# Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity

Research output: Contribution to journalArticlepeer-review

280 Scopus citations

## Abstract

Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ord inary differential equations: the sign of the spectral bound of the variational matrix at 0 decides whether, at low density, the population becomes extinct or grows. Another important threshold parameter is the reproduction number R, which is the spectral radius of a related positive matrix. As is well known, the spectral bound and R - 1 have the same sign provided that the matrices have a particular form. The relation between spectral bound and reproduction number extends to models with infinite-dimensional state space and then holds between the spectral bound of a resolvent-positive closed linear operator and the spectral radius of a positive bounded linear operator. We also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete-time models. We illustrate the general theory by applying it to an epidemic model with distributed susceptibility, population models with age structure, and, using evolution semigroups, to time-heterogeneous population models.

Original language English (US) 188-211 24 SIAM Journal on Applied Mathematics 70 1 https://doi.org/10.1137/080732870 Published - Jul 1 2009

## Keywords

• Age-structure
• Evolution semigroups
• Evolutionary systems
• Exponential growth bound
• Integrated semigroups
• Laplace transform
• M matrices
• Next generation operator
• Operator semigroups
• Quasi-positive matrices
• Resolvent-positive operators
{"ft_lang_label":"__label__en","ft_lang_prob":0.8616368,"math_prob":0.880232,"size":1757,"snap":"2022-27-2022-33","text_gpt3_token_len":367,"char_repetition_ratio":0.13633771,"word_repetition_ratio":0.007905139,"special_character_ratio":0.1906659,"punctuation_ratio":0.056603774,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97536045,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T08:14:19Z\",\"WARC-Record-ID\":\"<urn:uuid:f0801bdc-8da1-43f5-b3d4-2fa765fe2b6e>\",\"Content-Length\":\"55509\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3768a047-e94d-40ce-8af0-60e471d6412b>\",\"WARC-Concurrent-To\":\"<urn:uuid:d93ffb57-36fa-4250-b4d6-65391360851f>\",\"WARC-IP-Address\":\"3.90.122.189\",\"WARC-Target-URI\":\"https://asu.pure.elsevier.com/en/publications/spectral-bound-and-reproduction-number-for-infinite-dimensional-p\",\"WARC-Payload-Digest\":\"sha1:JPVEQ2GPOZRUFDLWQUXB6N4ER6SSPZKF\",\"WARC-Block-Digest\":\"sha1:JZPEWTVDRMWLPBZ4KDA76JYRP7RYBQUT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882570913.16_warc_CC-MAIN-20220809064307-20220809094307-00742.warc.gz\"}"}