New results on bifurcation for fractional-order octonion-valued neural networks involving delays

This work chiefly explores fractional-order octonion-valued neural networks involving delays. We decompose the considered fractional-order delayed octonion-valued neural networks into equivalent real-valued systems via Cayley-Dickson construction. By virtue of Lipschitz condition, we prove that the solution of the considered fractional-order delayed octonion-valued neural networks exists and is unique. By constructing a fairish function, we confirm that the solution of the involved fractional-order delayed octonion-valued neural networks is bounded. Applying the stability theory and basic bifurcation knowledge of fractional order differential equations, we set up a sufficient condition remaining the stability behaviour and the appearance of Hopf bifurcation for the addressed fractional-order delayed octonion-valued neural networks. To illustrate the justifiability of the derived theoretical results clearly, we give the related simulation results to support these facts. Simultaneously, the bifurcation plots are also displayed. The established theoretical results in this work have important guiding significance in devising and improving neural networks.

Stability Analysis of SARS-CoV-2 with Heart Attack Effected Patients and Bifurcation

The aim of this study is to analyze and investigate the SARS-CoV-2 (SC-2) transmission with effect of heart attack in United Kingdom with advanced mathematical tools. Mathematical model is converted into fractional order with the help of fractal fractional operator (FFO). The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the SC-2 system is verified and test the proposed system with flip bifurcation. Also system is investigated for global stability using Lyponove first and second derivative functions. The existence, boundedness, and positivity of the SC-2 is checked which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects of heart attack in united kingdom. Solutions for fractional order system are derived with the help of advanced tool FFO for different fractional values to verify the combine effect of COVID-19 and heart patients. Simulation are carried out to see symptomatic as well as a symptomatic effects of SC-2 in the United Kingdom as well as its global effects, also show the actual behavior of SC-2 which will be helpful to understand the outbreak of SC-2 for heart attack patients and to see its real behavior globally as well as helpful for future prediction and control strategies.

Mathematical study of polycystic ovarian syndrome disease including medication treatment mechanism for infertility in women

Among women of reproductive age, PCOS (polycystic ovarian syndrome) is one of the most prevalent endocrine illnesses. In addition to decreasing female fertility, this condition raises the risk of cardiovascular disease, diabetes, dyslipidemia, obesity, psychiatric disorders and other illnesses. In this paper, we constructed a fractional order model for polycystic ovarian syndrome by using a novel approach with the memory effect of a fractional operator. The study population was divided into four groups for this reason: Women who are at risk for infertility, PCOS sufferers, infertile women receiving therapy (gonadotropin and clomiphene citrate), and improved infertile women. We derived the basic reproductive number, and by utilizing the Jacobian matrix and the Routh-Hurwitz stability criterion, it can be shown that the free and endemic equilibrium points are both locally stable. Using a two-step Lagrange polynomial, solutions were generated in the generalized form of the power law kernel in order to explore the influence of the fractional operator with numerical simulations, which shows the impact of the sickness on women due to the effect of different parameters involved.

Being wheelchair-bound and being bedridden: Two concept analyses

AIM

Analysis of the concepts and development of a conceptual definition of being wheelchair-bound and being bedridden.

DESIGN

Concept analysis.

METHODS

Walker and Avant´s concept analysis method was used. A thematic analysis guided the determination of the attributes, antecedents and consequences.

RESULTS

Being wheelchair-bound and being bedridden are defined as permanent states in which people are bound to an object. Being passively bound to a wheelchair and being bedridden both mean an increasing restriction of the life-space. Being passive wheelchair-bound often represents a preliminary stage to being bedridden. Both concepts have six attributes: in need of help, powerlessness, life-space confinement, mobility limitation, endurance and weakness. They differ in the main feature maintaining an independent sitting position. Physical immobility and physiological instability are antecedents with the following influencing factors: illness, complexity, burden, endogenous/exogenous booster. The consequences are the progression of inactivity and all related physical and psycho-social problems.

Study of COVID-19 mathematical model of fractional order via modified Euler method

Our main goal is to develop some results for transmission of COVID-19 disease through Bats-Hosts-Reservoir-People (BHRP) mathematical model under the Caputo fractional order derivative (CFOD). In first step, the feasible region and bounded ness of the model are derived. Also, we derive the disease free equilibrium points (DFE) and basic reproductive number for the model. Next, we establish theoretical results for the considered model via fixed point theory. Further, the condition for Hyers-Ulam’s (H-U) type stability for the approximate solution is also established. Then, we compute numerical solution for the concerned model by applying the modified Euler’s method (MEM). For the demonstration of our proposed method, we provide graphical representation of the concerned results using some real values for the parameters involve in our considered model.

The burden of boundedness and the implication for nursing: A scoping review

Background

In the COVID‐19 pandemic, many people experienced temporal boundedness in different ways (e.g., home, country, persons, and rules). However, being bound is also a permanent experience for chronically ill or handicapped people with sometimes serious consequences. To be able to recognize the phenomenon, a clear definition is necessary. In the literature, though, boundedness shows up as a very multifaceted phenomenon.

Objectives

Exploring and conceptualizing the phenomenon of boundedness taking into account the various forms and the consequences for nursing.

