry, or symmetry breaking, in geometry and morphology. Additionally, mathematical models are useful predictive tools able to reliably forecast the future system evolution or its response to specific inputs. More importantly, concerning biomedical systems, such models can even become prescriptive tools, allowing effective, sometimes optimal, intervention strategies for the treatment and control of pathological states to be planned. The application of mathematical physics, nonlinear analysis, systems and control theory to the study of biological and medical systems results in the formulation of new challenging problems for the scientific community. This Special Issue includes innovative contributions of experienced researchers in the field of mathematical modelling applied to biology and medicine

Artificial Intelligence, Mathematical Modeling and Magnetic Resonance Imaging for Precision Medicine in Neurology and Neuroradiology

La tesi affronta la possibilità di utilizzare metodi matematici, tecniche di simulazione, teorie fisiche riadattate e algoritmi di intelligenza artificiale per soddisfare le esigenze cliniche in neuroradiologia e neurologia al fine di descrivere e prevedere i patterns e l’evoluzione temporale di una malattia, nonché di supportare il processo decisionale clinico. La tesi è suddivisa in tre parti. La prima parte riguarda lo sviluppo di un workflow radiomico combinato con algoritmi di Machine Learning al fine di prevedere parametri che favoriscono la descrizione quantitativa dei cambiamenti anatomici e del coinvolgimento muscolare nei disordini neuromuscolari, con particolare attenzione alla distrofia facioscapolo-omerale. Il workflow proposto si basa su sequenze di risonanza magnetica convenzionali disponibili nella maggior parte dei centri neuromuscolari e, dunque, può essere utilizzato come strumento non invasivo per monitorare anche i più piccoli cambiamenti nei disturbi neuromuscolari oltre che per la valutazione della progressione della malattia nel tempo. La seconda parte riguarda l’utilizzo di un modello cinetico per descrivere la crescita tumorale basato sugli strumenti della meccanica statistica per sistemi multi-agente e che tiene in considerazione gli effetti delle incertezze cliniche legate alla variabilità della progressione tumorale nei diversi pazienti. L'azione dei protocolli terapeutici è modellata come controllo che agisce a livello microscopico modificando la natura della distribuzione risultante. Viene mostrato come lo scenario controllato permetta di smorzare le incertezze associate alla variabilità della dinamica tumorale. Inoltre, sono stati introdotti metodi di simulazione numerica basati sulla formulazione stochastic Galerkin del modello cinetico sviluppato. La terza parte si riferisce ad un progetto ancora in corso che tenta di descrivere una porzione di cervello attraverso la teoria quantistica dei campi e di simularne il comportamento attraverso l'implementazione di una rete neurale con una funzione di attivazione costruita ad hoc e che simula la funzione di risposta del modello biologico neuronale. E’ stato ottenuto che, nelle condizioni studiate, l'attività della porzione di cervello può essere descritta fino a O(6), i.e, considerando l’interazione fino a sei campi, come un processo gaussiano. Il framework quantistico definito può essere esteso anche al caso di un processo non gaussiano, ovvero al caso di una teoria di campo quantistico interagente utilizzando l’approccio della teoria wilsoniana di campo efficace.The thesis addresses the possibility of using mathematical methods, simulation techniques, repurposed physical theories and artificial intelligence algorithms to fulfill clinical needs in neuroradiology and neurology. The aim is to describe and to predict disease patterns and its evolution over time as well as to support clinical decision-making processes. The thesis is divided into three parts. Part 1 is related to the development of a Radiomic workflow combined with Machine Learning algorithms in order to predict parameters that quantify muscular anatomical involvement in neuromuscular diseases, with special focus on Facioscapulohumeral dystrophy. The proposed workflow relies on conventional Magnetic Resonance Imaging sequences available in most neuromuscular centers and it can be used as a non-invasive tool to monitor even fine change in neuromuscular disorders and to evaluate longitudinal diseases’ progression over time. Part 2 is about the description of a kinetic model for tumor growth by means of classical tools of statistical mechanics for many-agent systems also taking into account the effects of clinical uncertainties related to patients’ variability in tumor progression. The action of therapeutic protocols is modeled as feedback control at the microscopic level. The controlled scenario allows the dumping of uncertainties associated with the variability in tumors’ dynamics. Suitable numerical methods, based on Stochastic Galerkin formulation of the derived kinetic model, are introduced. Part 3 refers to a still-on going project that attempts to describe a brain portion through a quantum field theory and to simulate its behavior through the implementation of a neural network with an ad-hoc activation function mimicking the biological neuron model response function. Under considered conditions, the brain portion activity can be expressed up to O(6), i.e., up to six fields interaction, as a Gaussian Process. The defined quantum field framework may also be extended to the case of a Non-Gaussian Process behavior, or rather to an interacting quantum field theory in a Wilsonian Effective Field theory approach

