Lyapunov functions for tuberculosis models with fast and slow progression

C Connell Mccluskey

doi:10.3934/mbe.2006.3.603

C Connell Mccluskey

Open Access

https://doi.org/10.3934/mbe.2006.3.603

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Abstract

The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptotically stable.

Highlights

According the World Health Organization, one third of the world’s population is infected with tuberculosis (TB), leading to between two and three million deaths each year
The Lyapunov function which follows (with A and B given by equation (2.9)) does work for p = 1, but it is necessary to do the calculation separately for that case
The only invariant set contained in C is the singleton {Q∗}. This shows that each solution which intersects R5>0 limits to the endemic equilibrium Q∗, giving the following result

Summary

Introduction

According the World Health Organization, one third of the world’s population is infected with tuberculosis (TB), leading to between two and three million deaths each year. The global dynamics of both the three-dimensional model and the five-dimensional model of fast and slow progression in [1] are resolved through the use of Lyapunov functions. Global stability, tuberculosis, fast and slow progression.

Results

Conclusion