Stability Theory

During the period November 15, 1991 through July 15, 1992, substantial progress :ras made on the research grant DE-FGO3-91ER54115. Three papers have been published in Journal of Nuclear Materials, and two other papers have been submitted for publication. In addition, two Ph.D. students formulated the topics of their theses, and made good progress towards the degree. The basic displacement damage process in SiC has been fully explored, and the mechanisms identified. Major modifications have been made to the theory of damage dosimetry in Fusion, Fission and Ion Simulation studies of SiC. For the first time, calculations of displacements per atoms in SiC can be made in any irradiation environment. Applications to irradiations in fusion first wall neutron spectra (ARIES and PROMETHEUS) ,as well as in fission spectra ( HIFIR and FFTF) are given. Nucleation of helium-filled cavities in SiC was studied, using concepts of stability t_eory to determine the size of the critical nucleus under continuous generation of helium and displacement damage, lt is predicted that a bimodal distribution of cavity sizes is likely to occur in heavily irradiated SiC. A study of the chemical compatibility of SiC composite structures with fusion reactor coolants at high-temperatures was undertaken, lt was shown that SiC itself is chemically very stable in helium coolants in the temperature range 500-1000 °C. However, current fiber/matrix interfaces, such as C and BN are not. The fracture mechanics of high-temperature matrix cracks with bridging fibers is now in progress. A fundamentally unique approach to study the propagation and interaction of cracks in a composite was initiated. The main focus of our research during the following period will be : (1) Theory and experiments for the micro-mechanics of high-temperature failure; and (2) Analysis of radiation damage and microstructure evolution.

COVID pandemic has catalysed the development of novel coronavirus vaccines across pharmaceutical companies and research organizations. In a situation where the vaccine is still unavailable and the disease is spreading exponentially, an effective control measures seems to be the need of an hour to at least contain the size of the disease. In this context, age related transmissibility of COVID-19 has become a public health concern. This disease has caused the most severe health issues for adults over the age of 60 with particularly fatal results for those 80 years and old. In this situation an age structured modelling could be helpful in understanding the spread of the disease across different age groups. In this study initially, we propose an age structured model and calculate the equilibrium points and basic reproduction number. Later we propose an optimal control problem to understand the roles of treatment in controlling the epidemic. From the Stability analysis we see that the infection free equilibrium remains asymptotically stable whenever R0 < 1 and as R0 crosses unity we have the infected equilibrium to be stable. From the sensitivity analysis parameters u11, b1, β1, d1 and µ were found to be sensitive. Findings from the Optimal Control studies suggests that the infection among the adult population(age ≥ 30) is least considering the second control u12 whereas, when both the controls u11 and u12 are considered together the infectives is minimum in case of young populations(age ≤ 30). The cumulative infected population reduced the maximum when the second control was considered followed by considering both the controls together. The control u12 was effective for mild epidemic (R0 ∈ (1, 2)) whereas control u11 was found to be highly effective when epidemic was severe (R0 ∈ (2, 7)) for the population of age group (≤ 30). Whereas for age group (≥ 30) the control u12 was highly effective for the entire range of basic reproduction number. The effect of saturation level in treatment is also explored numerically.

This note provides a sufficient condition for steady states arising from economic optimal control models to be locally asymptotically stable. In particular, we show that if some submatrix of the matrix of eigenvectors of the corresponding Jacobian is invertible, the stable manifold is locally y-parametrizable.

