Quasi-stability and continuity of attractors for nonlinear system of wave equations

In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

Reduction principle at work

The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

The mutual singularity of the relative multifractal measures

M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

Effect of population migration and punctuated lockdown on the spread of infectious diseases

One of the critical measures to control infectious diseases is a lockdown. Once past the lockdown stage in many parts of the world, the crucial question now concerns the effects of relaxing the lockdown and finding the best ways to implement further lockdown(s), if required, to control the spread. With the relaxation of lockdown, people migrate to different cities and enhance the spread of the disease. This work presents the population migration model for n-cities and applies the model for migration between two and three cities. The reproduction number is calculated, and the effect of the migration rate is analyzed. A punctuated lockdown is implemented to simulate a protocol of repeated lockdowns that limits the resurgence of infections. A damped oscillatory behavior is observed with multiple peaks over a period.

A General Multipatch Model of Ebola Dynamics

A model for the transmission dynamics of Ebola virus in a multipatch network setting is studied. The model considers the contribution to the dynamics by people who are susceptible, infectious, isolated, deceased but still infectious and not yet buried, as well as the dynamics of the pathogen at interacting nodes or patches. Humans can move between patches carrying the disease to any patch in a region of n communities (patches). Both direct and indirect transmission are accounted for in this model. Matrix and graph-theoretic methods and some combinatorial identities are used to construct appropriate Lyapunov functions to establish global stability results for both the disease-free and the endemic equilibrium of the model. While the model is focused on Ebola, it can be adapted to the study of other disease epidemics, including COVID-19, currently affecting all countries in the world.

Analysis of infectious disease transmission and prediction through SEIQR epidemic model

In literature, various mathematical models have been developed to have a better insight into the transmission dynamics and control the spread of infectious diseases. Aiming to explore more about various aspects of infectious diseases, in this work, we propose conceptual mathematical model through a SEIQR (Susceptible-Exposed-Infected-Quarantined-Recovered) mathematical model and its control measurement. We establish the positivity and boundedness of the solutions. We also compute the basic reproduction number and investigate the stability of equilibria for its epidemiological relevance. To validate the model and estimate the parameters to predict the disease spread, we consider the special case for COVID-19 to study the real cases of infected cases from [2] for Russia and India. For better insight, in addition to mathematical model, a history based LSTM model is trained to learn temporal patterns in COVID-19 time series and predict future trends. In the end, the future predictions from mathematical model and the LSTM based model are compared to generate reliable results.

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

In this paper, we prove some existence results of solutions for a class of nonlocal initial value problem for nonlinear fractional hybrid implicit differential equations under generalized Hilfer fractional derivative. The result is based on a fixed point theorem on Banach algebras. Further, examples are provided to illustrate our results.

Polynomial stability of the wave equation with distributed delay term on the dynamical control

Using the frequency domain approach, we prove the rational stability for a wave equation with distributed delay on the dynamical control, after establishing the strong stability and the lack of uniform stability.

Faedo-Galerkin approximation of mild solutions of fractional functional differential equations

In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.

Prey-predator model in drainage system with migration and harvesting

In this paper, we consider a prey-predator model with a reserve region of predator where generalist predator cannot enter. Based on the intake capacity of food and other factors, we introduce the predator population which consumes the prey population with Holling type-II functional response; and generalist predator population consumes the predator population with Beddington-DeAngelis functional response. The density-dependent mortality rate for prey and generalist predator are considered. The equilibria of proposed system are determined. Local stability for the system are discussed. The environmental carrying capacity is considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighbourhood at an interior equilibrium point. Here the fishing effort is used as a control parameter to harvest the generalist predator population of the system. With the help of this control parameter, a dynamic framework is developed to investigate the optimal utilization of resources, sustainability properties of the stock and the resource rent. Finally, we present a numerical simulation to verify the analytical results, and the system is analyzed through graphical illustrations. The main findings with future research directions are described at last.