Education campaign and family understanding affect stability and qualitative behavior of an online game addiction model for children and youth in Thailand

Online game addiction has become a large problem worldwide, and it could give negative impacts on children in many ways such as physical health, learning, emotion, and behavior. Online game addiction is mostly found in children and youth at the age of 15 to 24 years. Therefore, in this paper, we have developed a mathematical model of online game addiction to explore the effects of education campaign and family understanding on online game addiction in Thailand. Analysis of this model reveals two main equilibria, addiction‐free (AFE) and addiction‐present (APE) ones. Results show that the AFE is locally asymptotically stable when the value of basic reproduction number (R₀) is less than the unity; otherwise, there is a unique endemic equilibrium point, and it is locally stable when satisfies the Routh‐Hurwitz criterion. Further, the conditions of AFE to be globally stable are demonstrated. Finally, the results of sensitivity analysis and numerical simulation show that the effectiveness of both education campaign and family understanding is an important factor in reducing the value of R₀ and holds great promise for lowering the number of children and youth who addict to online game in Thailand.     

Optimal control problems with terminal functionals represented as the difference of two convex functions

Two control problems for a state-linear control system are considered: the minimization of a terminal functional representable as the difference of two convex functions (d.c. functions) and the minimization of a convex terminal functional with a d.c. terminal inequality contraint. Necessary and sufficient global optimality conditions are proved for problems in which the Pontryagin and Bellman maximum principles do not distinguish between locally and globally optimal processes. The efficiency of the approach is illustrated by examples.     

考虑红利支付与提前退休的最优投资组合

研究了在经济代理人通过不可逆退休时间选择来调整劳动时间框架下的最优消费和投资问题,主要考虑风险资产派发红利的情形.运用随机控制方法,求解使得消费-闲暇预期效用最大化的最优策略.最优投资组合及最优退休时刻表明,代理人在为提前退休积累财富的同时,也能最佳享受消费和闲暇所带来的快乐.     

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型;利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指教值说明了控制的有效性.     

Local stability of an SIR epidemic model and effect of time delay

We describe an SIR epidemic model with a discrete time lag, analyse the local stability of its equilibria as well as the effects of delay on the reproduction number and on the dynamical behaviour of the system. The model has two equilibria—a necessary condition for local asymptotic stability is given. The proofs are based on linearization and the application of Lyapunov functional approach. An upper bound of the critical time delay for which the model remains valid is derived. Numerical simulations are carried out to illustrate the effect of time delay which tends to reduce the epidemic threshold. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal control of an elastic string with viscous damping

We study bilinear optimal control of a wave equation with one spatial dimension. The problem describes oscillations of an elastic string with viscous damping, and the damping coefficient is taken as the control. The objective functional involves driving the state solution close to a desired profile and incurring a cost on the control. The optimal control is characrerized in terms of an optimality system.     

Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching

Medical treatment and vaccination decisions are often sequential and uncertain. Markov decision process is an appropriate means to model and handle such stochastic dynamic decisions. This paper studies the near‐optimality of a stochastic SIRS epidemic model that incorporates vaccination and saturated treatment with regime switching. The stochastic model takes white noises and color noise into account. We first prove some priori estimates of the susceptible, infected, and recovered populations. Moreover, we establish some sufficient and necessary conditions of the near‐optimality by Pontryagin stochastic maximum principle. Our results show that the two kinds of environmental noises have great impacts on the infectious diseases. Finally, we illustrate our conclusions through numerical simulations.     

Global stability of an SIS epidemic model with delays

In this paper, we consider the global dynamics of the S(E)IS model with delays denoting an incubation time. By constructing a Lyapunov functional, we prove stability of a disease‐free equilibrium E₀ under a condition different from that in the recent paper. Then we claim that R₀≤1 is a necessary and sufficient condition under which E₀ is globally asymptotically stable. We also propose a discrete model preserving positivity and global stability of the same equilibria as the continuous model with distributed delays, by means of discrete analogs of the Lyapunov functional.     

Optimal control and basic reproduction numbers for a compartmental spatial multipatch dengue model

Dengue is a vector‐borne viral disease increasing dramatically over the past years due to improvement in human mobility. In this work, a multipatch model for dengue transmission dynamics is studied, and by that, the control efforts to minimize the disease spread by host and vector control are investigated. For this model, the basic reproduction number is derived, giving a choice for parameters in the endemic case. The multipatch system models the host movement within the patches, which coupled via a residence‐time budgeting matrix P. Numerical results confirm that the control mechanism embedded in incidence rates of the disease transmission effectively reduces the spread of the disease.     

