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The model also yields closed expressions for important features, such as the critical time to the singularity, in terms of basic demographic parameters. Our analysis flags a few important issues needing further research, such as the stochastic character of the emergence of variants. Greater understanding of the HEG process will yield dividends in other fields, since modern societies exhibit HEG at several levels, such as human population growth, economic indicators and technological innovation. For this reason, more research on HEG remains imperative, especially on HEG in the presence of limited resources.<\/jats:p>","DOI":"10.1007\/s00285-026-02437-8","type":"journal-article","created":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T06:44:53Z","timestamp":1784270693000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Hyper-exponential growth in an epidemic model: some explicit solutions and their significance"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9759-6792","authenticated-orcid":false,"given":"John M.","family":"Halley","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,17]]},"reference":[{"key":"2437_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3389\/frph.2024.1364352","volume":"6","author":"RJ Aitken","year":"2024","unstructured":"Aitken RJ (2024) What is driving the global decline of human fertility? 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