Volume 31 - Issue 1 - June 2006

Health expectancies

C. R. Heathcote


This expository paper discusses a particular approach to the estimation of the expected occupation times of finite state space stochastic processes in discrete time. In the public health context these occupation times are known as health expectancies, since they are the future times that a person of given age is expected to spend in different states of health. An assumption is that aggregate data is available as published by official statistical agencies, and this is typically the case with health data. A large sample logistic regression methodology is described and used to estimate age-dependent probabilities of states and hence the expectancies.

HIV, antiretroviral therapy, and dynamic programming

Frederick H. Chen


An epidemiological model of HIV transmission with endogenously determined risk behavior is presented, and its predictions regarding the impact of improving access to antiretroviral therapy (ART) are derived. It is shown that if antiretroviral treatment decreases the transmission probability by a higher percentage than the treatment-induced reduction in AIDS mortality rate, then HIV prevalence may be lowered by expanding ART coverage. Otherwise, increasing access to treatment will worsen the epidemic. Although expanding ART coverage can result in disease eradication if the transmission probability is lowered sufficiently, the conditions under which eradication obtains are more stringent than those predicted by conventional epidemiological models with exogenously specified individual behavior.

Solving some delay differential equations with computer algebra

Jane M. Heffernan and Robert M. Corless


We explore the use of a computer algebra system to solve some very simple linear delay differential equations (DDEs). Though simple, some of these DDEs are useful in their own right, and may also be helpful as test problems for more general methods. The solution methods included in this paper are of interest because they are widely used in studies involving the solution to DDEs, and can be easily automated using tools provided by computer algebra. In this paper we give detailed descriptions of the classical method of steps, the Laplace transform method, and a novel least-squares method, followed by some discussion on the limitations and successes of each.

A pest immigration process controlled by an intermittent predator

E. G. Kyriakidis and T. D. Dimitrakos


An infinite-state Markov decision model is considered for the control of a simple immigration process, which represents a pest population, by an intermittent predator. It is assumed that the predator may leave the habitat before capturing all the pests. The cost rate caused by the pests is an increasing function of their population size, while the cost rate of the controlling action is constant. A sequence of suitable finite-state Markov decision models is constructed such that the optimal average-cost policies in the sequence converge to the optimal average-cost policy in the original model. There is strong numerical evidence that the optimal policy introduces the predator if and only if the pest population is greater than or equal to some critical size.

An exact solution to a Kolmogorov-type model for two interacting populations

Max-Olivier Hongler and Roger Filliger


We present an exact solution to a model for two species interacting according to a nonlinear Kolmogorov-type equation with a structure similar to that of predator–prey models. We obtain simple analytical expressions for the parametric equations of the cycles, the cycles in the phase space, and the Hamiltonian and the corresponding integrating factor, showing that the dynamics are globally conservative. We then propose a general method for constructing similar classes of evolutionary models. As a particular example, we derive the evolutionary equation describing the time-dependent behavior of a two-player, two-strategy asymmetric game.

An exact solution for an M/M/1 queue with piecewise-constant rates

P. R. Parthasarathy and R. Sudhesh


A time-dependent solution for the number of customers in an M/M/1 queue with piecewise-constant arrival and departure rates is derived in closed form. An explicit expression is also found for the time-dependent mean of the queue length, and numerical illustrations provided for three simple cases.

The product Cauchy distribution

Saralees Nadarajah and Samuel Kotz


A new Cauchy distribution with finite mean and finite variance is introduced. Various structural properties of this distribution are derived, including its cumulative density function, moments and maximum likelihood estimates and the Fisher information matrix.