General modeling considerations

SPREAD

231

OF EPIDEMICS

REPRODUCTION NUMBERS AND STABILITY OF EQUILIBRIA FOR DETERMINISTIC AND STOCHASTIC SI MODELS JOHN

A. JACQUEZ,

University

of Michigan,

Ann Arbor,

Michigan

In recent work Simon and Jacquez relate conditions for local and global stability of the equilibria and basic reproduction numbers, for deterministic SI models for homogeneous and heterogeneous populations. In this paper it is shown that for homogeneous populations corresponding deterministic and stochastic models give the same threshold conditions in terms of the basic reproduction numbers, for global stability of the disease-free equilibria. Under what conditions is that still true for heterogeneous populations? Jacquez, J. A., Reproduction numbers and thresholds in stochastic epidemic models. I. Homogeneous populations, Michigan HIV Group Preprint no. 9 (1990). Jacquez, J. A. and C. P. Simon, AIDS: the epidemiologi~al significance of two different mean rates of partner change, IlMA J. Mu&. App/. Biol. Med. (in press). Jacquez, J. A., C. I’. Simon, and J. S. Koopman, The reproductive number in deterministic models of contagious disease, Co~~eff~s on Thuor. Biol. (in press). Simon, C. P. and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations, Michigan HIV Group Preprint no 7 (1990).

THE ROLE OF HOMOGENEOUS EQUATIONS IN MODELING DEMOGRAPHIC AND EPIDEMIOLOGICAL PROCESSES KARL

HADELER,

In modeling tion size and Homogeneous account. Some

Institut

fur Biologic,

Tubingen,

West Germany

epidemiologi~al processes, the estimation of actual populaactual contact rate and transmission rates is important. equation models are suited to take density effects into results on homogeneous models are presented.

Busenberg, S. N. and K. P. Hadeler, Demography and epidemics, Mat/z. Biosci. (in press). Hadeler, K. P., Periodic solutions of homogeneous equations, 1. Diff. Eq. (in press). Hadeler, K. P., Homogeneous delay equations and models for pair formation (submitted to J. Math. Biol.).

BASIC STOCHASTIC

MODELS

DYNAMIC

FOR AN EPIDEMIC

MODELS

AKI? SVENSSON,

Stockholm

University,

Stockholm,

Sweden

A simple model for the intensity of infection during an epidemic closed population is studied. It is shown that the size of an epidemic

in a (i.e.,