{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T01:17:19Z","timestamp":1773969439622,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:00:00Z","timestamp":1732665600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:00:00Z","timestamp":1732665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Innovation Capability Support Program of Shaanxi Province","award":["Program No. 2023-CX-TD-61"],"award-info":[{"award-number":["Program No. 2023-CX-TD-61"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12071268, 12301634"],"award-info":[{"award-number":["12071268, 12301634"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Biol Phys"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s10867-024-09667-1","type":"journal-article","created":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:33:50Z","timestamp":1732667630000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Analysis of tuberculosis model with indirect environmental transmission and optimal control"],"prefix":"10.1007","volume":"51","author":[{"given":"Xianyi","family":"Zhao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hui","family":"Cao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Danfeng","family":"Pang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,11,27]]},"reference":[{"key":"9667_CR1","doi-asserted-by":"crossref","unstructured":"Bussi, C., Gutierrez, M.G.: Mycobacterium tuberculosis infection of host cells in space and time. Fems. Microbiol. Rev. 43(4), 341-361 (2019). https:\/\/doi.org\/10.1093\/femsre\/fuz006","DOI":"10.1093\/femsre\/fuz006"},{"key":"9667_CR2","unstructured":"World Health Organization: Global Tuberculosis Report 2023. https:\/\/www.who.int\/tb\/publications\/global_report\/en\/ (2024). Accessed 5 November 2024"},{"key":"9667_CR3","doi-asserted-by":"crossref","unstructured":"Millington, K.A., White, R.G., Lipman, M., McQuaid, C.F., Hauser, J., Wooding, V., Potter, J., Abubakar, I., Wingfield, T.: The 2023 UN high-level meeting on tuberculosis: renewing hope, momentum, and commitment to end tuberculosis. Lancet Respir. Med. 12(1), 10-13 (2024). https:\/\/doi.org\/10.1016\/S2213-2600(23)00409-5","DOI":"10.1016\/S2213-2600(23)00409-5"},{"key":"9667_CR4","unstructured":"Tshilombo, K.V.: Survival and re-aerosolization in dust of Mycobacterium smegmatis \u2013 a surrogate for Mycobacterium tuberculosis. Stellenbosch University, Stellenbosch (2017)"},{"key":"9667_CR5","doi-asserted-by":"crossref","unstructured":"Ding, Z., Li, Y., Wang, X., Li, H., Cai, Y., Wang, B., Wang, K., Wang, W.: The impact of air pollution on the transmission of pulmonary tuberculosis. Math. Biosci. Eng. 17(4), 4317-4327 (2020). https:\/\/doi.org\/10.3934\/mbe.2020238","DOI":"10.3934\/mbe.2020238"},{"key":"9667_CR6","doi-asserted-by":"crossref","unstructured":"Mtetwa, H.N., Amoah, I.D., Kumari, S., Bux, F., Reddy, P.: The source and fate of Mycobacterium tuberculosis complex in wastewater and possible routes of transmission. BMC Public Health 22(1), 145 (2022). https:\/\/doi.org\/10.1186\/s12889-022-12527-z","DOI":"10.1186\/s12889-022-12527-z"},{"key":"9667_CR7","unstructured":"Davies, P.D.O., Grange, J.M.: Factors affecting susceptibility and resistance to tuberculosis. Thorax 56(2), 23-29 (2001)"},{"issue":"4","key":"9667_CR8","first-page":"3","volume":"2","author":"S Kanchan","year":"2015","unstructured":"Kanchan, S., Surya, K., Ajay, V.: Role of environmental factors in transmission of tuberculosis. Dyna. Human Health. 2(4), 3\u20136 (2015)","journal-title":"Dyna. Human Health."},{"key":"9667_CR9","doi-asserted-by":"crossref","unstructured":"Issarow, C.M., Mulder, N., Wood, R.: Environmental and social factors impacting on epidemic and endemic tuberculosis: a modelling analysis. Roy. Soc. Open Sci. 5(1), 170726 (2018). https:\/\/doi.org\/10.1098\/rsos.170726","DOI":"10.1098\/rsos.170726"},{"key":"9667_CR10","doi-asserted-by":"crossref","unstructured":"Cai, Y., Zhao, S., Niu, Y., Peng, Z., Wang, K., He, D., Wang, W.: Modelling the effects of the contaminated environments on tuberculosis in Jiangsu, China. J. Theor. Biol. 508, 110453 (2021). https:\/\/doi.org\/10.1016\/j.jtbi.2020.110453","DOI":"10.1016\/j.jtbi.2020.110453"},{"key":"9667_CR11","doi-asserted-by":"crossref","unstructured":"Li, Q., Wang, F.: An