Nontrivial Equilibrium Research Articles

Mathematical modeling and projections of a vector-borne disease with optimal control strategies: A case study of the Chikungunya in Chad

In this work, we extend existing models of vector-borne diseases by including density-dependent rates and some existing control mechanisms to decrease the disease burden in the human population. We begin by analyzing the vector model dynamics and by determining the offspring reproductive number denoted by N as well as the trivial and nontrivial equilibria. Using theory of cooperative systems and the general theory of Lyapunov, we prove that, although there is a possibility that the trivial equilibrium coexists with a positive equilibrium, it remains globally asymptotically stable whenever N ≤ 1 . The fact that the non-trivial equilibrium is globally asymptotically stable permits us to reduce the study of the full model to the study of a reduced model whenever N > 1 . Thus, we analyze the reduced model by computing the basic reproduction number R 0 , equilibrium points as well as asymptotic stability of each equilibrium point. We also explore the nature of the bifurcation for the disease-free equilibrium from R 0 = 1 . By the application of the centre manifold theory, we prove that the backward bifurcation phenomenon can occur in our model, which means that the necessary condition R 0 < 1 is not sufficient to guarantee the final extinction of the disease in human populations. To calibrate our model, we estimate model parameters on clinical data from the last Chikungunya epidemic which occurred in Chad, using the non-linear least-square method. We find out that R 0 = 1.8519 , which means that we are in an endemic state since R 0 > 1 . To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA) using a global method. It follows that the density-dependent death rate of mosquitoes and the average number of mosquito bites are key parameters in the disease dynamics. Following this, we thus formulate an optimal control model by including in the autonomous model, four time-dependent control functions to fight the disease spread. Pontryagin’s maximum principle is used to characterize our optimal controls. Numerical simulations, using parameter values of Chikungunya transmission dynamics, and efficiency analysis, are conducted to determine the better control strategy which guaranteed the final extinction of the disease in human populations.

Post-buckling behaviour of thin-walled regular polygonal tubes subjected to bending

This work presents and discusses numerical results concerning the elastic post-buckling behaviour and imperfection sensitivity of regular convex polygonal cross-section (RCPS) tubular beams buckling in local, distortional and mixed local–distortional modes, a topic currently lacking research. This study is carried out in the framework of Generalised Beam Theory (GBT) geometrically non-linear analyses, enriched with a branch switching technique, and takes advantage of the GBT intrinsic modal nature to shed new light on the mechanics underlying the post-buckling behaviour of these members. Due to the small half-wavelength of all the buckling phenomena addressed, only simply supported members under uniform bending are investigated. In particular, this work investigates the post-buckling behaviour and imperfection sensitivity of RCPS beams (i) exhibiting several wall numbers (6, 8, 10, 14, 20, 30) with distinct combinations of circumradius-to-thickness ratios (ii) having distinct lengths, and (iii) containing critical-mode initial geometrical imperfections with several amplitudes. Relevant displacement profiles and modal participation diagrams are provided along trivial and non-trivial equilibrium paths, in order to draw meaningful conclusions concerning the post-buckling behaviour of RCPS tubes under bending. For comparison and validation purposes, ABAQUS shell finite element results are also presented.

Analysis of a mosquito life cycle model

The gonotrophic cycle of mosquitoes conditions the frequency of mosquito-human contacts. The knowledge of this important phenomenon in the mosquito life cycle is a fundamental element in the epidemiological analysis of a communicable disease such as mosquito-borne diseases.In this work, we analyze a deterministic model of the complete life cycle of mosquitoes which takes into account the principal phases of female mosquitoes' gonotrophic cycle, and the Sterile Insect technique combined with the use of insecticide as control measures to fight the proliferation of mosquitoes. We compute the corresponding mosquito reproductive number N ∗ and prove the global asymptotic stability of trivial equilibrium. We prove that the model admits two non-trivial equilibria whenever N^{∗} is greater than another threshold, N_c, which the total number of sterile mosquitoes depends on. Numerical simulations, using mosquito parameters of the Aedes species, are carried out to illustrate our analytical results and permit to show that the strategy which consists in combining the sterile insect technique with adulticides, when it is well done, effectively combats the proliferation of mosquitoes.

