In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model’s ability to reproduce measured data on realistic problems.

This article specically deals with the asymptotic synchronization of non-identical complex dynamic fractional order networks with uncertainty. Initially, by using the Riemann-Liouville derivative, we developed a model for the general non-identical complex network, and based on the properties of fractional order calculus and the direct Lyapunov method in fractional order, we proposed that drive and response system if nonidentical complex networks ensuring asymp-totic synchronization by using neoteric control. Second, taking into account the uncertainties of non-identical complex networks in state matrices and evaluating theirrequirements forasymptotic synchronization. In addition, to explain the eectiveness of the proposed approach, two numerical simulations are given.

In this paper, using the Lyapunov-like functions, the practical stability with respect to part of the variables of fractional-order nonlinear systems depending on a small parameter is studied.

This paper is devoted to an approximate analytical solution of an integro-differential model describing the process of nucleation and growth of particles in crystallizers, taking into account the thermal-mass exchange with the environment and the removal of product crystals from the metastable medium. The method developed in this work for solving model equations (kinetic equation for the particle size distribution function and balance equations for temperature/impurity concentration) is based on using the saddle point method for calculating the Laplace-type integral. It is shown that the degree of metastability of the liquid decreases with time at a fixed value of the mass inflow from the outside (heat flow to the outside). The crystal size distribution function has the form of an irregular bell-shaped curve, which increases with the intensification of heat and mass exchange with the environment.

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

As the pandemic of Coronavirus Disease 2019 (COVID-19) rages throughout the world, accurate modeling of the dynamics thereof is essential. However, since the availability and quality of data varies dramatically from region to region, accurate modeling directly from a global perspective is difficult, if not altogether impossible. Nevertheless, via local data collected by certain regions, it is possible to develop accurate local prediction tools, which may be coupled to develop global models. In this study, we analyze the dynamics of local outbreaks of COVID-19 via a coupled system of ordinary differential equations (ODEs). Utilizing the large amount of data available from the ebbing outbreak in Hubei, China as a testbed, we estimate the basic reproductive number, R0 of COVID-19 and predict the total cases, total deaths, and other features of the Hubei outbreak with a high level of accuracy. Through numerical experiments, we observe the effects of quarantine, social distancing, and COVID-19 testing on the dynamics of the outbreak. Using knowledge gleaned from the Hubei outbreak, we apply our model to analyze the dynamics of outbreak in Turkey. We provide forecasts for the peak of the outbreak and the total number of cases/deaths in Turkey, for varying levels of social distancing, quarantine, and COVID-19 testing.

This work analyzes the existence of solution and approximate controllability for higher order non-linear fractional integro-differential systems with Riemann-Liouville derivatives in Banach spaces. Firstly, the definition of mild solution for the system is derived. Then a set of sufficient conditions for the existence of mild solution and approximate controllability of the system is obtained. The discussions are based on fixed point approach, and the theory of convolution and fractional resolvent. To illustrate the feasibility of developed theory, an example is given.

A graph is called a partial cube if it can be embedded into a hypercube isometrically. In this paper, we study a class of Cayley graphs —Cayley graphs generated by transpositions and show that a Cayley graph Γ generated by transpositions is a partial cube if and only if Γ is a bubble sort graph. This result enhances a result of Alahmadi et al. [Math. Meth. Appl. Sci. 39 (2016), 4856–4865]: BSn is a partial cube. As a corrollary, we give the analytical expressions of the Wiener indices of bubble sort graphs.

A Kalman reduced form is obtained for linear systems over Hermite rings. This reduced form gives information of the set of assignable polynomials to a given linear system.

This paper deals with the stability analysis and controller design for linear systems with time-varying delays and parameter uncertainties. By choosing appropriate augmented Lyapunov-Krasovskii functionals, a set of Linear Matrix inequalities is derived to get advanced feasible region of stability, and controller gain matrices which guarantee the asymptotic stability of the concerned systems within maximum bound of time-delays and its time-derivative. To further reduce the conservatism of stabilization criterion a recently developed mathematical technique which constructed a new augmented zero equality is applied. Finally, two numerical examples are utilized to show the validity and superiority of the proposed methods.