Two-target game model of an air combat with fire-and-forget all-aspect missiles

An air combat duel between similar aggressive fighter aircraft, both equipped with the same type of guided missiles, is formulated as a two-target differential game using the dynamic model of the game of two identical cars. Each of the identical target sets represents the effective firing envelope of an all-aspect fire-and-forget air-to-air missile. The firing range limits depend on the target aspect angle and are approximated by analytical functions. The maximum range, computed by taking into account the optimal missile avoidance maneuver of the target, determines the no-escape firing envelope. The solution consists of the decomposition of the game space into four regions: the respective winning zones of the two opponents, the draw zone, and the region where the game terminates by a mutual kill. The solution provides a new insight for future air combat analysis.This paper is based on the first author's D.Sc. Thesis. The research was supported by NASA Cooperative Agreement NCCW-4.     

Modeling and analysis of a modified Leslie–Gower type three species food chain model with an impulsive control strategy

This paper describes a modified Leslie–Gower type three species food chain model with harvesting. We have incorporated impulsive control strategy to the system. Theories of impulsive differential equations, small amplitude perturbation skills and comparison technique are used to study dynamical behavior of the system. Sufficient conditions are derived to ensure global stability of the lowest-level prey and mid-level predator eradication periodic solution. Sufficient conditions are also derived to examine the permanence of the system. Numerical simulations are carried out to verify the analytical results, and the system is analyzed through graphical illustrations. It is observed that the stability of the system exhibits several states, ranging from stable situation to cyclic oscillatory behavior, under different favorable conditions. These results are useful to study the dynamic complexity of ecological systems. The computation of the largest Lyapunov exponent demonstrates the chaotic dynamic nature of the system. The qualitative nature of strange attractor is examined. It is to be noted that the harvesting effort can cause a stable equilibrium to become unstable and even a switching of stabilities.     

Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination

This paper deals with global dynamics of an SIRS epidemic model for infections with non permanent acquired immunity. The SIRS model studied here incorporates a preventive vaccination and generalized non-linear incidence rate as well as the disease-related death. Lyapunov functions are used to show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one, and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one.     

一类具有扩散和Beddington-DeAngelis反应函数的病毒模型的全局稳定性

研究了一类具有扩散和Beddington-DeAngelis反应函数的病毒模型.通过构造Lyapunov函数,证明了模型的感染平衡点是全局渐近稳定的.     

Global asymptotic properties of an SEIRS model with multiple infectious stages

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.     

Optimal control applied on an HIV‐1 within‐host model

The treatment of human immunodeficiency virus (HIV) remains a major challenge, even if significant progress has been made in infection treatment by ‘drug cocktails’. Nowadays, research trend is to minimize the number of pills taken when treating infection. In this paper, an HIV‐1 within host model where healthy cells follow a simple logistic growth is considered. Basic reproduction number of the model is calculated using next generation matrix method, steady states are derived; their local, as well as global stability, is discussed using the Routh–Hurwitz criteria, Lyapunov functions and the Lozinskii measure approach. The optimal control policy is formulated and solved as an optimal control problem. Numerical simulations are performed to compare several cases, representing a treatment by Interleukin2 alone, classical treatment by multitherapy drugs alone, then both treatments at the same time. Objective functionals aim to (i) minimize infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood. Copyright © 2015 John Wiley & Sons, Ltd.     

Local and global stability analysis of a two prey one predator model with help

In this paper we propose and study a three dimensional continuous time dynamical system modelling a three team consists of two preys and one predator with the assumption that during predation the members of both teams of preys help each other and the rate of predation of both teams are different. In this work we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functional. At the end, numerical simulations are performed to substantiate our analytical findings.     

Optimal control theory of an age-structured coronavirus disease model and the dynamical analysis of the underlying ordinary differential equation model having constant parameters

This paper addresses the dynamics of COVID-19 using the approach of age-structured modeling. A particular case of the model is presented by taking into account age-free parameters. The sub-model consisting of ordinary differential equations (ODEs) is investigated for possible equilibria, and qualitative aspects of the model are rigorously presented. In order to control the spread of the disease, we considered two age- and time-dependent non-pharmaceutical control measures in the age-structured model, and an optimal control problem using a general maximum principle of Pontryagin type is achieved. Finally, sample simulations are plotted which support our theoretical work.