Horst R. Thieme

Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations

Conditions are presented under which the solutions of asymptotically autonomous differential equations have the same asymptotic behavior as the solutions of the associated limit equations. An example displays that this does not hold in general.

Persistence under relaxed point-dissipativity (with application to an endemic model)

An approach to persistence theory is presented which focuses on the concept of uniform weak persistence. By using the most elementary dynamical systems concepts only, it can be shown that uniform weak persistence implies uniform strong persistence. This even holds under relaxed point dissipativity. Uniform weak persistence can be proved by the method of fluctuation or by analyzing the boundary flow for acyclicity with point dissipativity being only required in a neighborhood of the boundary. The approach is illustrated for a model describing the spread of a fatal infectious disease in a population that would grow exponentially without the disease. Sharp conditions are derived for both host and disease persistence and for host limitation by the disease.

A competitive exclusion principle for pathogen virulence

For a modified Anderson and May model of host parasite dynamics it is shown that infections of different levels of virulence die out asymptotically except those that optimize the basic reproductive rate of the causative parasite. The result holds under the assumption that infection with one strain of parasite precludes additional infections with other strains. Technically, the model includes an environmental carrying capacity for the host. A threshold condition is derived which decides whether or not the parasites persist in the host population.

Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models

Abstract The theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time-delayed reaction and diffusion population models.

On the Formulation and Analysis of General Deterministic Structured Population Models. II. Nonlinear Theory

Abstract—We define a linear physiologically structured population model by two rules, one for reproduction and one for “movement” and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R0 and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step.

Uniform persistence and permanence for non-autonomous semiflows in population biology.

Conditions are presented for uniform strong persistence of non-autonomous semiflows, taking uniform weak persistence for granted. Turning the idea of persistence upside down, conditions are derived for non-autonomous semiflows to be point-dissipative. These results are applied to time-heterogeneous models of S-I-R-S type for the spread of infectious childhood diseases. If some of the parameter functions are asymptotically almost periodic, an almost sharp threshold result is obtained for uniform strong endemicity versus extinction in terms of asymptotic time averages. Applications are also presented to scalar retarded functional differential equations modeling one species population growth.

How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?

Epidemiological and behavioral factors crucial to the dynamics of HIV/AIDS include long and variable periods of infectiousness, variable infectivity, and the processes of pair formation and dissolution. Most of the recent mathematical work on AIDS models has concentrated on the effects of long periods of incubation and heterogeneous mixing in the transmission dynamics of HIV. This paper explores the role of variable infectivity in combination with a variable incubation period in the dynamics of HIV transmission in a homogeneously mixing population. The authors keep track of an individuals infection-age, that is, the time that has passed since infection, and assume a nonlinear functional relationship between mean sexual activity and the size of the sexually active population that saturates at high population sizes. The authors identify a basic reproductive number Ro and show that the disease dies out if

Asymptotically autonomous semiflows: Chain recurrence and lyapunov functions

R_0 1

Spectral Bound and Reproduction Number for Infinite-Dimensional Population Structure and Time Heterogeneity

the disease persists in the population, and the incidence rate converge...

Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations

From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.