Mathematical Analysis of the Concomitance of Illicit Drug Use and Banditry in a Population

One of the majors global health and social problem facing the world today is the use of illicit drug and the act banditry. The two problems have resulted into lost of precious lives, properties and even a devastating effects on the economy of some countries where such acts were been practiced. Of interest in this work is to study the global stability of illicit drug use spread dynamics with banditry compartment using a dynamical system theory approach. Illicit drug use and banditry reproduction number was evaluated analytically, which measures the potential spread of the illicit drug use and banditry in the population. The system exhibits supercritical bifurcation property, telling us that local stability of an illicit drug and banditry-present equilibrium exist and it is unique. In addition, the illicit drug and banditry-free and illicit drug and banditry-present equilibria were shown to be global asymptotically stable, this was achieved by construction of suitable Lyapunov functions. Sensitivity analysis was carried out to know the impact of each parameter on the dynamical spread of illicit drug use and banditry in a population. Numerical simulations were used to validate the obtained quantitative results, and examine the effects of some key parameters on the system. It was discovered that, to reduce the burden of banditry in the population, stringent control measures must be put in place to reduce the use of illicit drug in a population. Suggested control measures to use in curtail the menace of the illicit drug use and banditry were recommends.

Revisiting dynamics of interacting quintessence

We apply the tools of the dynamical system theory in order to revisit and uncover the structure of a nongravitational interaction between pressureless dark matter and dark energy described by a scalar field $$\phi $$ ϕ . For a coupling function $$Q = -(\alpha d\rho _m/dt + \beta d\rho _\phi /dt )$$ Q = - ( α d ρ m / d t + β d ρ ϕ / d t ) , where t is the cosmic time, we have found that it can be rewritten in the form $$Q = 3H (\alpha \rho _m + \beta (d\phi /dt)^2 )/(1-\alpha +\beta )$$ Q = 3 H ( α ρ m + β ( d ϕ / d t ) 2 ) / ( 1 - α + β ) , so that its dependence on the dark matter density and on the kinetic term of the scalar field is linear and proportional to the Hubble parameter. We analyze the scenarios $$\alpha =0$$ α = 0 , $$\alpha = \beta $$ α = β and $$\alpha = -\beta $$ α = - β , separately and in order to describe the cosmological evolution we have calculated various observables. A notable result of this work is that, unlike for the noninteracting scalar field with exponential potential where five critical points appear, in the case studied here, with the exception of the matter dominated solution, the remaining singular points are transformed into scaling solutions enriching the phase space. It is shown that for $$\alpha \ne 0$$ α ≠ 0 , a separatrix arises modifying prominently the structure of the phase space. This represents a novel feature no mentioned before in the literature.

Mathematical modelling of oscillating patterns for chronic autoimmune diseases

Many autoimmune diseases are chronic in nature, so that in general patients experience periods of recurrence and remission of the symptoms characterizing their specific autoimmune ailment. In order to describe this very important feature of autoimmunity, we construct a mathematical model of kinetic type describing the immune system cellular interactions in the context of autoimmunity exhibiting recurrent dynamics. The model equations constitute a non-linear system of integro-differential equations with quadratic terms that describe the interactions between self-antigen presenting cells, self-reactive T cells and immunosuppressive cells. We consider a constant input of self-antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with predisposition for this condition. We also consider the natural death of all cell populations involved in our model, caused by their interaction with cells of the host environment. We derive the macroscopic analogue and show positivity and well-posedness of the solution, and then we study the equilibria of the corresponding dynamical system and their stability properties. By applying dynamical system theory, we prove that steady oscillations may arise due to the occurrence of a Hopf bifurcation. We perform some numerical simulations for our model, and we observe a recurrent pattern in the solutions of both the kinetic description and its macroscopic analogue, which leads us to conclude that this model is able to capture the chronic behaviour of many autoimmune diseases.

A trajectory-based loss function to learn missing terms in bifurcating dynamical systems

Missing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations. With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. The choice of the loss function, which compares the training data trajectory in state space and the current estimated solution trajectory of the UDE to optimize the solution, plays a crucial role within this approach. The Mean Square Error as loss function contains the risk of a reconstruction which completely misses the dynamical behavior of the training data. By contrast, our suggested trajectory-based loss function which optimizes two largely independent components, the length and angle of state space vectors of the training data, performs reliable well in examples of systems from neuroscience, chemistry and biology showing Saddle-Node, Pitchfork, Hopf and Period-doubling bifurcations.

Global Directed Dynamic Behaviors of a Lotka-Volterra Competition-Diffusion-Advection System

This paper investigates the problem of the global directed dynamic behaviors of a Lotka-Volterra competition-diffusion-advection system between two organisms in heterogeneous environments. The two organisms not only compete for different basic resources, but also the advection and diffusion strategies follow the dispersal towards a positive distribution. By virtue of the principal eigenvalue theory, the linear stability of the co-existing steady state is established. Furthermore, the classification of dynamical behaviors is shown by utilizing the monotone dynamical system theory. This work can be seen as a further development of a competition-diffusion system.

Wall Shear Stress Topological Skeleton Analysis in Cardiovascular Flows: Methods and Applications

A marked interest has recently emerged regarding the analysis of the wall shear stress (WSS) vector field topological skeleton in cardiovascular flows. Based on dynamical system theory, the WSS topological skeleton is composed of fixed points, i.e., focal points where WSS locally vanishes, and unstable/stable manifolds, consisting of contraction/expansion regions linking fixed points. Such an interest arises from its ability to reflect the presence of near-wall hemodynamic features associated with the onset and progression of vascular diseases. Over the years, Lagrangian-based and Eulerian-based post-processing techniques have been proposed aiming at identifying the topological skeleton features of the WSS. Here, the theoretical and methodological bases supporting the Lagrangian- and Eulerian-based methods currently used in the literature are reported and discussed, highlighting their application to cardiovascular flows. The final aim is to promote the use of WSS topological skeleton analysis in hemodynamic applications and to encourage its application in future mechanobiology studies in order to increase the chance of elucidating the mechanistic links between blood flow disturbances, vascular disease, and clinical observations.

Assessing the future occurrence of severe thunderstorm environments in Europe: frequency, hotspots and impacts

<p>We study the future frequency of atmospheric environments leading to severe thunderstorms over Europe under different climate change scenarios in CMIP5 models. Our method is founded on dynamical system theory, which makes it possible to detect future atmospheric configurations that are close analogues of past events in the class of interest.</p><p>We rely on the EM-DAT and the European Severe Weather Database to select an ensemble of past events leading to significant damage or disruption, including severe thunderstorms, hail storms, derechos and tornadoes, between 1950 and 2020. We consider the geopotential height field at 500 hPa in ERA5 data as a dynamical proxy of the corresponding configurations. Then, we leverage extreme value theory to detect close dynamic analogues in the output of CMIP5 climate projection models under two emission scenarios, namely RCP4.5 and RCP8.5.</p><p>First of all, we assess possible differences between the trends in severe weather frequency due to different radiative forcing. Then, we study the spatial structure of such trends, highlighting regions where the occurrence of such phenomena could see a sharp increase or decrease. Finally, we estimate potential future impact of such phenomena where the information about economic damage is available in EM-DAT.</p><p> </p>