Global properties of a delayed SIR epidemic model with multiple parallel infectious stages

Discrete state-structured epidemic models with distributed delays

In this paper, we propose and investigate a class of discrete state-structured epidemic models with distributed delays. The model incorporates a general delay incidence form in which new infections distribute into different disease states, and infected individuals switch among different states. We establish that the global dynamics are completely determined by basic reproduction number [Formula: see text]: if [Formula: see text], the disease free equilibrium is globally asymptotically stable; if [Formula: see text], the disease free equilibrium is unstable, and there is a unique endemic equilibrium which is globally asymptotically stable.

Global stability properties of a class of renewal epidemic models

We investigate the global dynamics of a general Kermack-McKendrick-type epidemic model formulated in terms of a system of renewal equations. Specifically, we consider a renewal model for which both the force of infection and the infected removal rates are arbitrary functions of the infection age, [Formula: see text], and use the direct Lyapunov method to establish the global asymptotic stability of the equilibrium solutions. In particular, we show that the basic reproduction number, [Formula: see text], represents a sharp threshold parameter such that for [Formula: see text], the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when [Formula: see text], i.e. when it exists.

Dynamics of an HIV Model with Multiple Infection Stages and Treatment with Different Drug Classes

Highly active antiretroviral therapy can effectively control HIV replication in infected individuals. Some clinical and modeling studies suggested that viral decay dynamics may depend on the inhibited stages of the viral replication cycle. In this paper, we develop a general mathematical model incorporating multiple infection stages and various drug classes that can interfere with specific stages of the viral life cycle. We derive the basic reproductive number and obtain the global stability results of steady states. Using several simple cases of the general model, we study the effect of various drug classes on the dynamics of HIV decay. When drugs are assumed to be 100% effective, drugs acting later in the viral life cycle lead to a faster or more rapid decay in viremia. This is consistent with some patient and experimental data, and also agrees with previous modeling results. When drugs are not 100% effective, the viral decay dynamics are more complicated. Without a second population of long-lived infected cells, the viral load decline can have two phases if drugs act at an intermediate stage of the viral replication cycle. The slopes of viral load decline depend on the drug effectiveness, the death rate of infected cells at different stages, and the transition rate of infected cells from one to the next stage. With a second population of long-lived infected cells, the viral load decline can have three distinct phases, consistent with the observation in patients receiving antiretroviral therapy containing the integrase inhibitor raltegravir. We also fit modeling prediction to patient data under efavirenz (a nonnucleoside reverse-transcriptase inhibitor) and raltegravir treatment. The first-phase viral load decline under raltegravir therapy is longer than that under efavirenz, resulting in a lower viral load at initiation of the second-phase decline in patients taking raltegravir. This explains why patients taking a raltegravir-based therapy were faster to achieve viral suppression than those taking an efavirenz-based therapy. Taken together, this work provides a quantitative and systematic comparison of the effect of different drug classes on HIV decay dynamics and can explain the viral load decline in HIV patients treated with raltegravir-containing regimens.

Stability analysis for delayed viral infection model with multitarget cells and general incidence rate

In this paper, the sharp threshold properties of a (2n + 1)-dimensional delayed viral infection model are investigated. This model combines with n classes of uninfected target cells, n classes of infected cells and nonlinear incidence rate h(x, v). Two kinds of distributed time delays are incorporated into the model to describe the time needed for infection of uninfected target cells and virus replication. Under certain conditions, it is shown that the basic reproduction number is a threshold parameter for the existence of the equilibria, uniform persistence, as well as for global stability of the equilibria of the model.