Lyapunov functionals for multistrain models with infinite delay

Yoji Otani; Tsuyoshi Kajiwara; Toru Sasaki

doi:10.3934/dcdsb.2017025

Article Contents

2017, Volume 22, Issue 2: 507-536. Doi: 10.3934/dcdsb.2017025

This issue Previous Article Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks Next Article Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data

Lyapunov functionals for multistrain models with infinite delay

Graduate School of Environmental and Life Science, Okayama University, Okayama, 700-8530, Japan

Received: January 31, 2016

Revised: September 30, 2016

Published: December 2017

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Abstract

We construct Lyapunov functionals for delay differential equation models of infectious diseases in vivo to analyze the stability of the equilibria. The Lyapunov functionals contain the terms that integrate over all previous states. An appropriate evaluation of the logarithm functions in those terms guarantees the existence of the integrals. We apply the rigorous analysis for the one-strain models to multistrain models by using mathematical induction.

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Mathematics Subject Classification: Primary:92D25;Secondary:34D20.

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Figure 1. $x^* \ne \hat{x}_i$ for all $i\in J$: in this case $K_J=\{1,2\}$

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Figure 2. $x^* = \hat{x}_i$ for some $i\in J$: in this case $K_J=\{1,2,3\}, x^* = \hat{x}_3$

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