In the present paper we study the stability properties of nonlinear perturbed neutral fractional autonomous linear differential systems with distributed delay. It is shown that if the zero solution of the linear part of the nonlinear perturbed system is globally asymptotically stable, then the zero solution of the perturbed nonlinear system is globally asymptotically stable too. The results are based on a formula for the integral presentation of the general solution of a linear autonomous neutral system with distributed delay which was proved by the authors in a previous work. In what follows we use the following notations: N, R and C are the sets of natural, real and complex numbers, respectively;

Recently, the world is facing the terror of the novel corona-virus, termed as COVID-19. Various health institutes and researchers are continuously striving to control this pandemic. In this article, the SEIAR (susceptible, exposed, infected, symptomatically infected, asymptomatically infected and recovered) infection model of COVID-19 with a constant rate of advection is studied for the disease propagation. A simple model of the disease is extended to an advection model by accommodating the advection process and some appropriate parameters in the system. The continuous model is transposed into a discrete numerical model by discretizing the domains, finitely. To analyze the disease dynamics, a structure preserving non-standard finite difference scheme is designed. Two steady states of the continuous system are described i.e., virus free steady state and virus existing steady state. Graphical results show that both the steady states of the numerical design coincide with the fixed points of the continuous SEIAR model. Positivity of the state variables is ensured by applying the M-matrix theory. A result for the positivity property is established. For the proposed numerical design, two different types of the stability are investigated. Nonlinear stability and linear stability for the projected scheme is examined by applying some standard results. Von Neuman stability test is applied to ensure linear stability. The reproductive number is described and its pivotal role in stability analysis is also discussed. Consistency and convergence of the numerical model is also studied. Numerical graphs are presented via computer simulations to prove the worth and efficiency of the quarantine factor is explored graphically, which is helpful in controlling the disease dynamics. In the end, the conclusion of the study is also rendered.

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

A Caputo-type fractional-order mathematical model for "metapopulation cholera transmission" was recently proposed in [Chaos Solitons Fractals 117 (2018), 37-49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease.

Swine influenza (H1N1) continues to be a major public health problem of interest because of the high potential for transmission and the ability to have rapid mutagenesis and to cause persistence in infected hosts. Here, we introduce and discuss a novel within-host fractional-order mathematical model including environmental viral transmission and nonlinear dynamics of the infection through saturated incidence functions. The model includes healthy epithelial cells, infected cells, free virus particles, and environmental viral load, and the Caputo fractional derivative, accounting for memories of delayed biological responses and viral replication. First, we demonstrate the well-posedness of the model by establishing existence, uniqueness, positivity, and boundedness of solutions. We come up with an invariant region to guarantee that all state variables are biologically significant for time. The basic reproduction number R0 is obtained based on the next-generation matrix methodology and is shown to have two additively related contributions related to the route of either the direct viral transmission and/or the environmental feedback pathways. Analyses of equilibria stability are performed with linearization, Routh-Hurwitz criteria, and the fractional order stability theory. We found that viral dynamics outside the host significantly increase the effective reproduction number and can prolong infection by reinfection mechanisms. Collectively, these results illustrate that environmental transmission effects, nonlinear infection, and memory exert significant contributions to an infection process on infection outcomes. The model provides an integrated perspective of swine flu physiology, and could serve as a framework for effective control programmes.

This paper considers the problem of robust non-fragile control for a class of two-dimensional (2-D) discrete uncertain systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model under controller gain variations. The parameter uncertainty is assumed to be norm-bounded. The problem to be addressed is the design of non-fragile robust controllers via state feedback such that the resulting closed-loop system is asymptotically stable for all admissible parameter uncertainties and controller gain variations. A sufficient condition for the existence of such controllers is derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. Finally, a numerical example is illustrated to show the contribution of the main result.

This paper considers the problem of robust non-fragile control for a class of two-dimensional (2-D) discrete uncertain systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model under controller gain variations. The parameter uncertainty is assumed to be norm-bounded. The problem to be addressed is the design of non-fragile robust controllers via state feedback such that the resulting closed-loop system is asymptotically stable for all admissible parameter uncertainties and controller gain variations. A sufficient condition for the existence of such controllers is derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. Finally, a numerical example is illustrated to show the contribution of the main result.

