Global dynamics of a staged-progression model with amelioration for infectious diseases

Abstract

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R 0 . If R 0 ≤ 1, then the disease-free equilibrium P 0 is globally asymptotically stable, and the disease always dies out. If R 0 > 1, P 0 is unstable, a unique endemic equilibrium P * is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated.

A graph-theoretic approach to the method of global Lyapunov functions

“…Compared to results in [4,10], the group structure in system (1.3) greatly increases the complexity exhibited in the derivatives of the Lyapunov function V . The key to our analysis is a complete description of the patterns exhibited in the derivative • V using graph theory.…”

“…[3,9] and the references therein. Recently, this form of Lyapunov functions was applied to several single-group epidemic models and used to prove the global stability of a unique endemic equilibrium [4,10]. In the present paper, to further demonstrate the applicability of this form of Lyapunov functions, we apply them to a class of n-group (n ≥ 2) epidemic models of SEIR type with bilinear incidence, described by the following system of equations:…”

Stability of an SEIR epidemic model with independent stochastic perturbations

“…Tuberculosis and HIV are examples of infections that are commonly modeled by including such a latent class. Among others, the paper [4] of Guo and Li presents a model which provides for several levels of intensity of infection. There are also models with multiple stages for other diseases, such as for instance the HIV model in Ref.…”

Global dynamics of a two-strain avian influenza model

“…Consider the non-linear Lyapunov function (non-linear functions of this type have been used in [8,12,16]) for system (18)…”