Some limitations on the use of the basic reproduction number

Abstract

We briefly review some history and some of the main mathematical properties of the basic reproduction number, a concept encountered frequently in mathematical biology. We discuss some limitations on the practical use of basic reproduction numbers. One stems from the known fact that basic reproduction numbers are not unique. Another is related to the fact that basic reproduction numbers and the spectral radius of their associated linear operator could respond in opposite ways when controlled.

KEYWORDS:

• Basic reproduction number
• invariant cone
• stability
• control

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 39A06
• 39A60
• 92-08
• 15B48
• 47D06

Acknowledgments

We would like to thank Horst Thieme and Odo Diekmann for commenting on a preliminary draft of this paper, and for pointing out several relevant references. We are also grateful to the organizers of the CMBS Conference: Interface of Mathematical Biology and Linear Algebra for hosting a wonderful meeting at the University of Central Florida in Orlando, FL in May of 2022. Our special thanks go to Zhisheng Shuai for his tireless efforts which made this conference so successful. We would also like to thank several conference participants who contributed to certain aspects of the work presented here. They include Keith Carlson, Alberto Condori, Leah LeJeune, Archana Neupane Timsina, Alexsander Popovic and Merennage Sadun Salgado.

Disclosure statement

No potential conflict of interest was reported by the author(s).