Searcharxiv⌕ Search

arXiv subjects

Manh Tuan Hoang

Publications and source records attributed to Manh Tuan Hoang.

16 recordsLinked to original sources
arXiv2026

A Mathematical Study of an SIS Epidemic Model: Global Asymptotic Stability Analysis and Construction of Nonstandard Numerical Schemes

The aim of this work is to provide a rigorous mathematical analysis of a well-known SIS epidemic model with a saturating contact rate. First, we establish the global asymptotic stability (GAS) of the model's equilibria by employing a suitable Lyapunov function in combination with the Poincaré--Bendixson theorem and the Bendixson--Dulac criterion. The resulting GAS results improve upon previous findings for this SIS model and may also be applicable to its extensions with more general saturating contact rates. Second, we construct families of first- and second-order nonstandard finite difference (NSFD) schemes that preserve the positivity and asymptotic stability properties of the SIS model for arbitrary step sizes. All these schemes are formulated within Mickens' framework; however, compared with their first-order counterparts, the second-order schemes employ a more elaborate construction that combines a weighted nonlocal approximation of the right-hand side with suitably renormalized denominator functions. The weighted nonlocal approximation guarantees dynamic consistency, whereas the denominator functions ensure second-order convergence. Finally, we conduct numerical experiments to validate the theoretical results and demonstrate the advantages of the proposed NSFD schemes. The numerical results are in good agreement with the theoretical results. The approach developed in this work is applicable not only to other epidemiological systems but also, more generally, to mathematical models arising in a wide range of real-world applications.

math.NA↗
arXiv2026

A simple second-order nonstandard numerical method for a general class of dynamical systems and its applications

In this work, we consider a class of continuous-time autonomous dynamical systems that model various important phenomena and processes encountered in real-world situations. We construct a second-order nonstandard finite difference (NSFD) method that simultaneously preserves two essential properties of the dynamical systems for all finite step sizes, namely the positivity of the solutions, the set of equilibrium points and their asymptotic stability. This NSFD method is constructed based on an appropriate choice of nonstandard denominator functions and a weighted discretization of the right-hand side functions. Under easily-verified conditions, the denominator functions guarantee second-order convergence, whereas the weights ensure the dynamic consistency. By taking advantage of the specific structure of the right-hand side functions, a simple discretization is utilized instead of the nonlocal discretization approaches commonly used in previous works. This simplifies the construction of the proposed NSFD method and, in particular, makes its asymptotic stability analysis easier. As an illustration and an important application, we apply the constructed second-order NSFD method to a well-known two-stage structured species model with recruitment. Consequently, a simple second-order NSFD scheme for the considered two-stage structured species model is derived, improving upon a first-order NSFD scheme constructed in a previous work. Numerical experiments demonstrate the advantages of the second-order NSFD scheme over a standard second-order numerical method, namely, the explicit trapezoidal method. The proposed NSFD method is simple and can be applied to a broad class of dynamical system models arising in both theory and applications. Moreover, it can be readily combined with the Richardson extrapolation technique to improve its accuracy.

math.NA↗
arXiv2026

Global Dynamics of a Pharmacokinetic Compartment Model for Human Ethanol Metabolism and Its Generalization

In this work, we revisit a continuous-time two-compartment pharmacokinetic model of human ethanol metabolism originally proposed by Levitt and Levitt. We first establish the positivity and boundedness of the solutions, investigate the existence and uniqueness of a positive equilibrium, and analyze its local and global asymptotic stability. As a result, the global dynamics of the ethanol metabolism model is completely characterized, thereby complementing and extending the analytical results reported in the original benchmark study. Second, we extend the original continuous-time model by replacing the Michaelis--Menten metabolism rate with a general class of metabolism-rate functions that includes many well-known monotone and nonmonotone forms. This extension enhances the flexibility of the model and enables it to capture a wider range of realistic metabolic scenarios. We then investigate the global dynamics of the generalized continuous-time model. Finally, numerical experiments are conducted to support the theoretical findings. The numerical results provide further evidence for the theoretical results.

