Recent publications

In recent years, fractional calculus has witnessed tremendous progress in variousareas of sciences and mathematics [...]

Background and objective:
In this work, a mathematical model based on differential equations is proposed to describe the propagation of polio in a human population. The motivating system is a compartmental nonlinear model which is based on the use of ordinary differential equations and four compartments, namely, susceptible, exposed, infected and vaccinated individuals.
Methods:
In this manuscript, the mathematical model is extended in order to account for spatial diffusion in one dimension. Nonnegative initial conditions are used, and we impose homogeneous Neumann conditions at the boundary. We determine analytically the disease-free and the endemic equilibria of the system along with the basic reproductive number.
Results:
We establish thoroughly the nonnegativity and the boundedness of the solutions of this problem, and the stability analysis of the equilibrium solutions is carried out rigorously. In order to confirm the validity of these results, we propose an implicit and linear finite-difference method to approximate the solutions of the continuous model.
Conclusions:
The numerical model is stable in the sense of von Neumann, it yields consistent approximations to the exact solutions of the differential problem, and that it is capable of preserving unconditionally the positivity of the approximations. For illustration purposes, we provide some computer simulations that confirm some theoretical results derived in the present manuscript.

Tumor invasion follows a complex mechanism which involves cell migration and proliferation. To study the processes in which primary and secondary metastases invade and damage the normal cells, mathematical models are often extremely useful. In this manuscript, we present a mathematical model of acid-mediated tumor growth consisting of radially symmetric reaction-diffusion equations. The assumption on the radial symmetry of the solutions is imposed here in view that tumors present spherical symmetry at the microscopic level. Moreover, we consider various empirical mechanisms which describe the propagation of tumors by considering cancer cells, normal cells, and the concentration of H + ions. Among other assumptions, we suppose that these components follow logistic-type growth rates. Evidently, this is an important difference with respect to various other mathematical models for tumor growth available in the literature. Moreover, we also add competition terms of normal and tumor cells growth. We carry out a balancing study of the equations of the model, and a numerical model is proposed to produce simulations. Various practical remarks derived from our assumptions are provided in the discussion of our the simulations. Response to Reviewers: % In LaTeX format. Please, compile the file. \documentclass[letterpaper,10pt]{article} \usepackage{color} \usepackage{amssymb} \setlength{\leftmargini}{0.3in} % \setlength{\parindent}{0.5in} % \setlength{\topmargin}{-0.75in} % Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation

A system of two partial differential equations with fractional diffusion is considered in this study. The system extends the conventional Zakharov system with unknowns being nonlinearly coupled complex- and real-valued functions. The diffusion is understood in the Riesz sense, and suitable initial–boundary conditions are imposed on an open and bounded domain of the real numbers. It is shown that the mass and Higgs’ free energy of the system are conserved. Moreover, the total energy is proven to be dissipated, and that both the free and the total energy are non-negative. As a corollary from the conservation of energy, we find that the solutions of the system are bounded throughout time. Motivated by these properties on the solutions of the system, we propose a numerical model to approximate the fractional Zakharov system via finite-difference approaches. Along with this numerical model for solving the continuous system, discrete analogues for the mass, the Higgs’ free energy and the total energy are we provided. Furthermore, utilizing Browder’s fixed-point theorem, we establish the solubility of the discrete model. It is shown that the discrete total mass and the discrete free energy are conserved, in agreement with the continuous case. The discrete energy functionals (both the discrete free energy and the discrete total energy) are proven to be non-negative functions of the discrete time thoroughly the boundedness of the numerical solutions. Properties of consistency, stability and convergence of the scheme are also studied rigorously. Numerical simulations illustrate some of the anticipated theoretical features of our finite-difference solution procedure.