Multistability and circuit implementation of tabu learning two-neuron model: application to secure biomedical images in IoMT.

In this paper, the dynamics of a non-autonomous tabu learning two-neuron model is investigated. The model is obtained by building a tabu learning two-neuron (TLTN) model with a composite hyperbolic tangent function consisting of three hyperbolic tangent functions with different offsets. The possibility to adjust the compound activation function is exploited to report the sensitivity of non-trivial equilibrium points with respect to the parameters. Analysis tools like bifurcation diagram, Lyapunov exponents, phase portraits, and basin of attraction are used to explore various windows in which the neuron model under the consideration displays the uncovered phenomenon of the coexistence of up to six disconnected stable states for the same set of system parameters in a TLTN. In addition to the multistability, nonlinear phenomena such as period-doubling bifurcation, hysteretic dynamics, and parallel bifurcation branches are found when the control parameter is tuned. The analog circuit is built in PSPICE environment, and simulations are performed to validate the obtained results as well as the correctness of the numerical methods. Finally, an encryption/decryption algorithm is designed based on a modified Julia set and confusion–diffusion operations with the sequences of the proposed TLTN model. The security performances of the built cryptosystem are analyzed in terms of computational time (CT = 1.82), encryption throughput (ET = 151.82 MBps), number of cycles (NC = 15.80), NPCR = 99.6256, UACI = 33.6512, χ2-values = 243.7786, global entropy = 7.9992, and local entropy = 7.9083. Note that the presented values are the optimal results. These results demonstrate that the algorithm is highly secured compared to some fastest neuron chaos-based cryptosystems and is suitable for a sensitive field like IoMT security.

Compressible viscous heat-conducting surface wave without surface tension

In this paper, we consider a three-dimensional compressible viscous heat-conducting fluid in a horizontally periodic domain, bounded above by a free surface and below by a rigid bottom. The motion of the fluid is governed by the full compressible, gravity-driven Navier–Stokes equations with appropriate boundary conditions. On the free surface, the effect of surface tension is neglected, and the temperature is assumed to satisfy the Robin boundary condition. Motivated by Y. Guo and I. Tice [Anal. PDE 6, 287–369 (2013), Arch. Ration. Mech. Anal. 207, 459–531 (2013), and Anal. PDE 6, 1429–1533 (2013)], we establish the global well-posedness of this free boundary problem, provided that the initial data are close to a nontrivial equilibrium state.

Student Group Dynamic Model Based on Understanding in Mathematics Subjects

This study discusses the interaction of students with a mathematical modeling point of view. This interaction involves students who understand and do not understand mathematics subject matter. The interaction process between groups is modeled in a two-dimensional system of differential equations. Variable A is the percentage of students who understand the material, and variable B is the percentage of students who do not understand the material. The dynamic analysis results obtained by one trivial equilibrium point and three non-trivial equilibrium points exist with several conditions. Based on the stability analysis of the non-trivial equilibrium point, it is found that the conditions without students do not understand mathematics subject matter. This condition is the goal of this study, which involves interaction between students; it can increase the learning process's success.

Existence of equilibrium in an overlapping‐generations model with government debt

AbstractThe Diamond overlapping generations (OLG) model with government debt has been widely utilized in economics, but the existence of the steady‐state equilibrium is either assumed or illustrated numerically. This paper provides easily checkable conditions for the existence and uniqueness of steady‐state equilibrium in this model. By checking the first derivatives of the production and utility functions and their interactions, we can determine whether the model has a nontrivial equilibrium or not. We show that the level of government debt, production technology, individual preference, and the growth rate are important for the existence of equilibrium. If government debt exceeds a certain level, equilibrium will not exist. Given technology, preference, and the growth rate, the upper‐bounds of the government debt‐output ratio in equilibrium can be determined based on our results.

