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This paper deals with time-fractional stochastic Navier-Stokes equations, which are characterized by the coexistence of stochastic noise and a fractional power of the Laplacian. We establish sufficient conditions for the existence and approximate controllability of a unique mild solution to time-fractional stochastic Navier-Stokes equations. Using a fixed point technique, we first demonstrate the existence and uniqueness of a mild solution to the equation under consideration. We then establish approximate controllability results by using the concepts of fractional calculus, semigroup theory, functional analysis and stochastic analysis.

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of $\kappa(\xi)$-Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space $\psi$-fractional $\mathcal{H}^{\mu,\nu;\,\psi}_{\kappa(\xi)}(\Lambda)$. In this sense, we present a few special cases and remark on the outcomes explored., Comment: 18 pages

In this paper we study the protection of N-qubit Greenberger-Horne- Zeilinger (GHZ) state and generalized N-qubit GHZ states in amplitude damping channel by means of quantum weak measurement and flip operations. We derive the explicit formulas of the performances of the protection scheme: average fidelity, average probability and the average quantum fisher information (QFI). Moreover, the analytical results for maximizing the average fidelity and probability are obtained. We show that our scheme can effectively protect the average QFI of phase for GHZ states and generalized GHZ states. The proposed scheme has the merit of protecting GHZ state and the QFI of phase against heavy amplitude damping noise. Further we show that for some generalize GHZ state, the proposed scheme can protect the state with probability one and fidelity more than 99%., Comment: 18 pages, 6 figures

One of the crucial tasks in quantum systems is to reduce the effects of decoherence due to the unavoidable interactions between a system and its environment. Many protection schemes have been proposed recently, among them the weak measurement quantum measurement reversal (WMQMR), weak measurement-based quantum feedback control (QFBC) and quantum feed-forward control (QFFC) are reviewed in this paper. By considering weak measurement, the aim is to find a balance between information gain and disturbance of the system caused by the measurement. We classify different types of measurement and give the definition of noise sources and their effects on the state of the system. Finally, we compare and analyze the performance of the discussed protection schemes for different noise sources by numerical simulations., Comment: 22 pages, 6 figures

We introduce a digital twin of the classical compartmental SIR (Susceptible, Infected, Recovered) epidemic model and study the interrelation between the digital twin and the system. In doing so, we use Stieltjes derivatives to feed the data from the real system to the virtual model which, in return, improves it in real time. As a byproduct of the model, we present a precise mathematical definition of solution to the problem. We also analyze the existence and uniqueness of solutions, introduce the concept of Main Digital Twin and present some numerical simulations with real data of the COVID-19 epidemic, showing the accuracy of the proposed ideas.

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models., Comment: This is a preprint of a paper whose final and definite form is published by 'Chaos Solitons Fractals' (ISSN: 0960-0779). Paper Submitted 04/June/2020; Revised 22/July/2020 and 02/Aug/2020; Accepted 04/Jan/2021

The SEIR model is a compartmental model used to simulate the dynamics of an epidemic. In this chapter, we introduce two control functions in the compartmental SEIR model representing vaccination and plasma transfusion. Optimal control problems are proposed to study the effects of these two control measures, on the reduction of infected individuals and increase of recovered ones, with minimal costs. Up to our knowledge, the plasma transfusion treatment has never been considered as a control strategy for epidemics mitigation. The proposed vaccination and treatment strategies may have a real application in the challenging and hard problem of controlling the COVID-19 pandemic., Comment: This is a preprint of a paper whose final and definite form is published in 'Analysis of Infectious Disease Problems (Covid-19) and Their Global Impact', Springer Nature Singapore Pte Ltd. Submitted 15/July/2020; revised 02/Oct/2020; accepted 08/Oct/2020

The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to "normal life" and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool., Comment: Paper whose final and definite form is published Open Access by 'Scientific Reports', ISSN 2045-2322, Springer Nature. Submitted 01/Sept/2020; Revised 16/Dec/2020; Accepted 27/Jan/2021; Published by [www.nature.com/scientificreports], see [https://doi.org/10.1038/s41598-021-83075-6]

This article concerns a class of beam equations with damping on rectangular tori. When the generators satisfy certain relationship, by excluding some value of two model parameters, we prove that such models admit small amplitude quasi-periodic travelling wave solutions with two frequencies, which are continuations of two rotating wave solutions with one frequency. This result holds not only for an isotropic torus, but also for an anisotropic torus. The proof is mainly based on a Lyapunov--Schmidt reduction together with the implicit function theorem., Comment: 22 pages

We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China., Comment: Submitted 27-March-2020; Revised 10-April-2020; Accepted 22-April-2020 to 'Chaos, Solitons & Fractals', ISSN: 0960-0779

In many mathematical types of research, in order to solve the fuzzy fractional differential equations, we should transform these problems into crisp corresponding problems and by solving them the approximate solution can be obtained. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler's method is applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, the convergence and the stability of the generalized Euler's method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.

The aim of this research is to apply a novel technique based on the embedding method to solve the n*n fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained by transforming the n*n FSLE to the crisp system. In this paper, Ezzati's method to solve the FSLEs is modified and improved. Several theorems are proved to show the number of operations for presented method are less than the methods of Friedman and Ezzati. In order to show the advantages of scheme, two applicable algorithms are presented and several examples are solved by applying them. Also, some graphs of obtained results are demonstrated which show the solutions are in the fuzzy form., Comment: 20 pages, 4 figures

We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are established using approximating solutions and the theory of maximal monotone operators. Using the De Giorgi-Nash-Moser technique, we prove that the solutions are bounded and H\"{o}lder continuous for all positive time., Comment: 28 pages