Background and objective:
In this manuscript, we consider a compartmental model to describe the dynamics of propagation of an infectious disease in a human population. The population considers the presence of susceptible, exposed, asymptomatic and symptomatic infected, quarantined, recovered and vaccinated individuals. In turn, the mathematical model considers various mechanisms of interaction between the sub-populations in addition to population migration.
Methods:
The steady-state solutions for the disease-free and endemic scenarios are calculated, and the local stability of the equilibium solutions is determined using linear analysis, Descartes' rule of signs and the Routh-Hurwitz criterion. We demonstrate rigorously the existence and uniqueness of non-negative solutions for the mathematical model, and we prove that the system has no periodic solutions using Dulac's criterion. To solve this system, a nonstandard finite-difference method is proposed.
Results:
As the main results, we show that the computer method presented in this work is uniquely solvable, and that it preserves the non-negativity of initial approximations. Moreover, the steady-state solutions of the continuous model are also constant solutions of the numerical scheme, and the stability properties of those solutions are likewise preserved in the discrete scenario. Furthermore, we establish the consistency of the scheme and, using a discrete form of Gronwall's inequality, we prove theoretically the stability and the convergence properties of the scheme. For convenience, a Matlab program of our method is provided in the appendix.
Conclusions:
The computer method presented in this work is a nonstandard scheme with multiple dynamical and numerical properties. Most of those properties are thoroughly confirmed using computer simulations. Its easy implementation make this numerical approach a useful tool in the investigation on the propagation of infectious diseases. From the theoretical point of view, the present work is one of the few papers in which a nonstandard scheme is fully and rigorously analyzed not only for the dynamical properties, but also for consistently, stability and convergence.

Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, boundedness of the solutions outside a set of zero measure is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempy sets is a positive extended real number.

A dissipation-preserving numerical scheme is proposed and theoretically analyzed for the first time to solve the Caputo-Riesz time-space-fractional generalized nonlinear Klein-Gordon equation. More concretely, we investigate a multidimensional nonlinear wave equation with generalized potential, involving Caputo temporal fractional derivatives and Riesz spatial fractional operators. We consider different initial conditions and impose homogeneous Dirichlet data on the boundary of a finite-dimensional hyper cube. Our scheme is applied to a fractional extension of the well-known Klein-Gordon equation, as well as the sine-Gordon and double sine-Gordon equations, describing the dynamics of 1-dimensional lattices. In the first stage, we transform the original equation into a system of two differential equations, introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. Our approach uses the L 1 scheme to approximate Caputo-type temporal derivatives, and fractional-order centered differences to approximate the Riesz spatial operators. A discrete form of the continuous energy is proposed, and the discrete operator is shown to satisfy a conservative law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. Finally, we carry out a number of numerical simulations to verify the validity of our theoretical results.

In this work, we design and analyze a discrete model to approximate the solutions of a parabolic partial differential equation in multiple dimensions. The mathematical model considers a nonlinear reaction term and a space-dependent diffusion coefficient. The system has a Gibbs' free energy, we establish rigorously that it is non-negative under suitable conditions, and that it is dissipated with respect to time. The discrete model proposed in this work has also a discrete form of the Gibbs' free energy. Using a fixed-point theorem, we prove the existence of solutions for the numerical model under suitable assumptions on the regularity of the component functions. We prove that the scheme preserves the positivity and the dissipation of the discrete Gibbs' free energy. We establish theoretically that the discrete model is a second-order consistent scheme. We prove the stability of the method along with its quadratic convergence. Some simulations illustrating the capability of the scheme to preserve the dissipation of Gibb's energy are presented.

