Andrei Korobeinikov

Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models

Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced. Global stability of the endemic equilibrium states of the models is thereby established.

A Lyapunov function for Leslie-Gower predator-prey models

A Lyapunov function for continuous time Leslie-Gower predator-prey models is introduced. Global stability of the unique coexisting equilibrium state is thereby established.

Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate

We consider two models for the spread of an infection with a free-living infective stage, where parasite reproduction and virulence (parasite-induced mortality) depend on the parasite dose to which the host is exposed and are given by unspecified non-linear functions of the number of the free pathogen particles, and the incidence rate is non-linear. We study the impact of these non-linearities with the focus on the global properties of these models. We consider a very general form of the non-linearities: we assume that the virulence and the parasite reproduction rates are given by unspecified non-linear functions of the number of the free pathogen particles and that the incidence rate is an unspecified function of the number of susceptible hosts and free pathogen particles; all these functions are constrained by a few biologically feasible conditions. We construct Lyapunov functions that enable us to find biologically realistic conditions which are sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number, this equilibrium state can be either positive, where parasite endemically persists, or infection free.

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

We consider global properties of compartment SIR and SEIR models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number R0, this state can be either endemic (R0>1), or infection-free (R0≤1).

Stability of ecosystem: global properties of a general predator–prey model

Establishing the conditions for the stability of ecosystems and for stable coexistence of interacting populations is a problem of the highest priority in mathematical biology. This problem is usually considered under specific assumptions made regarding the functional forms of non-linear feedbacks. However, there is growing understanding that this approach has a number of major deficiencies. The most important of these is that the precise forms of the functional responses involved in the model are unknown in detail, and we can hardly expect that these will be known in feasible future. In this paper, we consider the dynamics of two species with interaction of consumer-supplier (prey-predator) type. This model generalizes a variety of models of population dynamics, including a range of prey-predator models, SIR and SIRS epidemic models, chemostat models, etc. We assume that the functional responses that are usually included in such models are given by unspecified functions. Using the direct Lyapunov method, we derive the conditions which ensure global asymptotic stability of this general model. It is remarkable that these conditions impose much weaker constraints on the system properties than that are usually assumed. We also identify the parameter that allows us to distinguish between existence and non-existence of the coexisting steady state.

Lyapunov functions for SIR and SIRS epidemic models

In this paper, we construct a new Lyapunov function for a variety of SIR and SIRS models in epidemiology. Global stability of the endemic equilibrium states of these systems is established.

Global properties of the three-dimensional predator-prey Lotka-Volterra systems

The global properties of the classical three-dimensional Lotka-Volterra two prey-one predator and one prey-two predator systems, under the assumption that competition can be neglected, are analysed with the direct Lyapunov method. It is shown that, except for a pathological case, one species is always driven to extinction, and the system behaves asymptotically as a two-dimensional predator-prey Lotka-Volterra system. The same approach can be easily extended to systems with many prey species and one predator, or many predator species and one prey, and the same conclusion holds. The situation considered is common for New Zealand wild life, where indigenous and introduced species interact with devastating consequences for the indigenous species. According to our results the New Zealand indigenous species are definitely driven to extinction, not only in consequence of unsuccessful competition, but even when competition is absent. This result leads to a better understanding of the mechanism of natural selection, and gives a new insight into pest control practice.

Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility

We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.

Global stability of a population dynamics model with inhibition and negative feedback.

Reactions or interactions with the rate which is inhibited by the product or a by-product of the reaction are fairly common in biology and chemical kinetics. Biological examples of such interactions include selfpoisoning of bacteria, the non-lytic immune response and the antiviral (and in particular antiretroviral) therapy. As a case study, in this notice, we consider global asymptotic properties for a simple model with negative feedback (the Wodarz model) where the interaction is inhibited by a by-product of the reaction. The objective of this notice is an extending of a technique that was developed during last decade for the global analysis of models with positive feedback to the systems, where the feedback is negative. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of this model. This result enables us to evaluate the comparative impacts of the lytic and nonlytic components, the efficiency of the antiviral therapy and the possibility of self-poisoning for bacteria. The same approach can be successfully applied to more complex models with negative feedback.

HIV evolution and progression of the infection to AIDS.

In this paper, we propose and discuss a possible mechanism, which, via continuous mutations and evolution, eventually enables HIV to break from immune control. In order to investigate this mechanism, we employ a simple mathematical model, which describes the relationship between evolving HIV and the specific CTL response and explicitly takes into consideration the role of CD4(+)T cells (helper T cells) in the activation of the CTL response. Based on the assumption that HIV evolves towards higher replication rates, we quantitatively analyze the dynamical properties of this model. The model exhibits the existence of two thresholds, defined as the immune activation threshold and the immunodeficiency threshold, which are critical for the activation and persistence of the specific cell-mediated immune response: the specific CTL response can be established and is able to effectively control an infection when the virus replication rate is between these two thresholds. If the replication rate is below the immune activation threshold, then the specific immune response cannot be reliably established due to the shortage of antigen-presenting cells. Besides, the specific immune response cannot be established when the virus replication rate is above the immunodeficiency threshold due to low levels of CD4(+)T cells. The latter case implies the collapse of the immune system and beginning of AIDS. The interval between these two thresholds roughly corresponds to the asymptomatic stage of HIV infection. The model shows that the duration of the asymptomatic stage and progression of the disease are very sensitive to variations in the model parameters. In particularly, the rate of production of the naive lymphocytes appears to be crucial.