This companion appendix provides a standalone numerical supplement to the
current reciprocal quantum hydrodynamics manuscript. The goal is narrow and
explicit: to convert the paper’s one-dimensional periodic Weaver verification protocol
into an executable, auditable simulation model. The supplement records the dimen-
sionless verification system, the numerical scheme, the diagnostic contract, a baseline
full-system run, a modal verdict sweep across the mobility exponents n= 0.5,1.0,1.5,
and a nodal ϵ-descent stress test. The main numerical findings are that mass is
preserved to machine precision in the reported runs, the high-mode modal response
exhibits the predicted threshold ordering across the n= 1 transition, and decreasing
ϵ strongly amplifies the anomaly magnitude near nodal structure while remaining
controlled in the present regularized reduced solver. These results strengthen the
manuscript’s verification and falsifiability claims, while leaving the singular ϵ→0
limit, higher-dimensional non-normality diagnostics, and experimental calibration as
open targets.

We formulate a regularized reciprocal extension of quantum hydrodynamics motivated
by action-reaction symmetry in pilot-wave theory and by Sutherland-type post-quantum
back-reaction schemes. The reciprocal anomaly is sourced by the imaginary part of the
local momentum weak value. In the formal unregularized limit this gives the minimal
closure ∆p proportional to Im(pw ). Since that reduced closure is singular at nodes and
generically behaves as an effectively open dynamics, we construct a bounded completion on
the torus by introducing: (i) a nodal regularization Aε =−(ℏ/2m)∇ln(ρ+ ε), (ii) a smooth
information-limited mobility controlled by a nonnegative global admissibility functional
built from relative entropy and regularized Fisher information, and (iii) a transported
Eulerian memory field encoding cumulative reciprocal activity. The resulting model is a
globally coupled nonlinear PDE for (ρ,S,M) with Hamiltonian advection, bounded reciprocal
transport, capillarity-regularized quantum pressure, and local memory loading. We prove
exact identities for weak-value decomposition, mass conservation, reciprocal-current entropy-
gradient form, and memory-energy balance. We then state an energy-controlled local classical
well-posedness theorem for fixed ε>0, based on a high-regularity Sobolev–capillarity energy.
A modal asymptotic result shows that if the mobility exponent satisfies n>1, ultraviolet
amplification is suppressed in a Fisher-dominated fixed-amplitude scaling regime, with an
extended condition for amplitude-decaying spectra. In reciprocal equilibrium the model
reduces to a regularized Madelung-type quantum hydrodynamics, recovering the standard
unregularized system formally as ε →0 away from nodal singularities. The framework
remains a candidate closure rather than an established physical theory, but its mathematical
content is explicit enough to be analyzed, simulated, and falsified.

We study Max-Product and Max-Plus Systems with Markovian Jumps and focus on stochastic stability problems. At first, a Lyapunov function is derived for the asymptotically stable deterministic Max-Product Systems. This Lyapunov function is then adjusted to derive sufficient conditions for the stochastic stability of Max-Product systems with Markovian Jumps. Many step Lyapunov functions are then used to derive necessary and sufficient conditions for stochastic stability. The results for the Max-Product systems are then applied to Max-Plus systems with Markovian Jumps, using an isomorphism and almost sure bounds for the asymptotic behavior of the state are obtained. A numerical example illustrating the application of the stability results on a production system is also given.

This note formalizes δ as the deterministic transition function governing admissible state evolution within the IDDA architecture. The model distinguishes execution, holding, blocking, and controlled reset as distinct outcomes of a unified transition rule. A key design property is that blocking is not terminal: inadmissible progression triggers a controlled reset to a designated safe baseline s 0 , preserving operational continuity. The repeated application of δ induces a piecewise-stable state trajectory under admissibility constraints. The result is a compact formal layer connecting admissibility, execution, and recovery.

