A note on dynamics of an age-of-infection cholera model.

Ran Zhang,Jinliang Wang,Toshikazu Kuniya

doi:10.3934/mbe.2016.13.227

Abstract

A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if i(a,t)/i*(a)=0 on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.

Highlights

In this note, we will revisit an age-of-infection cholera model which is presented and studied in [1] by F
The results presented in [1] gave a global attracting analysis of equilibria of model (1), but leaving out the necessary arguments, including relative compactness of orbit generated by system (1) and uniform persistence, which are two major challenges in applying the main results in [7] to particular models
The object of this note is to show that, under some assumptions, system (1) can be reformulated as a Volterra integral equation in order to apply functional analysis theory, and we present some results about uniform persistence and about the existence of global attractors

Summary

Introduction

We will revisit an age-of-infection cholera model which is presented and studied in [1] by F. For any positive equilibrium P ∗ = (S∗, i∗(a), p∗(b)) of system (1), it should satisfy the following equations The object of this note is to show that, under some assumptions, system (1) can be reformulated as a Volterra integral equation in order to apply functional analysis theory, and we present some results about uniform persistence and about the existence of global attractors.

Results

Conclusion