Stochastic Stability and Analytical Solution with Homotopy Perturbation Method of Multicompartment Non-Linear Epidemic Model with Saturated Rate

In this work, we consider a nonlinear epidemic model with a saturated incidence rate. we consider a population of size N(t) at time t, this population is divided into six subclasses, with N(t)=S(t)+I(t)+I₁(t)+I₂(t)+I₃(t)+Q(t). Where S(t), I(t), I₁(t), I₂(t), I₃(t), and Q(t) denote the sizes of the population susceptible to disease, infectious members, and quarantine members, respectively. We have made the following contributions: 1. The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determined by the ratio called the basic reproductive number. 2. We find the analytical solution of the nonlinear epidemic model by Homotopy perturbation method. 3. Finally the stochastic stabilities. The study of its sections are justified with theorems and demonstrations under certain conditions. In this work, we have used the different references cited in different studies in the three sections already mentioned.

Nonlinear vibration analysis of a supercritical fluid-conveying pipe made of functionally graded material with initial curvature

In the view of the dynamic behaviors of fluid-conveying pipes become more complex in the supercritical case that it is quite common in engineering, we concentrate on the supercritical fluid-conveying pipe composed of functionally graded material and study the nonlinear vibration for the first time to provide an insight reference for the possible applications of the practice pipe. Firstly, the nonlinear dynamic equations are deduced by Hamiltonian’s principle based on the Euler-Bernoulli and Timoshenko theories. Then the non-trivial equilibrium expressions and the critical fluid velocities are derived analytically for the straight pipe and pipe with initial curvature. For the straight FGM pipe, we conclude that the critical fluid velocity of the Euler-Bernoulli model is larger than that of the Timoshenko model. For the FGM pipe with initial curvature, it is shown that the equilibrium deformations of Euler–Bernoulli model are larger than those of Timoshenko model. For a larger initial curvature, the critical fluid velocity becomes smaller, and the bifurcation occurs earlier for both pipe models. Moreover, Galerkin method is used to analysis the frequencies of the supercritical fluid-conveying FGM Timoshenko pipes based on the buckling configuration. We obtain that the frequencies increase as the power-law exponent and initial tension increase.

Non-trivial equilibriums and natural frequencies of a slightly curved pipe conveying supercritical fluid

Despite extensive use of curved pipes conveying high-speed fluids in many engineering fields, few researches have been conducted on the supercritical dynamics of curved pipes to explore new dynamic behavior. For the first time, this paper investigates the vibration characteristics of a slightly curved pipe conveying fluids in a supercritical range. The generalized Hamilton's principle is adopted to derive the governing equation. The non-trivial equilibriums and the critical flow velocities are then analytically obtained. The analytical predictions agree well with the numerical results obtained using the finite difference method and the differential quadrature element method. By introducing the coordinate transformation, the governing equation of the curved pipe is established for the vibration about the non-trivial equilibrium position. The natural frequencies of the pipe are obtained by the Galerkin truncation method and verified with the discrete Fourier transform. It is found that the critical velocities and the natural frequencies are highly dependent on the initial curvature. The research results show an interesting phenomenon that the natural frequency of the curved pipe may increase as the pipe length increases, but may not in monotonous manner. The obtained results provide useful information for further studies of fluid-conveying pipes with geometric imperfection.

Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction–diffusion neural networks with time delays under Dirichlet zero boundary value

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of globally exponentially stable (positive) stationary solution of delayed reaction–diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria in the classical meaning. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods. It is worth mentioning that some interesting mathematical problems are originally put forward in the last chapter.

Stability of Turing bifurcation in a weighted networked reaction–diffusion system

By introducing a weighted networked structure to the classical reaction–diffusion system, we investigate the Turing bifurcation which changes the trivial equilibrium to the nontrivial equilibrium. We show the existence of Turing bifurcation if the diffusion rate is large. By a weakly nonlinear analysis, we induce the amplitude equation of Turing bifurcation. By analyzing the amplitude equation, we show that the Turing bifurcation is stable.

Stability Analysis of Delayed Age-Structured Resource-Consumer Model of Population Dynamics With Saturated Intake Rate

This article studies nonlinear n-resource-consumer autonomous system with age-structured consumer population. The model of consumer population dynamics is described by a delayed transport equation, and the dynamics of resource patches are described by ODE with saturated intake rate. The delay models the digestion period of generalist consumer and is included in the calorie intake rate, which impacts the consumer’s fertility and mortality. Saturated intake rate models the inhibition effect from the behavioral change of the resource patches when they react to the consumer population growing or from the crowding effect of the consumer. The conditions for the existence of trivial, semi-trivial, and non-trivial equilibria and their local asymptotic stability were obtained. The local asymptotic stability/instability of non-trivial equilibrium of a system with depleted patches is defined by new derived criteria, which relate the demographic characteristics of consumers with their search rate, growth rate of resource in patches, and behavioral change of the food resource when consumer population grows. The digestion period of a generalist consumer does not cause local asymptotical instabilities of consumer population at the semi-trivial and nontrivial equilibria. These theoretical results may be used in the study of metapopulation dynamics, desert locust populations dynamics, prey-predator interactions in fisheries, etc. The paper uses numerical experiments to confirm and illustrate all dynamical regimes of the n-resource-consumer population.

Modeling the effect of reproductive toxicant in subclass and is emitted into the environment by the biological species itself

In this paper, a nonlinear model is analyzed to review of some of the chemical substances and this chemical substance produces harmful effects on the reproductive capability of the biological species. The chemical substance is the reproduction toxicant produced by the biological species itself. As the species is polluting its own environment therefore an increase in reproduction toxicant is directly proportional to the increase in population of the species. This shows various effects on the reproduction behavior of the species depending upon the concentration level or intake quantity by the biological species. This reproduction toxicant enters inside the body through inhalation, skin contact, or ingestion causing reproduction deformities in different parts of the body or the reproduction system of the species after undergoing various different mechanisms (structurally or chemically). Mathematical models have played a vital role in sketching the ecological problem to research and predicting the dynamical behavioral variation of both aquatic and terrestrial ecosystems. Some members of biological species get critically affected and show abnormalities in the reproduction behavior of biological species, like deformity, fecundity, endometriosis, reproductive cancer, etc. Various mathematical tools and stability theory of equation is employed to research the decreases within the biological population we conduct stability analysis on the nontrivial equilibrium solution of the model using Lyapunov’s direct method and eventually, we conduct numerical simulations so as to check the analytical outcomes with numerical computations, so on substantiate that there is a parameter range that the results are relevant. The reproduction process increases the population of a selected species which in turn increases the reproduction toxicant in the environment, and due to an increased level of reproduction toxicant, the reproduction in species gets lethal and sub-lethal effect because of which the population decreases below the carrying capacity. This paper is to draw attention of the viewer to put a control on the population of species including the human beings and let them to become more conscious in keep the environment free from toxicant effect.

Piezoelectrically induced nonlinear resonances for dynamic morphing of lightweight panels

A rich variety of resonance scenarios induced in thin elastic plates by periodic axial strains is investigated to describe the morphing capability of flexible, ultra-lightweight panels. The strain-induced excitation is provided by embedded piezoelectric strips actuated by time-varying voltages. The study deals with plates characterized by geometric and mechanical symmetry entailing cylindrical motions so that an ad hoc approximate nonlinear model of elastic beams is adopted to formulate the incremental equation of motion about the nonlinear equilibrium under dead loads. An asymptotic approach based on the method of multiple scales is employed to study the plate dynamics and, primarily, to investigate the effect of the plate slenderness on the nonlinearity of the lowest normal modes. A bifurcation analysis is carried out to investigate the wide array of resonances induced by the harmonic axial excitation, acting in nontrivial equilibrium states, which include primary, super-harmonic, sub-harmonic, and principal parametric resonances, respectively. For selected slendernesses of the plate, corresponding to nonlinear softening, hardening and quasi-linear behaviors, respectively, comprehensive parametric studies on the effect of the PZT cross-sectional size, the plate damping, and the piezoelectric coupling coefficient are carried out to identify the threshold voltages triggering the different resonances.