In the theory of stochastic processes a special role is played by results concerning the existence of invariant densities and the long-time behaviour of their distributions. These results can be formulated and proved in terms of stochastic semigroups induced by these processes. We consider two properties: asymptotic stability and sweeping. A stochastic semigroup induced by a stochastic process is asymptotically stable if the densities of one-dimensional distributions of this process converge to a unique invariant density. Sweeping is an opposite property to asymptotic stability and it means that the probability that trajectories of the process are in a set Z goes to zero. The main result presented here shows that under some conditions a substochastic semigroup can be decomposed into asymptotically stable parts and a sweeping part. This result and some irreducibility conditions allow us to formulate theorems concerning asymptotic stability, sweeping and the Foguel alternative. This alternative says that under suitable conditions a stochastic semigroup is either asymptotically stable or sweeping. Let the triple (X, Σ, m) be a σ-finite measure space. In the whole chapter, we will consider substochastic operators and semigroups on L 1 = L 1 (X, Σ, m) corresponding to transition kernels. The support of any measurable function f is defined up to a set of measure zero by the formula supp f = {x ∈ X : f (x) = 0}.

This paper is concerned with the robust stability and stabilization for a class of switched discrete-time systems with state parameter uncertainty. Firstly, a new matrix inequality considering uncertainties is introduced and proved. By means of it, a novel sufficient condition for robust stability and stabilization of a class of uncertain switched discrete-time systems is presented. Furthermore, based on the result obtained, the switching law is designed and has been performed well, and some sufficient conditions of robust stability and stabilization have been derived for the uncertain switched discrete-time systems using the Lyapunov stability theorem, block matrix method, and inequality technology. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R 0 is computed using the nextgeneration matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R 0 ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R 0 > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R 0 = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.

Classical systems theory explains stability through feedback-driven regulation toward equilibrium. From early cybernetics to contemporary control theory and dynamical systems, the dominant assumption has been that systems maintain coherence by detecting deviation and correcting it. Gradients generate movement, feedback reduces error, and over time, systems tend toward stable states or bounded oscillations around equilibrium conditions. This framework has proven remarkably powerful, offering a generalizable language for understanding regulation across physical, biological, and engineered domains. Existing work in thermodynamics and systems theory has already demonstrated that stability can emerge under non-equilibrium conditions. The present paper does not dispute these findings. Instead, it identifies and formalizes a specific subclass of such systems whose persistence depends on the structured retention and continuous exploitation of gradients. Yet a persistent tension remains. Many real systems do not behave in accordance with equilibrium-centered expectations. They do not fully dissipate gradients, nor do they reliably converge toward balanced states. Instead, they exhibit forms of stability that coexist with ongoing imbalance. Differences in potential, pressure, or distribution are not eliminated but maintained over extended durations. These systems remain operational-often highly productive-precisely because gradients persist rather than resolve. This suggests that equilibrium-based models, while foundational, are incomplete.

Celebrity worship is a form of almost obsessional participation in which people idolize and even start to “worship” their favorite celebrities. Twitter reveals that in 2021, Indonesia was ranked fourth as the country with the most number of Korean Pop (K-Pop) fans. In this study, we constructed a SEIRS mathematical model of celebrity worship tendency among K-Pop fans (CWKF) because there have not been specific mathematical models built on the problem of celebrity worship tendency among K-Pop fans. The CWKF free equilibrium point ( E 0 ), endemic equilibrium point ( E 1 ), and using the Next Generation Matrix for the Basic Reproduction Number ( R 0 ) are calculated through analysis. Using the Routh-Hurwitz criteria for equilibrium analysis, we found that E 0 is locally asymptotically stable if R 0 &lt; 1, and E 1 is locally asymptotically stable if R 0 &gt; 1. Secondly, we conduct numerical simulations using Maple software, and research data consisting of 204 individuals from various...

