Stability Properties Of Equilibrium Points Research Articles

Assessing the impact of information-induced self-protection on Zika transmission: A mathematical modeling approach

Abstract As per the World Health Organization’s (WHO’s) suggestions, personal protection via adopting precautionary measures is one of the most effective control aspects to avoid Zika infection in the absence of suitable medical treatment. This personal protection further can be enhanced and explored by propagating information about disease prevalence. Therefore, in this study, we wish to see the effect of information on Zika transmission by formulating a compartmental mathematical model that quantifies the effect of an individual’s behavioral response as self-protection due to information. Furthermore, the basic reproduction number was calculated using the next-generation matrix technique. The model analysis was carried out to determine the local and global stability properties of equilibrium points. In addition, the model shows the occurrence of forward bifurcation when the reproduction number crosses unity. To understand the impact of various model parameters, we conducted a sensitivity analysis using both the normalized sensitivity index and the partial rank correlation coefficient methods. Moreover, we performed numerical simulations to assess the influence of important parameters on the model’s behavior for Zika prevalence. Our study accentuates that as information-induced self-protection increases, the prevalence of Zika infection will be at a very minimum level, and this observation is in line with WHO suggestions.

A mathematical model of cell-mediated immune response to tumor.

Mathematical models of tumor-immune interactions provide an analytic framework for studying tumor-immune dynamics. In this paper, we present a mathematical model to describe tumor-immune cell interactions, focusing on the role of the natural killer (NK) cells and CD8+ cytotoxic T lymphocytes (CTLs) in immune surveillance. According to the experimental and clinical results, we determine part of the model parameters to reduce the model parameter space. Then we analyze the local geometric properties of the equilibria of model and carry out numerical simulations to verify the conditions for the stability properties of equilibrium points. Numerical results suggest that the host immune system alone is not fully effective against progression of tumor cells, and CTLs play a crucial role in immune surveillance.

Dynamics of a SIR Epidemic Model of Childhood Diseases with a Saturated Incidence Rate: Continuous Model and Its Nonstandard Finite Difference Discretization

A SIR epidemic model that describes the dynamics of childhood disease with a saturated incidence rate and vaccination program at a constant rate was investigated. For the continuous model we first show its basic properties, namely, the non-negativity and boundedness of solutions. Then we investigate the existence and both local and global stability of the equilibrium points. It was found that the existence and stability properties of equilibrium points fully determined the basic reproduction number. We also propose and analyze a discrete-time analogue of the continuous childhood diseases by applying a nonstandard finite difference method. It is shown that our discrete model preserves the dynamical properties of the corresponding continuous model, such as the positivity solutions, the population conservation law, the existence of equilibrium points and their global stability properties.

Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays

A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study.

Global dynamics of a ratio-dependent Holling–Tanner predator–prey system

A ratio-dependent Holling–Tanner predator–prey model is studied. By investigating the local stability properties of equilibrium points and periodic solutions of a transformed system, and by applying the Poincaré–Bendixson theorem, we are able to give a complete classification of the global dynamics of the ratio-dependent Holling–Tanner predator–prey model.

Dynamic Lyapunov functions

Lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems. The existence of Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. Nevertheless the actual computation (of the analytic expression) of the function may be difficult. Herein we propose an approach to avoid the computation of an explicit solution of the Lyapunov partial differential inequality, introducing the concept of Dynamic Lyapunov function. These functions allow to study stability properties of equilibrium points, similarly to standard Lyapunov functions. In the former, however, a positive definite function is combined with a dynamical system that render Dynamic Lyapunov functions easier to construct than Lyapunov functions. Moreover families of standard Lyapunov functions can be obtained from the knowledge of a Dynamic Lyapunov function by rendering invariant a desired submanifold of the extended state-space. The invariance condition is given in terms of a system of partial differential equations similar to the Lyapunov pde. Differently from the latter, however, in the former no constraint is imposed on the sign of the solution or on the sign of the term on the right-hand side of the equation. Several applications and examples conclude the paper.

Dynamics of prey threshold harvesting and refuge

The dynamics of a predator–prey model with continuous threshold prey harvesting and prey refuge is studied. One central question is how harvesting and refuge could directly affect the dynamics of the ecosystem, such as the stability properties of some coexistence equilibria and periodic solutions. Theoretical and numerical methods are used to investigate boundedness of solutions, existence of bionomic equilibria, as well as the existence and stability properties of equilibrium points and periodic solutions. Several bifurcations are also studied.

Dynamic Lyapunov Functions

AbstractLyapunov functions are a fundamental tool to investigate the stability properties of equilibrium points in linear or nonlinear systems. Unfortunately, even if the existence of Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems, the actual computation of the analytic expression of the function may be difficult or impossible. Herein we propose an approach to avoid the issue of finding an explicit solution of the Lyapunov partial differential inequality, providing a family of Lyapunov functions for linear and nonlinear systems.

Population-Level Consequences of Antipredator Behavior: A Metaphysiological Model Based on the Functional Ecology of the Leaf-Eared Mouse

We present a predator–prey metaphysiological model, based on the available behavioral and physiological information of the sigmodontine rodent Phyllotis darwini. The model is focused on the population-level consequences of the antipredator behavior, performed by the rodent population, which is assumed to be an inducible response of predation avoidance. The decrease in vulnerability is explicitly considered to have two associated costs: a decreasing foraging success and an increasing metabolic loss. The model analysis was carried out on a reduced form of the system by means of numerical and analytical tools. We evaluated the stability properties of equilibrium points in the phase plane, and carried out bifurcation analyses of rodent equilibrium density under varying conditions of three relevant parameters. The bifurcation parameters chosen represent predator avoidance effectiveness (A), foraging cost of antipredator behavior (C1′), and activity-metabolism cost (C4′). Our analysis suggests that the trade-offs involved in antipredator behavior plays a fundamental role in the stability properties of the system. Under conditions of high foraging cost, stability decreases as antipredator effectiveness increases. Under the complementary scenario (not considering the highest foraging costs), the equilibria are either stable when both costs are low, or unstable when both costs are higher, independent of antipredator effectiveness. No evidence of stabilizing effects of antipredator behavior was found.

Analysis and synthesis of neural networks with lower block triangular interconnecting structure

A qualitative analysis of Hopfield-type neural network models with lower block triangular interconnecting structure is presented. Such networks are viewed as interconnected systems and the results are phrased in terms of the qualitative properties of the subsystems of the networks and in terms of the properties of the interconnecting structure of the networks. The results address the stability properties of equilibrium points and estimates of trajectory properties. A design method is devised for this class of neural networks to establish desired relationships between scalar-valued analog input signals and output signals in binary form. The applicability of the design methodology is demonstrated by means of several specific examples, including the design of an A/D converter, the design of a resistor sorter, and the design of a resister tolerancer. Specific simulations show that the present design method offers significant improvements over other design techniques that employ Hopfield neural networks.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Deterministic single-locus density-dependent selection.

Density-regulated selection is considered for a single, multiallele gene locus and separated generations. Characteristics resulting from the basic assumption that the average population fitness decreases with increasing density are derived. Under this assumption, it proves to be necessary to distinguish between regions of allelic frequencies which imply limited population growth, unlimited growth, or ultimate extinction when the population stays in the respective region. Particular attention is given to the investigation of the region of limited growth and the 'carrying capacity function' theta defined on it. Relationships between theta and the average fitness (adaptive surface) in the non-density dependent model are explained. Besides stability properties of equilibrium points, more general characteristics concerning the asymptotic behavior of population trajectories are treated. In this context, the problems of sudden loss of alleles and of population extinction as a result of large fluctuations in density are discussed.