Il Hyo Jung

Stability analysis and optimal vaccination of an SIR epidemic model.

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.

Optimal treatment of an SIR epidemic model with time delay

In this paper the optimal control strategies of an SIR (susceptible-infected-recovered) epidemic model with time delay are introduced. In order to do this, we consider an optimally controlled SIR epidemic model with time delay where a control means treatment for infectious hosts. We use optimal control approach to minimize the probability that the infected individuals spread and to maximize the total number of susceptible and recovered individuals. We first derive the basic reproduction number and investigate the dynamical behavior of the controlled SIR epidemic model. We also show the existence of an optimal control for the control system and present numerical simulations on real data regarding the course of Ebola virus in Congo. Our results indicate that a small contact rate(probability of infection) is suitable for eradication of the disease (Ebola virus) and this is one way of optimal treatment strategies for infectious hosts.

Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence

The paper considers a model for the transmission dynamics of a vector-borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive number R0 and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

Presentation of Malaria Epidemics Using Multiple Optimal Controls

An existing model is extended to assess the impact of some antimalaria control measures, by reformulating the model as an optimal control problem. This paper investigates the fundamental role of three type of controls, personal protection, treatment, and mosquito reduction strategies in controlling the malaria. We work in the nonlinear optimal control framework. The existence and the uniqueness results of the solution are discussed. A characterization of the optimal control via adjoint variables is established. The optimality system is solved numerically by a competitive Gauss-Seidel-like implicit difference method. Finally, numerical simulations of the optimal control problem, using a set of reasonable parameter values, are carried out to investigate the effectiveness of the proposed control measures.

Existence of solutions for nonlinear inequalities in G-convex spaces

By using a fixed-point theorem in G-convex spaces due to the first author, an existence result for abstract nonlinear inequalities without any monotonicity assumptions is established. As a consequence of our result, we obtain some further generalizations of recent known results. As application, an existence theorem for perturbed saddle point problems is obtained in noncompact G-convex spaces.

Stabilization of the Kirchhoff type wave equation with locally distributed damping

We derive energy decay estimates of the Kirchhoff type wave equation with a localized damping term in a bounded domain. The damping coefficient function may act alive only on a neighborhood of some part of the boundary.

Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.

Pareto equilibria for constrained multiobjective games in locally L-convex spaces

Abstract In this paper, we introduce and study a class of constrained multiobjective games in locally L -convex spaces without linear structure. A new fixed-point theorem for a family of set-valued mappings and an existence theorem of solutions for a system of quasi-equilibrium problems are first proved in noncompact locally L-convex spaces. As applications, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact locally L -convex spaces. These theorems improve, unify, and generalize the corresponding results of the multiobjective games in recent literature.

Almost periodic solutions of fuzzy systems

We prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations. Moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system.

Optimal strategy of vaccination & treatment in an SIR epidemic model

In this work, we propose a susceptibleinfectedrecovered (SIR) epidemic model which describes the interaction between susceptible and infected individuals in a community and analyze the SIR epidemic model through the optimal control theory and mathematical analysis. In addition, we present some possible strategies to prevent the spread of some infection causing epidemic in the society. In order to do this, we introduce an optimal control problem with an objective functional, where two control functions, vaccination and treatment have been used as control measures for susceptible and infected individuals. We show the existence of an optimal control pair for the optimal control problem and derive the optimality condition. Finally we consider a smoking epidemic model to illustrate our theoretical results with some numerical simulations, which use real data collected in April and May 2004 from 300 male students at three vocational technical high schools in Korean metropolitan areas. Our analysis suggests that two control strategies are more effective than only one control strategy in controlling the increase of male student smokers in Korean metropolitan areas.