The main aim of this paper is to consider the classes of quasiasymptotically almost periodic functions and Stepanov quasi-asymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions, Stepanov asymptotically almost periodic functions and S-asymptotically ω-periodic functions with values in Banach spaces. We investigate the invariance of introduced properties under the action of finite and inifinite convolution products, providing also an illustrative application to abstract nonautonomous differential equations of first order. 2010 Mathematics Subject Classification. 47A16, 47B37, 47D06. Key words and phrases. Quasi-asymptotically almost periodic functions, Stepanov quasiasymptotically almost periodic functions, convolution products, evolution systems, abstract nonautonomous differential equations of first order.

The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.

In this paper, we analyze multi-dimensional (R X , B)-almost periodic type functions and multi-dimensional Bohr B-almost periodic type functions. The main structural characterizations and composition principles for the introduced classes of almost periodic functions are established. Several applications of our abstract theoretical results to the abstract Volterra integrodifferential equations in Banach spaces are provided, as well. Contents 2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. (R, B)-Multi-almost periodic type functions, (R X , B)-multi-almost periodic type functions, Bohr B-almost periodic type functions, composition principles, abstract Volterra integro-differential equations.

In this paper, we introduce and analyze several different notions of almost periodic type functions and uniformly recurrent type functions in Lebesgue spaces with variable exponent L p ( x ) . We primarily consider the Stepanov and Weyl classes of generalized almost periodic type functions and generalized uniformly recurrent type functions. We also investigate the invariance of generalized almost periodicity and generalized uniform recurrence with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract fractional differential inclusions in Banach spaces.

In this paper, we consider composition principles for generalized almost periodic functions. We prove several new composition principles for the classes of (asymptotically) Stepanov p-almost periodic functions and (asymptotically, equi-)Weyl p-almost periodic functions, where 1 ≤ p < ∞, and explain how we can use some of them in the qualitative analysis of solutions for certain classes of abstract semilinear Cauchy inclusions in Banach spaces.

In the paper under review, we introduce the notions of various types of generalized (asymptotical) almost periodicity with variable exponents. We define and thoroughly analyze an important subclass of (asymptotically) Stepanov almost periodic functions which contains all (asymptotically) almost periodic functions. We provide a great number of relevant applications to abstract Volterra integro-differential equations in Banach spaces. 2010 Mathematics Subject Classification. 34C27, 35B15, 46E30. Key words and phrases. Lebesgue spaces with variable exponents, generalized almost periodicity with variable exponents, generalized asymptotical almost periodicity with variable exponents, abstract Volterra integro-differential equations, abstract fractional differential equations. Please, insert the number of your grant here... The second named author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.

In this paper, we introduce and analyze several different notions of Weyl almost periodic functions and Weyl ergodic components in Lebesgue spaces with variable exponent L p(x) . We investigate the invariance of (asymptotical) Weyl almost periodicity with variable exponent under the actions of convolution products, providing also some illustrative applications to abstract fractional differential inclusions in Banach spaces. The introduced classes of generalized (asymptotically) Weyl almost periodic functions are new even in the case that the function p(x) has a constant value p ≥ 1, provided that the functions φ(x) and F (l, t) under our consideration satisfy φ(x) = x or F (l, t) = l (-1)/p .

In this paper, we introduce several various classes of (?, c)-almost periodic type functions and their Stepanov generalizations. We also consider the corresponding classes of (?, c)-almost periodic type functions depending on two variables and related composition principles. We provide several illustrative examples and applications to the abstract Volterra integro-differential equations in Banach spaces.

A constructive method for time-varying stabilization of smooth driftless controllable systems is developed. It provides time-varying homogeneous feedback laws that are continuous and smooth away from the origin. These feedbacks make the closed-loop system globally exponentially asymptotically stable if the control system is homogeneous with respect to a family of dilations and, using local homogeneous approximation of control systems, locally exponentially asymptotically stable otherwise. The method uses some known algorithms that construct oscillatory control inputs to approximate motion in the direction of iterated Lie brackets that we adapt to the closed-loop context.

