(SI10-057) Effect of Time-delay on an SIR Type Model For Infectious Diseases with Saturated Treatment

This study presents the complex dynamics of an SIR epidemic model incorporating a constant time-delay in incidence rate with saturated type of treatment rate. The system is studied to observe the effect of time lag in the asymptotic stability of endemic equilibrium states. We also establish global asymptotic stability of both disease-free and endemic equilibrium states by Lyapunov direct method with the help of suitable Lyapunov functionals. The existences of periodic solutions are ensured for the suitable choice of delay parameter. Finally, we perform numerical simulations supporting the analytical findings as well as to observe the effect of time-delay. The theoretical and numerical results together show delay can have both stabilizing and destabilizing effects on the system. Moreover, we observe that infection may die out from the population when the corresponding system without delay has two endemic equilibrium for appropriate choice of time-delay

Bifurcation analysis and optimal control of a network-based SIR model with the impact of medical resources

A new network-based SIR epidemic model, which incorporates the individual medical resource factor and public medical resource factor is proposed. It is verified that the larger the public medical resource factor, the smaller the control reproduction number, and the larger individual medical resource factor can weaken the spread of diseases. We found that the control reproduction number below unity is not enough to ensure global asymptotic stability of the disease-free equilibrium. When the number of hospital beds or the individual medical resource factor is small enough, the system will undergoes backward bifurcation. Moreover, the existence and uniqueness of the optimal control and two time-varying variables’s optimal solutions are obtained. On the scale-free network, the level of optimal control is also proved to be different for different degrees. Finally, the theoretical results are illustrated by numerical simulations. This study suggests that maintaining sufficient both public medical resources and individual medical resources is crucial for the control of infectious diseases

Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results

OPTIMAL CONTROL ANALYSIS OF A SEIV EPIDEMIC MODEL WITH VACCINATION AND EDUCATION

This paper discusses optimal control of a mathematical epidemic model governed by an ODE system with saturated incidence rate.  An epidemic model is developed using optimal control theory by dividing the population into Susceptible, Exposed, Infected, and Vaccinated (SEIV) sub populations.  In the model we assume that half of new born individual have been vaccinated. Optimal control is conducted by adding two control variables namely vaccination and education. The aim of optimal control is to minimize the density of exposed subpopulation, infected subpopulation, and the cost of control. Optimal control is obtained by applying Pontryagin minimum principle. Furthermore, the optimal control problem is solved numerically by using Forward-Backward Sweep method. Three approaches were used to conduct numerical simulations, applying vaccination control without education, applying education control without vaccination, and using both vaccination and education control. There are Numerical simulations show that vaccination and education are effective in reducing exposed and infected subpopulation

Bifurcation Analysis and Chaos Control in a Discrete Epidemic System

The dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interior R+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the system x→x+δ[rN(1-N/K)-βxy/N-(μ+m)x], y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method

Different types of backward bifurcations on account of an improvement in treatment efficiency

Understanding why there are multiple equilibrium points when R0 < 1 has been one of the main motivations to analyze existence of a backward bifurcation in epidemiological models. Existence of multiple endemic states is usually associated to branches of equilibrium points of the models, which could arise from either the disease-free equilibrium point if R0 = 1 or from an endemic equilibrium point if R0 > 1. In this work, an SIR model with a density-dependent treatment rate is analyzed. The nature of the point where backward bifurcation emerges is explained in function of the velocity of the per-capita treatment rate. Strategies for the control or eradication of the disease will be proposed in function of the efficiency of the treatment