Global dynamics of a staged progression model for infectious diseases

Abstract

We analyze a mathematical model for infectious diseases that progress through distinct stages within infected hosts. An example of such a disease is AIDS, which results from HIV infection. For a general n-stage stage-progression (SP) model with bilinear incidences, we prove that the global dynamics are completely determined by the basic reproduction number R0: 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; P0 is unstable, and a unique endemic equilibrium P(*) is globally asymptotically stable, and the disease persists at the endemic equilibrium. The basic reproduction numbers for the SP model with density dependent incidence forms are also discussed.

Global Stability of Equilibria in a Two-Sex HPV Vaccination Model

“…In systems with three or less dimensions, a typical global stability analysis invokes the Poincaré-Bendixson theorem and applies one of the available methods to rule out existence of periodic solutions. Recently, several researchers have successfully applied the Lyapunov direct method to prove the global stability of endemic states in a variety of epidemic and virus population in vivo models with systems of higher dimensions (Korobeinikov, 2004a(Korobeinikov, , 2004b(Korobeinikov, , 2006Korobeinikov and Maini, 2004;Korobeinikov and Wake, 2002;Iwasa et al, 2004;Guo and Li, 2006;McCluskey, 2006). The origin of the Lyapunov functions used in the analysis of these models goes back to Volterra (Harrison, 1979).…”

Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators

“…Recently, even large epidemic systems have been successfully treated by Volterra Lyapunov functions, first systems with an arbitrary number of disease stages, both finite [1,17,25,26,46] and distributed [39,42,[47][48][49][50][51], and then systems with an arbitrary, but finite number of subpopulations [13,18,19,35,36], and finally combinations of both [37]. In most of the latter cases, graph-theoretic methods are used (Metzler matrices are used in [13]).…”

Global Properties of SIR and SEIR Epidemic Models with Multiple Parallel Infectious Stages

“…This function, which has its origin in ecology, was extended to the models of epidemiology by Korobeinikov and Wake (2002) and Korobeinikov (2004a), and then effectively applied to a variety of compartment models, including the models with multiple progressive stages (Guo and Li, 2006;Korobeinikov, 2004b), and models with nonlinear incidence rates of different forms (Georgescu and Hsieh, 2006;Korobeinikov, 2006Korobeinikov, , 2007Korobeinikov, , 2008Maini, 2004, 2005;Korobeinikov and Petrovskii, 2008).…”