This paper presents a comprehensive cone-theoretic comparison principle for Caputo fractional differential equations, thereby addressing a key gap in the qualitative theory of fractional systems. Although classical comparison techniques based on scalar and vector Lyapunov functions have been extended to fractional settings, the more general and powerful framework of cone-valued Lyapunov functions has received little attention. In this work, we develop a complete theoretical foundation for this extension. We establish core results on cone-preserving fractional differential inequalities, prove the existence and characterization of maximal solutions with respect to arbitrary closed convex cones, and derive a general comparison principle for Caputo fractional systems. The central result is a unified comparison theorem that enables stability analysis of fractional systems using cone-valued Lyapunov functions. This approach incorporates both scalar and vector Lyapunov methods as special cases, and can be naturally extended to systems whose dynamics follow non-standard partial orderings. In all, our results provide a robust theoretical basis for studying complex fractional-order systems that lie beyond the reach of traditional comparison techniques.

Rabies is a viral disease of nervous system that is often transmitted to human beings through the bite or scratch of rabid animals. The uprising of in-security globally has forced several people to get dogs in their houses. In this paper the mathematical model of rabies transmission and control was formulated by incorporating vaccination class. The Disease Free Equilibrium (DFE) state of the model was obtain and used to compute the basic reproduction number 0 R . Local stability analysis of the DFE was carried out using Jacobian Matrix techniques. The DFE is locally asymptotically stable if

In this paper, a nine compartmental model for malaria transmission in children was developed and a threshold parameter called control reproduction number which is known to be a vital threshold quantity in controlling the spread of malaria was derived. The model has a disease free equilibrium which is locally asymptotically stable if the control reproduction number is less than one and an endemic equilibrium point which is also locally asymptotically stable if the control reproduction number is greater than one. The model undergoes a backward bifurcation which is caused by loss of acquired immunity of recovered children and the rate at which exposed children progress to the mild stage of infection.

Rabies is a viral disease of nervous system that is often transmitted to human beings through the bite or scratch of rabid animals. The uprising of in-security globally has forced several people to get dogs in their houses. In this paper the mathematical model of rabies transmission and control was formulated by incorporating vaccination class. The Disease Free Equilibrium (DFE) state of the model was obtain and used to compute the basic reproduction number 0 R . Local stability analysis of the DFE was carried out using Jacobian Matrix techniques. The DFE is locally asymptotically stable if

This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

In this paper, a nine compartmental model for malaria transmission in children was developed and a threshold parameter called control reproduction number which is known to be a vital threshold quantity in controlling the spread of malaria was derived. The model has a disease free equilibrium which is locally asymptotically stable if the control reproduction number is less than one and an endemic equilibrium point which is also locally asymptotically stable if the control reproduction number is greater than one. The model undergoes a backward bifurcation which is caused by loss of acquired immunity of recovered children and the rate at which exposed children progress to the mild stage of infection. Keywords: Malaria, Model, Backward Bifurcation, Local Stability.

The objective of this paper is to present a mathematical model formulated to investigate the dynamics of human immunodeficiency virus (HIV). The disease free equilibrium of the model was found to be locally and globally asymptotically stable. The endemic equilibrium point exists and it was discovered that the endemic equilibrium point is globally asymptotically stable; suggestion was also made in the research in regards to protection during sexual intercourse especially in a sexually active population.

This paper examines the transmission dynamics of avian influenza. A nonlinear mathematical model for the problem is formulated and analysed. For the prevalence of the disease and the ease of analysis, we considered the model in proportions of susceptible, infectious, isolated and recovered compartments. The basic reproduction number was computed and used to prove the stability of the disease free equilibrium states. It is proved that the basic reproduction number is a decreasing function of the culling rate of infected birds. It is further proved that the disease free equilibrium state is locally asymptotically stable whenever the basic reproduction number is less than unity.

