# Monotonicity of the Number of Passages in Linear Chains and of the Basic Reproduction Number in Epidemic Models

### J. Müller

Universität Tübingen, Germany### K.P. Hadeler

Universität Tübingen, Germany

## Abstract

models for infectious diseases, the basic reproduction number is the crucial parameter which determines the possibility of an outbreak. In simple situations it depends in a monotone way on the infectivity. Non-monotone behavior may occur in diseases where infectivity depends on time since infection and where transmission depends on social structure, as is shown by an example. A typical application is the HIV infection where transmission rates depend on existing pair bonds and infectivity changes drastically over time.

For a class of epidemic models with pair formation models and infectivity depending on time since infection it is shown that the basic reproduction number is a monotone function of infectivity. This observation is a consequence of a general result on a class of cyclic linear reaction chains with tridiagonal structure for which it is shown that the number of passages depends in a monotone way on the rates.

## Cite this article

J. Müller, K.P. Hadeler, Monotonicity of the Number of Passages in Linear Chains and of the Basic Reproduction Number in Epidemic Models. Z. Anal. Anwend. 19 (2000), no. 1, pp. 61–75

DOI 10.4171/ZAA/938