In this work, we propose an implicit finite-difference scheme to approximate the solutions of a generalization of the well-known Klein-Gordon-Zakharov system. More precisely, the system considered in this work is an extension to the spatially fractional case of the classical Klein-Gordon-Zakharov model, considering two different orders of differentiation and fractional derivatives of the Riesz type. The numerical model proposed in this work considers fractional-order centered differences to approximate the spatial fractional derivatives. The energy associated to this discrete system is a non-negative invariant, in agreement with the properties of the continuous fractional model. We establish rigorously the existence of solutions using fixed-point arguments and complex matrix properties. To that end, we use the fact that the two difference equations of the discretization are decoupled, which means that the computational implementation is easier that for other numerical models available in the literature. We prove that the method has square consistency in both time and space. In addition, we prove rigorously the stability and the quadratic convergence of the numerical model. As a corollary of stability, we are able to prove the uniqueness of numerical solutions. Finally, we provide some illustrative simulations with a computer implementation of our scheme.

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally we implemented the scheme computationally to confirm the validity of
the mass and energy conservation properties, obtaining satisfactory results.

Background and objective. We present and analyze a nonstandard numerical method to solve an epidemic model with memory that describes the propagation of Ebola-type diseases. The epidemiological system contemplates the presence of sub-populations of susceptible, exposed, infected and recovered individuals, along with nonlinear interactions between the members of those sub-populations. The system possesses disease-free and endemic equilibrium points, whose stability is studied rigorously.
Methods. To solve the epidemic model with memory, a nonstandard approach based on Grünwald–Letnikov differences is used to discretize the problem. The discretization is conveniently carried out in order to produce a fully explicit and non-singular scheme. The discrete problem is thus well defined for any set of non-negative initial conditions.
Results. The existence and uniqueness of the solutions of the discrete problem for non-negative initial data is thoroughly proved. Moreover, the positivity and the boundedness of the approximations is also theoretically elucidated. Some simulations confirm the validity of these theoretical results. Moreover, the simulations prove that the computational model is capable of preserving the equilibria of the system (both the disease-free and the endemic equilibria) as well as the stability of those points.
Conclusions. Both theoretical and numerical results establish that the computational method proposed in this work is capable of preserving distinctive features of an epidemiological model with memory for the propagation of Ebola-type diseases. Among the main characteristics of the numerical integrator, the existence and the uniqueness of solutions, the preservation of both positivity and boundedness, the preservation of the equilibrium points and their stabilities as well as the easiness to implement it computationally are the most important features of the approach proposed in this manuscript.

Background and objective: In this work, we analyze the {\color{blue}spatial-temporal} dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively.
Methods: We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established.
Results: We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain.
Conclusions: The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.

The present work is the first manuscript of the literature in which a numerical model that preserves the dissipation of free energy is proposed to solve a time-space fractional generalized nonlinear parabolic system. More precisely, we investigate an extension of the multidimensional heat equation with nonlinear reaction and fractional derivatives in space and time. The model considers homogeneous Dirichlet boundary conditions and initial data. The temporal fractional derivative is understood in the Caputo sense, and Riesz fractional derivatives are employed in the spatial variables. The system possesses a free energy functional and we prove mathematically that it is a decreasing function of time. Motivated by these facts, we propose a discretization of the space-time-fractional model which dissipates the discrete free energy. The discrete model is obtained using fractional-order centered differences to approximate the Riesz derivatives, and the L 1 scheme to estimate the Caputo derivatives. The numerical model is a consistent approximation of the continuous system with quadratic order in space, and at most second order in time. Simulations confirm the capability of the numerical model to dissipate the free energy of the continuous fractional system.

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross--Pitaevskii equations, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

This manuscript is devoted to studying approximations of a coupled Klein-Gordon-Zakharov system where different orders of fractional spatial derivatives are utilized. The fractional derivatives involved are in the Riesz sense. It is understood that such a modeling system possesses an energy functional which is conserved throughout the period of time considered, and that its solutions are uniformly bounded. Motivated by these facts, we propose two numerical models to approximate the underlying continuous system. While both approximations remain to be nonlinear, one of them is implicit and the other is explicit. For each of the discretized models, we introduce a proper discrete energy functional to estimate the total energy of the continuous system. We prove that such a discrete energy is conserved in both cases. The existence of solutions of the numerical models is established via fixed-point theorems. Continuing explorations of intrinsic properties of the numerical solutions are carried out. More specifically, we show rigorously that the two schemes constructed are capable of preserving the boundedness of the approximations and that they yield consistent estimates of the true solution. Numerical stability and convergence are likewise proved theoretically. As one of the consequences, the uniqueness of numerical solutions is shown rigorously for both discretized models. Finally, comparisons of the numerical solutions are provided, in order to evaluate the capabilities of these discrete methods to preserve the discrete energy of underlying systems.