Assuming a general distribution for the sojourn time in the infectious class, we consider an SIS type epidemic model formulated as a scalar integral equation. We prove that the endemic equilibrium of the model is globally asymptotically stable whenever it exists, solving the conjecture of Hethcote and van den Driessche (1995) for the case of nonfatal diseases.

The performance and stability of tokamak plasmas are strongly influenced by their geometric shaping. In compact spherical tokamaks (low aspect ratio), advanced shaping configurations featuring high elongation and negative triangularity (an inward-facing <D= shape) have demonstrated improved confinement and stability at high plasma pressure (high beta). This paper analyzes the role of geometry in achieving stable, high-beta operation, with emphasis on negative triangularity effects observed in experiments on devices like TCV and DIII-D. We formulate the equilibrium framework via the Grad3Shafranov equation and define key shaping parameters4 elongation (»), aspect ratio (A), triangularity (•)4along with the normalized beta (³_N) used to gauge pressure limits. The impacts of extreme shaping on vertical stability and bootstrap current fraction are examined. Negative triangularity plasmas (• < 0) are shown to suppress edge instabilities (notably avoiding ELMs) while maintaining H-mode-level confinement without an edge pedestal[1][2]. High-beta regimes in spherical tokamaks (A j 1.53 2.0, » > 2.5) are explored, including the Troyon limit on ³ and the engineering constraints of REBCO high-field magnets. Structural considerations such as Lorentz force management via bucking cylinders and strain-focused magnet design (as in the Kronos S.M.A.R.T. tape architecture) are discussed in enabling such shapes. Finally, we consider the relevance of these geometries to aneutronic fusion (D3He³, p3¹¹B), where shaped field-lines can facilitate direct energy conversion of charged fusion products. Negative triangularity and compact geometries emerge as promising strategies for next-generation fusion reactors, combining high confinement efficiency with manageable plasma3wall interactions and advanced fuel compatibility.

In previous researches control techniques are defined to let underactuated marine surface vessels follow a straight line path in formation and to let a single vessel follow a path while disturbed by ocean current. Combining both techniques, this report investigates straight line path following of underactuated marine surface vessels in formation including compensation of the ocean current. In this sense decentralized controllers based on the Line Of Sight guidance law are defined for all participants. Furthermore the formation will move with a given velocity and structure and in the end of the report the results are validated by simulations.

The purpose of this article is to formulate a simplified nonlinear fractional mathematical model to illustrate the dynamics of the new coronavirus . Based on the infectious characteristics of COVID-19, the population is divided into five compartments: susceptible S(t), asymptomatic infection I(t), unreported symptomatic infection U(t), reported symptomatic infections W(T) and recovered R(t), collectively referred to as (SIUWR). The existence, uniqueness, boundedness, and non-negativeness of the proposed model solution are established. In addition, the basic reproduction number R 0 is calculated. All possible equilibrium points of the model are examined and their local and global stability under specific conditions is discussed. The disease-free equilibrium point is locally asymptotically stable for R 0 leq1 and unstable for R 0 > 1. In addition, the endemic equilibrium point is locally asymptotically stable with respect to R 0 > 1. Perform numerical simulations using the Adams-Bashforth-Moulton-type fractional predictor-corrector PECE method to validate the analysis results and understand the effect of parameter variation on the spread of COVID-19. For numerical simulations, the behavior of the approximate solution is displayed in the form of graphs of various fractional orders. Finally, a brief conclusion about simulation on how to model transmission dynamics in social work.

This paper deals with results on existence of asymptotically almost automorphic solutions for a third order in time abstract differential equation which model, on one side, high intensity ultrasound in acoustic wave propagation, while on the other side, vibrations of flexible structures possessing internal material damping. We established the asymptotically almost automorphy of the output solution subject to the asymptotically almost automorphy of the input distur- bance.