math.DS↗
arXiv2026

On a Generalized Compartment Model for Ethanol Metabolism in the Human Body

We introduce a generalized continuous-time compartment model of ethanol metabolism in the human body that extends a recently developed framework. In the proposed model, we replace the Michaelis-Menten mechanism of the liver's ethanol metabolism rate with a general class of nonlinear rate functions. This modification provides greater modeling flexibility and enables the model to capture a wider range of hepatic ethanol metabolism dynamics. The qualitative behavior of the proposed ethanol metabolism model is analyzed rigorously. More specifically, we investigate the positivity and boundedness of solutions, as well as the global asymptotic stability (GAS) of the unique equilibrium point using an appropriate quadratic Lyapunov function. Second, we formulate a discrete-time counterpart of the proposed continuous-time model and investigate its dynamical properties. We show that, under an appropriate condition on the time step size, the discrete-time model faithfully reproduces the qualitative dynamical behavior of the corresponding continuous-time system. Lastly, we conduct a series of numerical experiments employing several ethanol metabolism rate functions to support the theoretical results.

math.DS↗
arXiv2026

A Generalized Hacker Dynamics Model with Nonlinear Incidence: Analysis and Positivity-Preserving Numerical Simulation

We propose and analyze a generalized compartment model for hacker dynamics in cybersecurity systems. This model is an extension of a recently introduced framework, replacing the bilinear interaction term with a broad class of nonlinear incidence functions. This provides greater modeling flexibility and allows for the description of a wider range of cyber-propagation and information-spreading processes. First, we investigate the qualitative dynamics of the model. We establish the positivity and boundedness of solutions, derive the basic reproduction number, and characterize the existence of hacker-free and hacker-present equilibria. We obtain local and global asymptotic stability results, yielding a complete description of the global dynamics in terms of the basic reproduction number. Next, we develop a second-order, positivity-preserving, nonstandard finite difference (NSFD) scheme for numerical simulations. Unlike many existing NSFD approaches, which are typically first-order accurate, the proposed method achieves second-order convergence while preserving positivity for arbitrary step sizes. Furthermore, the proposed method reproduces the asymptotic stability properties of the continuous model, making it suitable for long-time simulations. Numerical experiments validate the theoretical results and demonstrate the superior accuracy and qualitative performance of the proposed scheme compared to several first-order methods.

math.NA↗
arXiv2026

A Modified SIS Epidemic Model with Application to Health Insurance Pricing

In this work, we investigate a modified version of the classical SIS model that incorporates hospitalization for treatment and disease-induced mortality, aiming to more accurately capture the dynamics relevant to health insurance pricing models. More precisely, we introduce a new framework, referred to as the SISHD model, which considers both hospitalized individuals and disease-induced mortality. Dynamical properties of the proposed model are thoroughly analyzed, including positivity and boundedness of the solutions, the basic reproduction number, the existence and asymptotic stability of equilibrium points. Furthermore, we utilize the proposed model in the context of health insurance pricing, where the population sizes estimated from the SISHD model are used to determine appropriate insurance costs. Finally, numerical simulations are conducted to illustrate and validate the theoretical results.

math.DS↗
arXiv2025

A Generalized Second-Order Positivity-Preserving Numerical Method for Non-Autonomous Dynamical Systems with Applications

In this work, we propose a generalized, second-order, nonstandard finite difference (NSFD) method for non-autonomous dynamical systems. The proposed method combines the NSFD framework with a new non-local approximation of the right-hand side function. This method achieves second-order convergence and unconditionally preserves the positivity of solutions for all step sizes. Especially, it avoids the restrictive conditions required by many existing positivity-preserving, second-order NSFD methods. The method is easy to implement and computationally efficient. Numerical experiments, including an improved NSFD scheme for an SIR epidemic model, confirm the theoretical results. Additionally, we demonstrate the method's applicability to nonlinear partial differential equations and boundary value problems with positive solutions, showcasing its versatility in real-world modeling.

math.NA↗
arXiv2023

Nonstandard finite difference methods preserving general quadratic Lyapunov functions

In this work, we consider a class of dynamical systems described by ordinary differential equations under the assumption that the global asymptotic stability (GAS) of equilibrium points is established based on the Lyapunov stability theory with the help of quadratic Lyapunov functions. We employ the Micken's methodology to construct a family of explicit nonstandard finite difference (NSFD) methods preserving any given quadratic Lyapunov function $V$, i.e. they admit $V$ as a discrete Lyapunov function. Here, the proposed NSFD methods are derived from a novel non-local approximation for the zero vector function. Through rigorous mathematical analysis, we show that the constructed NSFD methods have the ability to preserve any given quadratic Lyapunov functions regardless of the values of the step size. As an important consequence, they are dynamically consistent with respect to the GAS of continuous-time dynamical systems. On the other hand, the positivity of the proposed NSFD methods is investigated. It is proved that they can also preserve the positivity of solutions of continuous-time dynamical systems. Finally, the theoretical findings are supported by a series of illustrative numerical experiments, in which advantages of the NSFD methods are demonstrated.