On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

This paper deals with global asymptotic behaviour of the dynamics for $N$-dimensional competitive Kolmogorov differential systems of equations $\frac{dx_i}{dt} =x_if_i(x), 1\leq i\leq N, x\in \R^N_+$. A theory based on monotone dynamical systems was well established by Morris W Hirsch (Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51--71). One of his theorems is outstanding and states the existence of a co-dimension 1 compact invariant submanifold $\Sigma$ that attracts all the nontrivial orbits under certain assumptions and, in practice, under the condition that the system is totally competitive (all $N^2$ entries of the Jacobian matrix $Df$ are negative). The submanifold $\Sigma$ has been called carrying simplex since then and the theorem has been well accepted with many hundreds of citations. In this paper, we point out that the requirement of total competition is too restrictive and too strong; we prove the existence and uniqueness of a carry simplex under the assumption of strong internal competition only (i.e. $N$ diagonal entries of $Df$ are negative), a much weaker condition than total competition. Thus, we improve the theorem significantly by dramatic cost reduction from requiring $N^2$ to $N$ negative entries of $Df$. As an example of applications of the main result, the existence and global attraction (repulsion) of a heteroclinic limit cycle for three-dimensional systems is discussed and two concrete examples are given to demonstrate the existence of such heteroclinic cycles

The effect of fishing on the evolution of North Sea cod

With the recent collapses of many major fish stocks and North Sea cod seeming to be the next on a long list, it has become apparent that our actions have a great effect on fish populations. This thesis looks at how fishing can not only have an immediate effect such as causing declines, but can also affect the evolution of a stock. A population model is built which is first examined for stability properties. A comparison is also made of the model under fishing and when fishing is absent. A measure of population fitness in terms of ability to invade other populations is then established. This measure is used to examine the sensitivity of the model to parameter values. This is also done for models which use different functions to model life history in order to determine the importance of model choice. Components of fishing mortality are considered with respect to their impact on the stock, for both the main model and the alternate models. Finally, spatial and seasonal considerations are added in a simple way to check if a single region model can be trusted to model the whole of the North Sea. It is found that although the model is sensitive to the choice of growth function, generally growth has the most effect on population fitness. It is also shown that the level of fishing has more impact on the fitness and yield of the stock than the initial capture length. Thus, it is more important to reduce fishing effort, than change aspects of the fishery such as mesh size in nets. Furthermore the spatial model shows that the establishment of reservoirs, or no-fishing zones, should be done carefully in order not to favour a decrease in growth rate of the stock

Local and nonlocal integro-differential reaction-diffusion models for protein dynamics in Alzheimer's disease

In this work, integro-differential reaction-diffusion models are presented for the description of the temporal and spatial evolution of the concentrations of Abeta and tau proteins involved in Alzheimer's disease. Initially, a local model is analysed: this is obtained by coupling with an interaction term two heterodimer models, modified by adding diffusion and Holling functional terms of the second type. We then move on to the presentation of three nonlocal models, which differ according to the type of the growth (exponential, logistic or Gompertzian) considered for healthy proteins. In these models integral terms are introduced to consider the interaction between proteins that are located at different spatial points possibly far apart. For each of the models introduced, the determination of equilibrium points with their stability and a study of the clearance inequalities are carried out. In addition, since the integrals introduced imply a spatial nonlocality in the models exhibited, some general features of nonlocal models are presented. Afterwards, with the aim of developing simulations, it is decided to transfer the nonlocal models to a brain graph called connectome. Therefore, after setting out the construction of such a graph, we move on to the description of Laplacian and convolution operations on a graph. Taking advantage of all these elements, we finally move on to the translation of the continuous models described above into discrete models on the connectome. To conclude, the results of some simulations concerning the discrete models just derived are presented

Doctor of Philosophy

dissertationMagnetic Resonance (MR) is a relatively risk-free and flexible imaging modality that is widely used for studying the brain. Biophysical and chemical properties of brain tissue are captured by intensity measurements in T1W (T1-Weighted) and T2W (T2-Weighted) MR scans. Rapid maturational processes taking place in the infant brain manifest as changes in co{\tiny }ntrast between white matter and gray matter tissue classes in these scans. However, studies based on MR image appearance face severe limitations due to the uncalibrated nature of MR intensity and its variability with respect to changing conditions of scan. In this work, we develop a method for studying the intensity variations between brain white matter and gray matter that are observed during infant brain development. This method is referred to by the acronym WIVID (White-gray Intensity Variation in Infant Development). WIVID is computed by measuring the Hellinger Distance of separation between intensity distributions of WM (White Matter) and GM (Gray Matter) tissue classes. The WIVID measure is shown to be relatively stable to interscan variations compared with raw signal intensity and does not require intensity normalization. In addition to quantification of tissue appearance changes using the WIVID measure, we test and implement a statistical framework for modeling temporal changes in this measure. WIVID contrast values are extracted from MR scans belonging to large-scale, longitudinal, infant brain imaging studies and modeled using the NLME (Nonlinear Mixed Effects) method. This framework generates a normative model of WIVID contrast changes with time, which captures brain appearance changes during neurodevelopment. Parameters from the estimated trajectories of WIVID contrast change are analyzed across brain lobes and image modalities. Parameters associated with the normative model of WIVID contrast change reflect established patterns of region-specific and modality-specific maturational sequences. We also detect differences in WIVID contrast change trajectories between distinct population groups. These groups are categorized based on sex and risk/diagnosis for ASD (Autism Spectrum Disorder). As a result of this work, the usage of the proposed WIVID contrast measure as a novel neuroimaging biomarker for characterizing tissue appearance is validated, and the clinical potential of the developed framework is demonstrated