{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T22:29:25Z","timestamp":1787783365686,"version":"build-2784847793"},"reference-count":67,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T00:00:00Z","timestamp":1768003200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T00:00:00Z","timestamp":1768003200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003989","name":"Ikerbasque, Basque Foundation for Science","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003989","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["FJC2021-046826-I"],"award-info":[{"award-number":["FJC2021-046826-I"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["RYC2021-031380-I"],"award-info":[{"award-number":["RYC2021-031380-I"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003086","name":"Eusko Jaurlaritza","doi-asserted-by":"publisher","award":["BERC 2022-2025"],"award-info":[{"award-number":["BERC 2022-2025"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["CEX2021-001142-S \/ MICIN \/ AEI \/ 10.13039\/501100011033"],"award-info":[{"award-number":["CEX2021-001142-S \/ MICIN \/ AEI \/ 10.13039\/501100011033"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s00285-026-02341-1","type":"journal-article","created":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T07:20:37Z","timestamp":1768029637000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Bifurcation analysis of a two-infection transmission model with explicit vector dynamics"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3786-9159","authenticated-orcid":false,"given":"Akhil Kumar","family":"Srivastav","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vanessa","family":"Steindorf","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bruno V.","family":"Guerrero","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nico","family":"Stollenwerk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bob W.","family":"Kooi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ma\u00edra","family":"Aguiar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,1,10]]},"reference":[{"key":"2341_CR1","doi-asserted-by":"publisher","unstructured":"Abdalgader T, Banerjee M, Zhang L (2022) Spatially weak syncronization of spreading pattern between aedes albopictus and dengue fever. Ecological Modelling 473, 110123 https:\/\/doi.org\/10.1016\/j.ecolmodel.2022.110123","DOI":"10.1016\/j.ecolmodel.2022.110123"},{"key":"2341_CR2","doi-asserted-by":"publisher","unstructured":"Aguiar M (2018) Dengue vaccination: a more ethical approach is needed. The Lancet Infectious Diseases 391 https:\/\/doi.org\/10.1016\/S0140-6736(18)30865-1","DOI":"10.1016\/S0140-6736(18)30865-1"},{"key":"2341_CR3","doi-asserted-by":"publisher","unstructured":"Aguiar M, al.:(2015) Dengue transmission during the 2014 fifa world cup in brazil. The Lancet Infectious Diseases 15 https:\/\/doi.org\/10.1016\/S1473-3099(15)00073-0","DOI":"10.1016\/S1473-3099(15)00073-0"},{"key":"2341_CR4","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/j.plrev.2023.05.003","volume":"46","author":"M Aguiar","year":"2023","unstructured":"Aguiar M, Anam V, Blyuss KB, Estadilla CDS, Guerrero BV, Knopoff D, Kooi BW, Mateus L, Srivastav AK, Steindorf V, Stollenwerk N (2023) Prescriptive, descriptive or predictive models: What approach should be taken when empirical data is limited? Reply to comments on \u201cMathematical models for Dengue fever epidemiology: A 10-year systematic review\u2019\u2019. Phys Life Rev 46:56\u201364. https:\/\/doi.org\/10.1016\/j.plrev.2023.05.003","journal-title":"Phys Life Rev"},{"key":"2341_CR5","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.plrev.2022.02.001","volume":"40","author":"M Aguiar","year":"2022","unstructured":"Aguiar M, Anam V, Blyuss KB, Estadilla CDS, Guerrero BV, Knopoff D, Kooi BW, Srivastav AK, Steindorf V, Stollenwerk N (2022) Mathematical models for dengue fever epidemiology: A 10-year systematic review. Phys Life Rev 40:65\u201392. https:\/\/doi.org\/10.1016\/j.plrev.2022.02.001","journal-title":"Phys Life Rev"},{"key":"2341_CR6","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1016\/j.jtbi.2011.08.043","volume":"289","author":"M Aguiar","year":"2011","unstructured":"Aguiar M, Ballesteros S, Kooi BW, Stollenwerk N (2011) The role of seasonality and import in a minimalistic multi-strain dengue model capturing differences between primary and secondary infections: Complex dynamics and its implications for data analysis. J Theor Biol 289:181\u2013196. https:\/\/doi.org\/10.1016\/j.jtbi.2011.08.043","journal-title":"J Theor Biol"},{"key":"2341_CR7","doi-asserted-by":"publisher","unstructured":"Aguiar M, Halstead SB, Stollenwerk N (2017) Consider stopping dengvaxia administration without immunological screening. Expert Rev Vaccines 16(4):301\u2013302. https:\/\/doi.org\/10.1080\/14760584.2017.1276831","DOI":"10.1080\/14760584.2017.1276831"},{"issue":"3","key":"2341_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1051\/mmnp\/20127301","volume":"7","author":"M Aguiar","year":"2012","unstructured":"Aguiar M, Kooi BW, Martins J, Stollenwerk N (2012) Scaling of stochasticity in dengue hemorrhagic fever epidemics. Math Model Nat Phenom 7(3):1\u201311. https:\/\/doi.org\/10.1051\/mmnp\/20127301","journal-title":"Math Model Nat Phenom"},{"key":"2341_CR9","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.ecocom.2012.09.001","volume":"16","author":"M Aguiar","year":"2013","unstructured":"Aguiar M, Kooi BW, Rocha F, Ghaffari P, Stollenwerk N (2013) How much complexity is needed to describe the fluctuations observed in dengue hemorrhagic fever incidence data? Ecol Complex 16:31\u201340. https:\/\/doi.org\/10.1016\/j.ecocom.2012.09.001","journal-title":"Ecol Complex"},{"issue":"4","key":"2341_CR10","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1051\/mmnp:2008070","volume":"3","author":"M Aguiar","year":"2008","unstructured":"Aguiar M, Kooi B, Stollenwerk N (2008) Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions. Mathematical Modelling of Natural Phenomena 3(4):48\u201370. https:\/\/doi.org\/10.1051\/mmnp:2008070","journal-title":"Mathematical Modelling of Natural Phenomena"},{"key":"2341_CR11","doi-asserted-by":"publisher","unstructured":"Aguiar M, Rocha F, Pessanha J, al.: (2015) Carnival or football, is there a real risk for acquiring dengue fever in brazil during holidays seasons? Sci Rep (5) https:\/\/doi.org\/10.1038\/srep08462","DOI":"10.1038\/srep08462"},{"key":"2341_CR12","doi-asserted-by":"publisher","first-page":"13621","DOI":"10.1007\/s11071-024-09710-9","volume":"112","author":"M Aguiar","year":"2024","unstructured":"Aguiar M, Steindorf V, Srivastav A, Stollenwerk N, Kooi BW (2024) Bifurcation analysis of a two infection sir-sir epidemic model with temporary immunity and disease enhancement. Nonlinear Dyn 112:13621\u201313639. https:\/\/doi.org\/10.1007\/s11071-024-09710-9","journal-title":"Nonlinear Dyn"},{"key":"2341_CR13","doi-asserted-by":"publisher","unstructured":"Aguiar M, Stollenwerk N (2017) Dengvaxia efficacy dependency on serostatus: A closer look at more recent data. Clin Infect Dis 66(4):641\u2013642. https:\/\/doi.org\/10.1093\/cid\/cix882. https:\/\/academic.oup.com\/cid\/article-pdf\/66\/4\/641\/23726074\/cix882.pdf","DOI":"10.1093\/cid\/cix882"},{"issue":"1","key":"2341_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5614\/cbms.2017.1.1.1","volume":"1","author":"M Aguiar","year":"2017","unstructured":"Aguiar M, Stollenwerk N (2017) