A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I_p S_q$ for the case p<1 are constructed. Global stability of the models is thereby established

Global asymptotic properties for a Leslie-Gower food chain model

We study global asymptotic properties of a continuous time Leslie-Gower food chain model. We construct a Lyapunov function which enables us to establish global asymptotic stability of the unique coexisting equilibrium state.Comment: 5 Pages, 1 figure. Keywords: Leslie-Gower model, Lyapunov function, global stabilit

Nonlinear incidence and stability of infectious disease models

In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds

Long-term coexistence for a competitive system of spatially varying gradient reaction-diffusion equations

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction-diffusion equations. In this work we consider a system of two reaction diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady-states (the time-independent solutions) and examine their stability and bifurcations

Stability of a stochastically perturbed model of intracellular single-stranded RNA virus replication

Replication of single-stranded RNA virus can be complicated, compared to that of double-stranded virus, as it require production of intermediate antigenomic strands that then serve as template for the genomic-sense strands. Moreover, for ssRNA viruses, there is a variability of the molecular mechanism by which genomic strands can be replicated. A combination of such mechanisms can also occur: a fraction of the produced progeny may result from a stamping-machine type of replication that uses the parental genome as template, whereas others may result from the replication of progeny genomes. F. Mart\'{\i}nez et al. and J. Sardany\'{e}s at al. suggested a deterministic ssRNA virus intracellular replication model that allows for the variability in the replication mechanisms. To explore how stochasticity can affect this model principal properties, in this paper we consider the stability of a stochastically perturbed model of ssRNA virus replication within a cell. Using the direct Lyapunov method, we found sufficient conditions for the stability in probability of equilibrium states for this model. This result confirms that this heterogeneous model of single-stranded RNA virus replication is stable with respect to stochastic perturbations of the environment

Cluster formation for multi-strain infections with cross-immunity

Many infectious diseases exist in several pathogenic variants, or strains, which interact via cross-immunity. It is observed that strains tend to self-organise into groups, or clusters. The aim of this paper is to investigate cluster formation. Computations demonstrate that clustering is independent of the model used, and is an intrinsic feature of the strain system itself. We observe that an ordered strain system, if it is sufficiently complex, admits several cluster structures of different types. Appearance of a particular cluster structure depends on levels of cross-immunity and, in some cases, on initial conditions. Clusters, once formed, are stable, and behave remarkably regularly (in contrast to the generally chaotic behaviour of the strains themselves). In general, clustering is a type of self-organisation having many features in common with pattern formation

Stryker Osteonics: Prosthetic Knee Joint

We examine, within a simple bearing model of a knee joint that only consideres pure sliding, the effect of the presence of a small vertical hole in the load area on the fluid film properties. The calculations indicate that fluid is entrapped in such a hole, which, for constant load, causes a smaller minimal film separation of the two surfaces. This will lower the horizontal friction, but may also bring about surface contact in high load situations