{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T17:37:39Z","timestamp":1791394659077,"version":"4.3.3"},"reference-count":56,"publisher":"The Royal Society","issue":"241","license":[{"start":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T00:00:00Z","timestamp":1786492800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of People\u2019s Republic of China","award":["12561091"],"award-info":[{"award-number":["12561091"]}]},{"name":"National Natural Science Foundation of People\u2019s Republic of China","award":["12361101"],"award-info":[{"award-number":["12361101"]}]},{"name":"Key Program of Natural Science in Gansu Province","award":["25JRRA148"],"award-info":[{"award-number":["25JRRA148"]}]},{"name":"Foundation for Innovative Fundamental Research Group Project of Gansu Province","award":["25JRRA805"],"award-info":[{"award-number":["25JRRA805"]}]},{"name":"Young Scientists Fund of Lanzhou Jiaotong University","award":["1200061424"],"award-info":[{"award-number":["1200061424"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,8,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The accurate estimation of time-varying transmission rates is fundamental for understanding infectious disease dynamics and implementing effective public health interventions. To this end, we propose an improved inverse method for estimating time-varying transmission rates in low-incidence settings, where conventional data pre-processing approaches often fail owing to sparse-case observations. To overcome this difficulty, we introduce an exponential B-spline interpolation approach that integrates both continuous and discrete inverse methods. In addition, we incorporate a non-standard finite-difference method, which enables the reconstruction of epidemic processes under non-equidistant time intervals caused by missing case data. This method ensures that transmission rate estimates remain non-negative and smooth, even when the observed data exhibit low cases or irregularly spaced time points. We apply this approach to several infectious disease models using real-world data from the People\u2019s Republic of China and Germany, including a scarlet fever model, a multi-strain influenza model and an age-structured influenza model, as well as a COVID-19 model for Germany. The results show that our method provides accurate transmission rate estimates, particularly in low-incidence infectious diseases, multi-group epidemic models and epidemic datasets with incomplete surveillance records, demonstrating its robustness and applicability across various epidemiological contexts. The improved inverse method offers a new perspective for epidemiological modelling and provides reliable technical support for related theoretical exploration and public health decision-making.<\/jats:p>","DOI":"10.1098\/rsif.2026.0044","type":"journal-article","created":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T00:09:52Z","timestamp":1786493392000},"source":"Crossref","is-referenced-by-count":1,"title":["An improved inverse method for estimating disease transmission rates in epidemics with low-incidence data"],"prefix":"10.1098","volume":"23","author":[{"given":"Shuanglin","family":"Jing","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Lanzhou Jiaotong University 1 , Lanzhou, Gansu, People\u2019s Republic of China"},{"name":"Gansu Center for Fundamental Research in Complex Systems Analysis and Control, Lanzhou Jiaotong University 2 , Lanzhou, Gansu, People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuting","family":"Huang","sequence":"additional","affiliation":[{"name":"The Third People\u2019s Hospital of Lanzhou 3 , Lanzhou, Gansu, People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9563-1483","authenticated-orcid":false,"given":"Hai-Feng","family":"Huo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Lanzhou Jiaotong University 1 , Lanzhou, Gansu, People\u2019s Republic of China"},{"name":"Gansu Center for Fundamental Research in Complex Systems Analysis and Control, Lanzhou Jiaotong University 2 , Lanzhou, Gansu, People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2026,8,12]]},"reference":[{"key":"2026081120094487800_bib1","doi-asserted-by":"crossref","first-page":"S336","DOI":"10.1137\/22M1498796","article-title":"Discrete inverse method for extracting disease transmission rates from accessible infection data","volume":"84","author":"Wang","year":"2024","journal-title":"SIAM J. 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