Global Stability Results Research Articles

TRANSMISSION AND CONTROL DYNAMICS OF ROTAVIRUS DIARRHEA MODEL WITH DOUBLE DOSE VACCINATION

This study introduces a six-compartmental mathematical model (S, , , E, I, R) to examine the impact of administering a double dose vaccine on the dynamic spread of diarrhea within a community. The mathematical analysis shows the existence of equilibrium points for both disease-free and endemic states in the model. The basic reproduction number was determined using the Next Generation Matrix. Analysis has shown that the basic reproduction number which indicates the disease-free equilibrium point is locally asymptotically stable. Also, using a suitable Lyapunov functional for the model system expressed in state variables and parameters defining the dynamic characteristics of spread and control strategies of the rotavirus diarrhea to obtain the global stability of disease-free equilibrium point over time. A numerical simulation was carried out by Wolfram Mathematica to show the effect of a second-dose vaccine. The inclusion of a double-dose vaccine has been found to have a significant effect on completely eliminating the outbreak of diarrhea. This is evidenced by the local and global stability results, which indicate that effective measures have been taken to prevent the reintroduction or transmission of the disease, and if there may be a risk of outbreaks or reemergence of the disease, very little continuous monitoring and intervention strategies are required to maintain control as this should be taken seriously by medical practitioners or policy health makers.

Stability analysis for viscoelastic fluid with thermorheological effects: Linear and nonlinear approaches

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D2-Chebyshev-τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes.

Modeling virus-stimulated proliferation of CD4[formula omitted] T-cell, cell-to-cell transmission and viral loss in HIV infection dynamics

Human immunodeficiency virus (HIV) can persist in infected individuals despite prolonged antiretroviral therapy and it may spread through two modes: virus-to-cell and cell-to-cell transmissions. Understanding viral infection dynamics is pivotal for elucidating HIV pathogenesis. In this study, we incorporate the loss term of virions, and both virus-to-cell and cell-to-cell infection modes into a within-host HIV model, which also takes into consideration the proliferation of healthy target cells stimulated by free viruses. By constructing suitable Lyapunov function and applying geometric methods, we establish global stability results of the infection free equilibrium and the infection persistent equilibrium, respectively. Our findings highlight the crucial role of the basic reproduction number in the threshold dynamics. Moreover, we use the loss rate of virions as the bifurcation parameter to investigate stability switches of the positive equilibrium, local Hopf bifurcation, and its global continuation. Numerical simulations validate our theoretical results, revealing rich viral dynamics including backward bifurcation, saddle–node bifurcation, and bistability phenomenon in the sense that the infection free equilibrium and a limit cycle are both locally asymptotically stable. These insights contribute to a deeper understanding of HIV dynamics and inform the development of effective therapeutic strategies.

Finite-Time Fault-Tolerant Control via Fully Actuated System Approaches.

In this article, a finite-time fault-tolerant controller based on the fully actuated system (FAS) theory is presented to realize system stabilization and trajectory tracking. Paralleling to first-order nonlinear state space theory, the high-order FAS (HOFAS) theory contains rich controller design approaches. The existing FAS approaches can only give general global asymptotic stability results. In order to enhance the applicability of FAS approaches in fast control systems, a parameterized FAS stabilization controller based on the homogeneity principle is established for global finite-time stability. Moreover, a finite-time FAS tracking controller based on a finite-time observer is proposed for a HOFAS model with process faults. The proposed observer can yield zero-value convergence of state estimation error and fault estimation error in a finite time, and the proposed fault-tolerant controller can yield zero-value convergence of tracking error in a finite time. The main results are proved theoretically and illustrated experimentally.

Neimark–Sacker bifurcation of two second-order rational difference equations

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.

A fractional order model for the transmission dynamics of shigellosis

Shigellosis, a highly contagious bacterial infection causing diarrhea, fever, and abdominal pain, necessitates a deep understanding of its transmission dynamics to devise effective control measures. Our study takes a novel approach, employing a fractional order framework to explore the influence of memory and control measures on Shigellosis transmission dynamics, thereby making a unique contribution to the field. The model is presented as a system of Caputo fractional differential equations capturing time constant controls. The Caputo derivatives are chosen for their inherent benefits. The qualitative features of the model, such as the solutions' existence and uniqueness, positivity, and boundedness, are thoroughly investigated. Moreover, the equilibria of the model are derived and analyzed for their stability using suitable theorems. In particular, local stability was proved through Routh's criteria, while global stability results were established in the Ulam-Hyers sense. The model is then solved numerically with the help of the predict-evaluate-correct-evaluate method of Adams-Bashforth-Moulton. The numerical results underscore the significant impact of memory on disease evolution, highlighting the novelty of integrating memory-related aspects into the meticulous planning of effective disease control strategies. Using fractional-order derivatives is more beneficial for understanding the dynamics of Shigellosis transmission than integral-order models. The advantage of fractional derivatives is their ability to provide numerous degrees of freedom, allowing for a broader range of analysis of the system's dynamic behaviour, including nonlocal solutions. Also, an investigation on the impacts of control measures via parameter variation is done; the findings show that applying treatment and sanitation minimizes disease eruption.