Physical theories successfully describe how systems evolve once laws and initial conditions are specified. Yet they rarely address a prior question: why do stable law-like regimes exist at all? The observable universe is dominated by persistent structures—particles, atoms, stars, and coherent interaction frameworks—despite the vast space of mathematically admissible configurations.

This paper proposes that selection may be structurally primitive. Rather than treating stability as a derivative consequence of fixed laws, we introduce a minimal formal hypothesis in which a stability-weighted measure acts on the space of admissible histories. Within this framework, law-regimes emerge as attractors under a deeper selection functional.

The proposal preserves standard dynamics while reinterpreting persistence as the primary explanatory feature. Possible implications for cosmology, statistical structure, and conscious systems are briefly outlined.

Background: This study proposes a simplified SIR-type compartmental mathematical model tailored to the HIV epidemic in Mizoram, a small state in India and is one of the highest-prevalence regions in the country. The model includes key processes such as antiretroviral treatment (ART), behaviour-change dynamics, recruitment, and natural mortality. Aims and Objectives: This study emphasizes the need for early treatment initiation and promotion of risk-reducing behaviour to control and curb the prevalence of HIV/AIDS especially in a highly prevalent region like Mizoram and may serve as a practical and flexible resource for policymakers and public health planners to evaluate the effectiveness of early interventions in high-prevalence populations. Materials and Methods: The basic reproduction number R0 is derived using the next-generation matrix method and local as well as global stability conditions for both disease-free and endemic equilibria are established via Lyapunov functions. Numerical simulations demonstrate the impact of transmission rate (β), treatment uptake (γ), and behavior-modification rate (µ) on epidemic outcomes. Results and Conclusion: It is observed that when γ and µ i.e., rate of treatment uptake and behaviour-modification rate respectively are increased effectively, R0 decreases and when R0<1, the disease-free equilibrium is globally asymptotically stable, indicating the possibility of disease eradication. When R0>1, the endemic equilibrium remains, indicating persistent transmission in the population. Although µ does not directly reduce transmission, it significantly boosts the recovered class when increased.

The transition between the core and scrape-off layer of a tokamak corresponds to a marked momentum shear layer, owing to sheath acceleration on limiters which drives near-sonic flows along the plasma magnetic field in the scrape-off layer, and a parallel shear flow instability can possibly be triggered. The possibility of this instability driven by the velocity gradient is investigated numerically, using a minimum model of particle and parallel momentum transport in the edge of a tokamak, in a computational domain modelling a limiter plasma with background turbulence modelled as an effective diffusion. It is found that unstable regions can exist in the vicinity of a limiter, in agreement with experimental findings, when momentum radial transport – and therefore coupling between SOL and core flows – is sufficiently weak. Instability is reinforced by core rotation, and is found to be maximum downstream of the limiter (with respect to the core plasma flow).

The cellular and dendritic growth of Zn–Cd dilute alloys is directly influenced by crystalline orientation, the primary orientation being the 〈 1 0 1 ¯ 0 〉 and eventually 〈0 0 0 1〉. In general, this information is absent during the description of planar to cellular transition. However, it has been experimentally shown that the critical velocity for the stability limit, and the associated wavelength of the first instabilities, are different depending on the growth direction. In this work the stability of the accelerated planar fronts of Zn–0.01%Cd samples, obtained using a Bridgman-like device, and the behavior against morphological stability theory has been compared.

Directional growth of dilute Al–0.2 wt.% Cu alloy was performed under conditions of controlled solidification. Proper metallographic analysis of the segregation patterns behind the solid–liquid (S–L) interface gives information about the effect of the local solidification conditions that determines the microsegregation. In order to explain the apparition of different solidification substructure as a function of the parameter ( G L / V C 0 ) ( G L , thermal gradient in the liquid in front of the S–L interface; V , solidification rate and C 0 , nominal composition of the alloy), as well as the small segregated instabilities detected at the cellular walls, a mechanism based on the morphological stability theory is proposed.