math.NA↗
arXiv2023

A generalized nonstandard finite difference method for a class of autonomous dynamical systems and its applications

In this work, a class of continuous-time autonomous dynamical systems describing many important phenomena and processes arising in real-world applications is considered. We apply the nonstandard finite difference (NSFD) methodology proposed by Mickens to design a generalized NSFD method for the dynamical system models under consideration. This method is constructed based on a novel non-local approximation for the right-side functions of the dynamical systems. It is proved by rigorous mathematical analyses that the NSFD method is dynamically consistent with respect to positivity, asymptotic stability and three classes of conservation laws, including direct conservation, generalized conservation and sub-conservation laws. Furthermore, the NSFD method is easy to be implemented and can be applied to solve a broad range of mathematical models arising in real-life. Finally, a set of numerical experiments is performed to illustrate the theoretical findings and to show advantages of the proposed NSFD method.

math.NA↗
arXiv2021

A new and simple condition for the global asymptotic stability of a malware spread model on WSNs

In a very recent work [J. D. Hernández Guillén, A. Martín del Rey, A mathematical model for malware spread on WSNs with population dynamics, Physica A: Statistical Mechanics and its Applications 545(2020) 123609], a novel theoretical model for the spread of malicious code on wireless sensor networks was introduced and analyzed. However, the global asymptotic stability (GAS) of the disease-endemic equilibrium (DEE) point was only resolved partially under technical hypotheses that are not only difficult to be verified but also restrict the space of feasible parameters for the model. In the present work, we use a simple approach to establish the complete GAS of the DEE point without the technical hypotheses proposed in the benchmark work. This approach is based on a suitable family of Lyapunov functions in combination with characteristics of Volterra-Lyapunov stable matrices. Consequently, we obtain a simple and easily-verified condition for the DEE point to be globally asymptotically stable. This result provides an important improvement for the results constructed in the benchmark work. In addition, the theoretical findings are supported by numerical and illustrative examples, which show that the numerical results are consistent with the theoretical ones.

math.DS↗
arXiv2021

A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD methods but also improve and extend some well-known results that have been published recently in [Applied Mathematics Letters 112(2021) 106775], [AIP Conference Proceedings 2302(2020) 110003] and [Applied Mathematics Letters 50(2015) 78-82]. Furthermore, as a simple but important application, we apply the constructed NSFD method for solving the logistic, sine, cubic, and Monod equations; consequently, the NSFD schemes constructed in the earlier work [Journal of Computational and Applied Mathematics 110(1999) 181-185] are improved significantly. Finally, we report some numerical experiments to support and illustrate the theoretical assertions as well as advantages of the constructed NSFD method.

math.NA↗
arXiv2017

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

In this paper, we construct explicit nonstandard Runge-Kutta (ENRK) methods which have higher accuracy order and preserve two important properties of autonomous dynamical systems, namely, the positivity and linear stability. These methods are based on the classical explicit Runge-Kutta methods, where instead of the usual $h$ in the formulas there stands a function $φ(h)$. It is proved that the constructed methods preserve the accuracy order of the original Runge-Kutta methods. The numerical simulations confirm the validity of the obtained theoretical results.

math.NA↗
arXiv2017

Exact finite difference schemes for three-dimensional linear systems with constant coefficients

In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for problems with periodic solutions on very large time interval demonstrate the efficiency and exactness of the EDS compared with high-order numerical methods. This result can be extended for constructing EDS for general systems of $n$ linear differential equations with constant coefficients and nonstandard finite difference (NSFD) schemes preserving stability properties for quasi-linear system of equations $\textbf{x}' = A\textbf{x }+ f(\textbf{x})$.

math.NA↗
arXiv2017

Nonstandard finite difference schemes for a general predator-prey system

In this paper we transform a continuous-time predator-prey system with general functional response and recruitment for both species into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. Especially, the global stability of a non-hyperbolic equilibrium point in a particular case of parameters is proved by the Lyapunov stability theorem. The performed numerical simulations confirmed the validity of the obtained theoretical results.

math.NA↗
arXiv2017

Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models

In this paper nonstandard finite difference (NSFD) schemes of two metapopulation models are constructed. The stability properties of the discrete models are investigated by the use of a generalization of Lyapunov stability theorem. Due to this result we have proved that the NSFD schemes preserve all properties of the metapopulation models. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. The method of Lyapunov functions proves to be much simpler than the standard method for studying stability of the discrete metapopulation model in our very recent paper.

math.NA↗