Methods

A scoping review using the framework of Arksey and O'Malley and the PRISMA statement (PRISMA‐ScR) to verify the fullness of the review.

Data Sources

Online dictionaries and theoretical and empirical publications in CINAHL, Medline via PubMed, PsycINFO, PsycArticles, Scopus, WISO. A total of 34 sources were included.

Results

Boundedness as a contextual concept is ambiguous. There are three basic causes: an acquired condition, personal obligations, arranged conditions, two principal courses: enduring and temporary, and seven types of being bound: to one or more person(s), to a place/position, to/in an object, to thoughts/opinions, to activities, to/in substances and to time. Examples of types are bedbound, culture‐bound, homebound, time‐bound, wheelchair‐bound and are particularly relevant for care. The consequences are manifold, physically, as well as mentally, and socially.

Conclusion

To reduce or avoid the burdens caused by boundedness, the concept must be implemented in nursing education and nursing practice. To this end, nursing research must further specify the types of boundedness in concept analyses and develop suitable interventions.

Homebound: A concept analysis

AIM

Analysis of the concept and development of a conceptual definition of homebound.

BACKGROUND

Homebound persons have a significantly higher mortality risk as well as physical and psychosocial burden. A clarification of the term is necessary to develop preventive measures.

DESIGN

Concept analysis.

DATA SOURCES

Scientific literature from electronic databases (CINAHL, Medline via PubMed, PsycINFO, PsycArticles, and Scopus) and definitions from online dictionaries.

REVIEW METHODS

Walker and Avant's method was applied to guide the concept analysis. To prevent arbitrary and empty results in determining the attributes, antecedents, and consequences a thematic analysis was carried out.

RESULTS

Homebound is defined as an enduring condition in which the life-space is reduced to the home, but moving around in the home is possible (by walking short distances alone or by holding on to furniture, or with the help of a stick, walker, or another person). Homebound has six attributes: in need of help in ADL/IADL and in leaving the life-space, powerlessness, life-space confinement, mobility limitation, endurance, and weakness. Physiological instability and physical immobility are antecedents with wide-ranging influencing factors as illness, complexity, burden, and endogenous/exogenous booster. Homebound has also wide-ranging consequences such as the progression of inactivity, physical, psychosocial, and/or spiritual problems.

CONCLUSIONS

The multidimensional concept of homebound modifies the concepts of mobility and immobility. Given the extensive consequences of homebound nurses play a central role in the prevention.

BOUNDEDNESS OF A CLASS OF SPATIALLY DISCRETE REACTION-DIFFUSION SYSTEMS

Although the spatially discrete reaction-diffusion equation is often used to describe biological processes, the effect of diffusion in this framework is not fully understood. In the spatially continuous case, the incorporation of diffusion can cause blow-up with respect to the L ∞ norm, and criteria exist to determine whether the system is bounded for all time. However, no equivalent criteria exist for the discrete reaction-diffusion system. Due to the possible dynamical differences between these two system types and the advantage of using the spatially discrete representation to describe biological processes, it is worth examining the discrete system independently of the continuous system. Therefore, the focus of this paper is on determining sufficient conditions to guarantee that the discrete reaction-diffusion system is bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and nonnegative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discrete reaction-diffusion system is bounded. These results are considered in the context of four example systems for which Lyapunov-like functions can and cannot be found.

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature.

Tumour chemotherapy strategy based on impulse control theory

Chemotherapy is a widely accepted method for tumour treatment. A medical doctor usually treats patients periodically with an amount of drug according to empirical medicine guides. From the point of view of cybernetics, this procedure is an impulse control system, where the amount and frequency of drug used can be determined analytically using the impulse control theory. In this paper, the stability of a chemotherapy treatment of a tumour is analysed applying the impulse control theory. The globally stable condition for prescription of a periodic oscillatory chemotherapeutic agent is derived. The permanence of the solution of the treatment process is verified using the Lyapunov function and the comparison theorem. Finally, we provide the values for the strength and the time interval that the chemotherapeutic agent needs to be applied such that the proposed impulse chemotherapy can eliminate the tumour cells and preserve the immune cells. The results given in the paper provide an analytical formula to guide medical doctors to choose the theoretical minimum amount of drug to treat the cancer and prevent harming the patients because of over-treating.This article is part of the themed issue 'Horizons of cybernetical physics'.

A non-local evolution equation model of cell-cell adhesion in higher dimensional space

A model for cell-cell adhesion, based on an equation originally proposed by Armstrong et al. [A continuum approach to modelling cell-cell adhesion, J. Theor. Biol. 243 (2006), pp. 98-113], is considered. The model consists of a nonlinear partial differential equation for the cell density in an N-dimensional infinite domain. It has a non-local flux term which models the component of cell motion attributable to cells having formed bonds with other nearby cells. Using the theory of fractional powers of analytic semigroup generators and working in spaces with bounded uniformly continuous derivatives, the local existence of classical solutions is proved. Positivity and boundedness of solutions is then established, leading to global existence of solutions. Finally, the asymptotic behaviour of solutions about the spatially uniform state is considered. The model is illustrated by simulations that can be applied to in vitro wound closure experiments.