Modeling and numerical simulations for a prey–predator model with interference among predators

In this paper, we modeled a prey–predator system with interference among predators using the Crowley–Martin functional response. The local stability and existence of Hopf bifurcation at the coexistence equilibrium of the system in the absence of diffusion are analyzed. Further, the stability of bifurcating periodic solutions is investigated. We derived the conditions for which nontrivial equilibrium is globally asymptotically stable. In addition, we study the diffusion driven instability, Hopf bifurcation of the corresponding diffusion system with zero flux boundary conditions and the Turing instability region regarding parameters are established. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are performed to illustrate the theoretical results.

Anisotropic Micropolar Fluids Subject to a Uniform Microtorque: The Unstable Case

We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that when the microstructure is inertially oblate (i.e. pancake-like) this equilibrium is nonlinearly asymptotically stable. Our proof employs a nonlinear energy method built from the natural energy dissipation structure of the problem. Numerous difficulties arise due to the dissipative-conservative structure of the problem. Indeed, the dissipation fails to be coercive over the energy, which itself is weakly coupled in the sense that, while it provides estimates for the fluid velocity and microstructure angular velocity, it only provides control of two of the six components of the microinertia tensor. To overcome these problems, our method relies on a delicate combination of two distinct tiers of energy-dissipation estimates, together with transport-like advection-rotation estimates for the microinertia. When combined with a quantitative rigidity result for the microinertia, these allow us to deduce the existence of global-in-time decaying solutions near equilibrium.

Global threshold dynamics of an age structured alcoholism model

We consider in this research an age-structured alcoholism model. The global behavior of the model is investigated. It is proved that the system has a threshold dynamics in terms of the basic reproduction number (BRN), where we obtained that alcohol-free equilibrium (AFE) is globally asymptotically stable (GAS) in the case [Formula: see text], but for [Formula: see text] we found that the system persists and the nontrivial equilibrium (EE) is GAS. Furthermore, the effects of the susceptible drinkers rate and the repulse rate of the recovers to alcoholics are investigated, which allow us to provide a proper strategy for reducing the spread of alcohol use in the studied populations. The obtained mathematical results are tested numerically next to its biological relevance.

Stability and bifurcation analysis of a tumor-immune system with two delays and diffusion.

A tumor-immune system with diffusion and delays is proposed in this paper. First, we investigate the impact of delay on the stability of nonnegative equilibrium for the model with a single delay, and the system undergoes Hopf bifurcation when delay passes through some critical values. We obtain the normal form of Hopf bifurcation by applying the multiple time scales method for determining the stability and direction of bifurcating periodic solutions. Then, we study the tumor-immune model with two delays, and show the conditions under which the nontrivial equilibria are locally asymptotically stable. Thus, we can restrain the diffusion of tumor cells by controlling the time delay associated with the time of tumor cell proliferation and the time of immune cells recognizing tumor cells. Finally, numerical simulations are presented to illustrate our analytic results.

Complexity in time-delay networks of multiple interacting neural groups

&lt;p style='text-indent:20px;'&gt;Coupled networks are common in diverse real-world systems and the dynamical properties are crucial for their function and application. This paper focuses on the behaviors of a network consisting of mutually coupled neural groups and time-delayed interactions. These interacting groups can include different sets of nodes and topological architecture, respectively. The local and global stability of the system are analyzed and the stable regions and bifurcation curves in parameter planes are obtained. Different patterns of bifurcated solutions arising from trivial and non-trivial equilibrium points are given, such as the coexistence of non-trivial equilibrium points and periodic responses and multiple coexisting periodic orbits. The bifurcation diagrams are shown and plenty of complex dynamic phenomena are observed, such as multi-period oscillations and multiple coexisting attractors.&lt;/p&gt;