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations which exhibit at most resemblance with turbulent flows. One of the mathematically well studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model namely feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus only finitely many modes of the shell model would suffice to stabilize the system. Later we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic riccati equation. Using the same riccati operator we establish asymptotic stability of the nonlinear system.

We are concerned with the global existence and decay rates of large solutions for the Poisson--Nernst--Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin--Lerner type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species is small enough. Moreover, the large solution is obtained for initial data belonging to the low regularity Besov spaces with different regularity and integral indices for the different charged species, which indicates more specific coupling relations between the negatively and positively charged species.

A spectral problem is considered in a domain $\Omega_{\varepsilon}$ that is the union of a domain $\Omega_{0}$ and a lot of thin trees situated $\varepsilon$-periodically along some manifold on the boundary of $\Omega_{0}.$ The trees have finite number of branching levels. The perturbed Robin boundary condition $\partial_{\nu}u^{\varepsilon} + \varepsilon^{\alpha_i} k_{i,m}u^{\varepsilon} = 0$ is given on the $i$th branching layer; $\{\alpha_i\}$ are real parameters. The asymptotic analysis of this problem is made as $\varepsilon\to0,$ i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

A Riemann-Liouville fractional Robin boundary-value problem is proposed to describe the fast heat transfer law both within isotropic materials and through the boundary of the materials in high temperature environment. The variational formulation of the fractional model is given, and further the energy estimation of the weak solution is deduced. The uniqueness theorem of weak solution is proved. A valid finite difference scheme is developed for the fractional model and numerical experiment is implemented. Numerical results indicate that the fractional model is applicable to discover the thermal superdiffusion in the thermal protective clothing(TPC) system and numerical algorithms are effective to improve the intelligence of TPC design.

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients related to each other. A unique relation between the Euler-Frobenius polynomial and B-spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and finite sums are derived. Moreover, A few recurrence relations containing these particular finite sums are given. Using these recurence relations, we give a solution of the problem which was given by Charalambides7, Exercise 30, p. 273. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

This article aims to investigate the numerical study of electroosmotic flow of the Eyring Powell fluid under the peristaltic mechanism with the influence of the porous medium in the micro-channel. The modified system is applied externally to an electrical field in the horizontal direction and to a magnetic field in the transverse direction. The flow of nanofluids is considered in the computation. The governing equations in the nano-fluid flow are modulated. Influence of lubrication theory approximation longequations are shortened. Reduced coupled nonlinear partial differential equations like velocity and energy equations are numerically solved using the powerful and well-known mathematical software MATHEMATICA by built in NDSolve command. The influence of various important parameters on the velocity and temperature profile is summarised by graphs.

The article is dedicated towards the study of fractional-order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions in Banach spaces. First, mild solution of the system is constructed and subsequently, its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using the concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out corrections in examples presented in previous articles.

In this paper, a deep dynamical analysis is made using tools from multidimensional real discrete dynamics of some derivative-free iterative methods with memory. They all have good qualitative properties, but one of them (due to Traub) shows the same behavior as Newton’s method on quadratic polynomials. Then, the same techniques are employed to analyze the performance of several multipoint schemes with memory, whose first step is Traub’s method, but their construction was made using different procedures. Therefore, their stability is analyzed, showing which is the best in terms of the wideness of basins of convergence or the existence of free critical points that would yield convergence towards different elements from the desired zeros of the nonlinear function. Therefore, the best stability properties are linked with the best estimations made in the iterative expressions rather than their simplicity. These results have been checked with a numerical and graphical comparison with many other known methods with and without memory, with different orders of convergence, with excellent performance.