Paul L. Salceanu

Persistence in a discrete-time, stage-structured epidemic model

Discrete-time SI and SIR epidemic models, formulated by Emmert and Allen [J. Differ. Equ. Appl., 10 (2004), pp. 1177–1199] for the spread of a fungal disease in a structured amphibian host population, are analysed. Criteria for persistence of the population as well as for persistence of the disease are established. Global stability results for host extinction and for the disease-free equilibrium are presented.

Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents.

This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence ina class of dissipative discrete-time dynamical systems on the positive orthant of R(m), generated by maps. Here a united approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of R(m+) to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.

Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets

Let M be a compact invariant set contained in a boundary hyperplane of the positive orthant of ℝ n for a discrete or continuous time dynamical system defined on the positive orthant. Using elementary arguments, we show that M is uniformly weakly repelling in directions normal to the boundary in which M resides provided all normal Lyapunov exponents are positive. This result is useful in establishing uniform persistence of the dynamics.

Persistence in a discrete-time stage-structured fungal disease model.

A discrete-time susceptible and infected (SI) epidemic model, with less than 100% vertical disease transmission, for the spread of a fungal disease in a structured amphibian host population, is analysed. Criteria for persistence of the population as well as the disease are established. Stability results for host extinction and for the disease-free equilibrium are presented. Bifurcation theory is used to establish existence of an endemic equilibrium.

Robust uniform persistence and competitive exclusion in a nonautonomous multi-strain SIR epidemic model with disease-induced mortality

A nonautonomous version of the SIR epidemic model in Ackleh and Allen (2003) is considered, for competition of

On a discrete selection–mutation model

n

Competitive Exclusion Through Discrete Time Models

infection strains in a host population. The model assumes total cross immunity, mass action incidence, density-dependent host mortality and disease-induced mortality. Sufficient conditions for the robust uniform persistence of the total population, as well as of the susceptible and infected subpopulations, are given. The first two forms of persistence depend entirely on the rate at which the population grows from the extinction state, respectively the rate at which the disease is vertically transmitted to offspring. We also discuss the competitive exclusion among the