Photosynthetic microbes are omnipresent in land and water. While they critically influence primary productivity in aquatic systems, their importance in terrestrial ecosystems remains largely overlooked. In terrestrial systems, photoautotrophs occur in a variety of habitats, such as sub-surface soils, exposed rocks, and bryophytes. Here, we study photosynthetic microbial communities associated with bryophytes from a boreal peatland and a tropical rainforest. We interrogate their contribution to bryophyte C uptake and identify the main drivers of that contribution. We found that photosynthetic microbes take up twice more C in the boreal peatland (~4.4 mg CO 2 .h ⁻¹ .m ⁻² ) than in the tropical rainforest (~2.4 mg CO 2 .h ⁻¹ .m ⁻² ), which corresponded to an average contribution of 4% and 2% of the bryophyte C uptake, respectively. Our findings revealed that such patterns were driven by the proportion of photosynthetic protists in the moss microbiomes. Low moss water content and light conditions were not favourable to the development of photosynthetic protists in the tropical rainforest, which indirectly reduced the overall photosynthetic microbial C uptake. Our investigations clearly show that photosynthetic microbes associated with bryophyte effectively contribute to moss C uptake despite species turnover. Terrestrial photosynthetic microbes clearly have the capacity to take up atmospheric C in bryophytes living under various environmental conditions, and therefore potentially support rates of ecosystem-level net C exchanges with the atmosphere.

Adolescents set different goals in their interactions with peers—some place high importance on demonstrating their social competence (demonstration approach), others on avoiding showing their incompetence (demonstration avoidance), and some youth prioritize both or care about neither. In this study, we examined whether associations between demonstration approach and adjustment would vary depending on the level of demonstration avoidance. Participants (N = 595; Mage = 16.02 years, SD = 0.33) completed self-reports of social goals (in ninth grade), and peer reports of aggression, prosocial behavior, popularity, and likeability (in sixth and ninth grade). The degree to which approach-oriented adolescents were more aggressive and popular varied depending on whether they were high or low in avoidance. Approach-aggression links became stronger as avoidance increased from low to high; the reverse was true for popularity. Our discussion focuses on understanding how fear of failure and negative evaluations, or lack thereof, can alter the adjustment outcomes for approach-oriented adolescents.

Most research on the development of personality traits like the Dark Triad (i.e., narcissism, Machiavellianism, and psychopathy) focuses on local effects like parenting style or attachment, but people live in a larger society that may set the stage for any local effects. Here we paired nation-level data on the traits from 49 nations with several milieu indicators (e.g., life expectancy, homicide rates) from three timepoints (and change among them) where the average participant (≈ 22yo) would have been a child (≈ 6yo), a pre-teen (≈ 11yo), and a teenager (≈ 16yo). Congruent with previous research, variance in narcissism was far more sensitive to variance in milieu conditions in general and across all three time points than variance in Machiavellianism or psychopathy. The milieu conditions differentiated the traits somewhat with income and education revealing negative correlations with narcissism, positive correlations with Machiavellianism, and null correlations with psychopathy. Sex differences in Machiavellianism and narcissism were correlated with homicide rates across the three timepoints. The evidence that changes in milieu conditions in ones' past predicts the traits was erratic, but larger sex differences in the traits were associated with decreased life expectancies and homicide rates between childhood and pre-teens.

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Tõnu Viik

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