Lyapunov functionals for a general time-delayed virus dynamic model with different CTL responses.

A time-delayed virus dynamic model is proposed with general monotonic incidence, different nonlinear CTL (cytotoxic T lymphocyte) responses [CTL elimination function pyg1(z) and CTL stimulation function cyg2(z)], and immune impairment. Indeed, the different CTL responses pose challenges in obtaining the dissipativeness of the model. By constructing appropriate Lyapunov functionals with some detailed analysis techniques, the global stability results of all equilibria of the model are obtained. By the way, we point out that the partial derivative fv(x,0) is increasing (but not necessarily strictly) in x>0 for the general monotonic incidence f(x,v). However, some papers defaulted that the partial derivative was strictly increasing. Our main results show that if the basic reproduction number R0≤1, the infection-free equilibrium E0 is globally asymptotically stable (GAS); if CTL stimulation function cyg2(z)=0 for z=0 and the CTL threshold parameter R1≤1<R0, then the immunity-inactivated infection equilibrium E1 is GAS; if the immunity-activated infection equilibrium E+ exists, then it is GAS. Two specific examples are provided to illustrate the applicability of the main results. The main results acquired in this paper improve or extend some of the existing results.

Viral infection dynamics with immune chemokines and CTL mobility modulated by the infected cell density.

We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state is globally asymptotically stable if . When , then becomes unstable, and another basic reproduction number of CTL response becomes the dynamic threshold in the sense that if , then the CTL-inactivated steady state is globally asymptotically stable; and if , then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.

A Complex Dynamic of an Eco-Epidemiological Mathematical Model with Migration

In this paper, we propose an eco-epidemiological mathematical model in order to describe the effect of migration on the dynamics of a prey–predator population. The functional response of the predator is governed by the Holling type II function. First, from the perspective of mathematical results, we develop results concerning the existence, uniqueness, positivity, boundedness, and dissipativity of solutions. Besides, many thresholds have been computed and used to investigate the local and global stability results by using the Routh–Hurwitz criterion and Lyapunov principle, respectively. We have also established the appearance of limit cycles resulting from the Hopf bifurcation. Numerical simulations are performed to explore the effect of migration on the dynamic of prey and predator populations.

Delay‐adaptive control of first‐order hyperbolic partial integro‐differential equations

AbstractWe develop a delay‐adaptive controller for a class of first‐order hyperbolic partial integro‐differential equations with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite‐dimensional backstepping technique to establish global stability results. Subsequently, the effectiveness of the proposed approach was validated through numerical simulations.

Mathematical modelling and analysis of Kidnapping dynamics

The problem of kidnapping as a social menace to a society is increasing in some African countries such as Nigeria. We therefore proposed a new deterministic mathematical model for the dynamics of Kidnapping in a community. This menace is considered like a two strains communicable disease with kidnapping propagation mission by kidnappers as one strain and adoption mission for kidnapped victims as the other strain to assess the impact of super-infection. The model exhibits four equilibrium points each of which is unique and asymptotically stable both locally and globally under certain conditions. We obtain the kidnapping propagation number where is the propagation number associated with strain . Another important threshold parameters associated with respective strains 1 and 2 are and . Indeed, we show that at most one strain invades the population if one of these parameters is less than unity. while the two strains coexist at endemic state when both and are greater than unity. The global stability results of the model equilibria are established by numerical simulations. This simulations results indicate that the super-infection destabilizes the coexistence equilibrium.

Global asymptotic stability for Gurtin-MacCamy’s population dynamics model

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker’s type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.

The dynamics of monkeypox disease under [formula omitted]–Hilfer fractional derivative: Application to real data

The mathematical model for monkeypox infection using the ψ–Hilfer fractional derivative is presented in this study. The integer order formulation is extended to the fractional order system by employing the ψ–Hilfer fractional derivative. The fractional order model analysis is provided. We investigate the model’s local asymptotical stability when R0<1. When R0>1, the global asymptotical stability result is displayed. We parameterize the model using recently reported cases of monkeypox infection in the United States. We calculated the basic reproduction using the estimated data and found it to be R0≈0.7121. We investigate the sensitivity of the monkeypox infection model and find the parameters that are sensitive R0. In general, we offer a numerical approach, and then for the monkeypox model, we present detailed findings. Some graphical outcomes for disease control in the United States are shown.

Global small solutions to the 3D compressible viscous non-resistive MHD system

Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is currently available for this 3D MHD system in the whole space [Formula: see text] or the periodic box [Formula: see text] even when the initial data is small or near a steady-state solution. This paper presents a global existence and stability result for smooth solutions to this 3D MHD system near any background magnetic field satisfying a Diophantine condition.

Formulation and mathematical analysis of a three-step model of anaerobic digestion with inhibition by ammonia

In this work, we are interested in the construction and mathematical analysis of a three-step anaerobic digestion model (Hydrolysis–Acidogenesis–Methanogenesis). Our aim is to investigate the qualitative properties of the system. In addition to the hypotheses used for formulating the well-known baseline two-step anaerobic model, we take into account the inhibition by ammonia together with the decay rate of acidogenic and methanogenic microorganisms, and consider the hydrolysis step to determine its effect on the growth of the methanogens. By so doing, we obtain a system of six ordinary differential equations describing the whole dynamics of the aforementioned phenomenon. As far as the mathematical analysis of this model is concerned, we first prove the existence of a unique positive and bounded global solution. Then we derive local and global stability results. The proofs of the main results rely essentially on some tools from the theory of ordinary differential equations. Finally, by performing some numerical simulations, we illustrate the theoretical results obtained and highlight the key parameters acting on the model’s behavior.

Global classical solutions to equatorial shallow-water equations

In this article, we study the equatorial shallow-water equations with slip boundary condition in bounded domain. By exploring the dissipative structures of the system, we obtaining a priori estimates of the solution for small initial data. Then the existence of classical global solutions and exponential stability results are given.&#x0D; For more inofrmation see https://ejde.math.txstate.edu/Volumes/2023/62/abstr.html

Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data

We consider the inverse problem of the simultaneous identification of the coefficients σ and q of the equation div(σ∇u)+qu=0 from the knowledge of the Cauchy data set. We assume that σ=γA, where A is a given matrix function and γ and q are unknown piecewise affine scalar functions. No sign, nor spectrum condition on q is assumed. We derive a result of global Lipschitz stability in dimension n≥3. The proof relies on the method of singular solutions and on the quantitative estimates of unique continuation.

Noisy prediction-based control leading to stability switch

Applying Prediction-Based Control (PBC) xn+1=(1−αn)f(xn)+αnxn with stochastically perturbed control coefficient αn=α+ℓξn+1, n∈N, where ξ are bounded identically distributed independent random variables, we globally stabilize the unique equilibrium K of the equation xn+1=f(xn) in a certain domain. In our results, the noisy control α+ℓξ provides both local and global stability, while the mean value α of the control does not guarantee global stability, for example, the deterministic controlled system can have a stable two-cycle, and non-controlled map be chaotic. In the case of unimodal f with a negative Schwarzian derivative, we get sharp stability results generalizing Singer’s famous statement ‘local stability implies global’ to the case of the stochastic control. New global stability results are also obtained in the deterministic settings for variable αn and, generally, continuous but not differentiable at K map f.

Mathematical modeling of pest resistance to insecticides in a heterogeneous environment

Insecticides are an effective tool for controlling grapevine moths such as Lobesia botrana. However, when massive use of insecticides, pests develop resistance because of selection of resistant individuals of the population and/or because of natural mutation. The main question is then to define a strategic pest management with respect to time and space. In the first part of this work, we present a model without spatial structure describing the dynamics of insects including its resistance to insecticides. In the second part, we include in the model dispersal between different space patches. Individuals can move from one patch to another at a fast time scale with respect to the demographical time scale. Using this model, we aim to understand the main characteristics leading to persistence or extinction of the insect pest population. We establish global stability results. Numerical simulations provide some interesting insights on the dynamics of the pest population.

Thermorheological effect on Rayleigh–Bénard magnetoconvection in a biviscous Bingham fluid with rough boundary condition on velocity and Robin boundary condition on temperature

AbstractThe thermorheological effect on the onset of Rayleigh–Bénard convection in a biviscous Bingham fluid in the presence of a horizontal magnetic field is investigated considering rough boundary conditions on velocity and Robin boundary conditions on temperature. The viscosity of the electrically conducting fluid is assumed to be sensitive to temperature variation. Linear and global nonlinear stability analyses are performed using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. A general interpretation is made from the results to show the effects of the magnetic field and the variable viscosity on the system's stability. The biviscous Bingham parameter and the Chandrasekhar number are shown to have a delay in the onset of convection, while the effect of temperature sensitivity is to advance the onset. It is found that the results of linear and global nonlinear stability are not in agreement, so the region of subcritical instability exists. Also, the results obtained for Rayleigh–Bénard convection agree pretty well with those of Platten and Legros and Siddheshwar et al. for the limiting cases.