Network Lyapunov Functions for Epidemic Models

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

Download Full-text

Lyapunov functions for SIR and SIRS epidemic models

Applied Mathematics Letters ◽

10.1016/j.aml.2009.11.014 ◽

2010 ◽

Vol 23 (4) ◽

pp. 446-448 ◽

Cited By ~ 30

Author(s):

Suzanne M. O’Regan ◽

Thomas C. Kelly ◽

Andrei Korobeinikov ◽

Michael J.A. O’Callaghan ◽

Alexei V. Pokrovskii

Keyword(s):

Lyapunov Functions ◽

Epidemic Models

Download Full-text

Large-Scale Epidemic Models and a Graph-Theoretic Method for Constructing Lyapunov Functions

The Dynamics of Biological Systems - Mathematics of Planet Earth ◽

10.1007/978-3-030-22583-4_3 ◽

2019 ◽

pp. 63-98

Author(s):

Michael Y. Li

Keyword(s):

Lyapunov Functions ◽

Large Scale ◽

Epidemic Models ◽

Theoretic Method ◽

Graph Theoretic

Download Full-text

Lyapunov Functions for a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate and Varying Population Sizes

Discrete Dynamics in Nature and Society ◽

10.1155/2014/472746 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Ying Wang ◽

Zhidong Teng ◽

Mehbuba Rehim

Keyword(s):

Incidence Rate ◽

Lyapunov Functions ◽

Epidemic Models ◽

Inductive Method ◽

Nonlinear Incidence Rate ◽

Nonlinear Incidence ◽

All Solutions ◽

Population Sizes ◽

Sufficient And Necessary Conditions ◽

Varying Population

We investigate the dynamical behaviors of a class of discrete SIRS epidemic models with nonlinear incidence rate and varying population sizes. The model is required to possess different death rates for the susceptible, infectious, recovered, and constant recruitment into the susceptible class, infectious class, and recovered class, respectively. By using the inductive method, the positivity and boundedness of all solutions are obtained. Furthermore, by constructing new discrete type Lyapunov functions, the sufficient and necessary conditions on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium are established.

Download Full-text