K.P. Hadeler

A core group model for disease transmission.

Models for sexually transmitted diseases generally assume that the size of the core group is fixed. Publicly available information on disease prevalence may influence the recruitment of new susceptibles into highly sexually active populations. It is assumed that the recruitment rate into the core population is low while disease prevalence is high, core group members mix only with each other, disease levels outside the core are negligible, and some core group members reduce their risk through the use of a partially effective vaccine or prophylactics. A demographic-epidemic model is formulated in which the combined size of the core and non-core population is constant. A simpler version models the epidemic in an isolated core population of constant size under the influence of educational programs and measures that reduce susceptibility. The threshold condition for an endemic infection is determined. Backward bifurcations, multiple infective stationary states, and hysteresis phenomena can be observed even in the simplified version. Abrupt changes in disease prevalence levels may result from small changes in the disease management parameters and do not occur in the absence of such a program. The general conclusion is that partially effective vaccination or education programs may increase the total number of cases while decreasing the relative frequency of cases in the core group. The study throws some new light on the role of the reproduction number in connection with elimination attempts. It shows that although the reproduction number defines the threshold for the spread of the disease in a susceptible population, it is of limited value when elimination of an existing epidemic is planned.

Backward bifurcation in epidemic control

For a class of epidemiological SIRS models that include public health policies, the stability at the uninfected state and the prevalence at the infected state are investigated. Backward bifurcation from the uninfected state and hysteresis effects are shown to occur for some range of parameters. In such cases, the reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. An explicit expression is given for this quantity. The phenomenon of subcritical bifurcation in epidemic modeling is also discussed in terms of group models, pair formation, and macroparasite infection.

On copositive matrices

Abstract A criterion for copositive matrices is given and for n = 3 the set of all copositive matrices is determined in terms of matrix elements. Copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations.

Parameter identification in epidemic models

Starting from a recent paper of Pollicott, Wang and Weiss we try to obtain improved representation formulas for the estimation of the time-dependent transmission rate of an epidemic in terms of either incidence or prevalence data. Although the formulas are (trivially) mathematically equivalent to previous formulas, the new representations need no additional estimates and they should be more stable numerically. We review the discrete time and the stochastic continuous time approach. We replace the assumption that recovery follows an exponential distribution and get estimates for the transmission rate for constant duration of the infectious phase.

Quiescence stabilizes predator–prey relations

The classical MacArthur Rosenzweig predator–prey system has a stable coexistence point or, if either the prey capacity is large or the predator mortality is low, a stable limit cycle. The question here is how the stability properties of the coexistence point change when the prey or the predator or both can go quiescent. It can be shown that a stable equilibrium stays stable, but an unstable equilibrium may become stable. The exact stability domain is determined. In general, increasing the duration of the quiescent phase of the prey or of the predator widens the stability window. Numerical studies show that limit cycles shrink when quiescent phases are introduced.

On the roots of cauchy polynomials

Abstract We discuss the butterfly-shaped region Mn in the complex plane which is defined as the set of all the roots of all normalized Cauchy polynomials of degree n. Besides the geometric structure, e.g. that the set Mn sb {1} is star-shaped with respect to the origin, some results concerning the boundary of Mn are presented.

Coupling and Quiescence

Suppose we have two vector fields (f,g: mathbb{R}^{n} rightarrow mathbb{R}^{n}) and the differential equations