epidemiological model for tuberculosis considering environmental transmission and reinfection. Math. 11(11), 2423 (2023). https:\/\/doi.org\/10.3390\/math11112423","DOI":"10.3390\/math11112423"},{"key":"9667_CR12","doi-asserted-by":"crossref","unstructured":"Gao, C., Zhang, T., Liao, Y., Wang, Y., Jiao, H., Wu, M., Cui, Q., Wang, K., Wang, L.: Modelling of tuberculosis dynamics incorporating indirect transmission of contaminated environment and infectivity of smear-negative individuals: a case study for Xinjiang, China. Acta Tropica 254, 107130 (2024). https:\/\/doi.org\/10.1016\/j.actatropica.2024.107130","DOI":"10.1016\/j.actatropica.2024.107130"},{"key":"9667_CR13","doi-asserted-by":"crossref","unstructured":"Kang, T.L., Huo, H.F., Xiang, H.: Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments. Math. Biosci. Eng. 21(4), 5308-5334 (2024). https:\/\/doi.org\/10.3934\/mbe.2024234","DOI":"10.3934\/mbe.2024234"},{"key":"9667_CR14","doi-asserted-by":"crossref","unstructured":"Wang, M., Huang, C., Shen, X., Zhang, Y., Zhang, Z., Li, J., Zhao, G., Pan, Q., Jiang, Y.: The study of tuberculosis outbreak in a high school \u2013 Shanghai, China, 2017-2018. J. Public Health 30, 1055-1062 (2022). https:\/\/doi.org\/10.1007\/s10389-020-01380-2","DOI":"10.1007\/s10389-020-01380-2"},{"key":"9667_CR15","unstructured":"Halim, N.A.: Tuberculosis Model: a Mathematical Analysis. University of Malaya (Malaysia) (2013)"},{"key":"9667_CR16","doi-asserted-by":"crossref","unstructured":"Dye, C., Scheele, S., Pathania, V., Raviglione, M.C.: Global burden of tuberculosis: estimated incidence, prevalence, and mortality by country. JAMA 282(7), 677-686 (1999). https:\/\/doi.org\/10.1001\/jama.282.7.677","DOI":"10.1001\/jama.282.7.677"},{"key":"9667_CR17","doi-asserted-by":"crossref","unstructured":"Yu, S., Ma, J., Jia, Z.: Estimating the incidence of tuberculosis \u2013 Shanghai, China, 2025-2050. China CDC Weekly 2(52), 995-998 (2020). https:\/\/doi.org\/10.46234\/ccdcw2020.266","DOI":"10.46234\/ccdcw2020.266"},{"key":"9667_CR18","doi-asserted-by":"crossref","unstructured":"Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1-2), 29-48 (2002). https:\/\/doi.org\/10.1016\/S0025-5564(02)00108-6","DOI":"10.1016\/S0025-5564(02)00108-6"},{"key":"9667_CR19","doi-asserted-by":"crossref","unstructured":"Zeraick Monteiro, N., Rodrigues Mazorche, S.: Limitations and applications in a fractional Barbalat\u2019s Lemma. Fract. Calc. Appl. Anal. 26(1), 253-275 (2023). https:\/\/doi.org\/10.1007\/s13540-022-00111-6","DOI":"10.1007\/s13540-022-00111-6"},{"key":"9667_CR20","volume-title":"The Mathematical Theory of Optimal Processes","author":"LS Pontryagin","year":"1986","unstructured":"Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. Gordon and Breach Science Publishers, New York (1986)"},{"key":"9667_CR21","doi-asserted-by":"crossref","unstructured":"Zhang, S., Li, X., Zhang, T., Fan, Y., Li, Y.: The experiences of high school students with pulmonary tuberculosis in China: a qualitative study. BMC Infect. Dis. 16, 1-8 (2016). https:\/\/doi.org\/10.1186\/s12879-016-2077-y","DOI":"10.1186\/s12879-016-2077-y"}],"container-title":["Journal of Biological Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10867-024-09667-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10867-024-09667-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10867-024-09667-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,27]],"date-time":"2025-12-27T14:21:27Z","timestamp":1766845287000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10867-024-09667-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,27]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["9667"],"URL":"https:\/\/doi.org\/10.1007\/s10867-024-09667-1","relation":{},"ISSN":["0092-0606","1573-0689"],"issn-type":[{"value":"0092-0606","type":"print"},{"value":"1573-0689","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,27]]},"assertion":[{"value":"13 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 November 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"N\/A","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}},{"value":"N\/A","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare no competing interests.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"4"}}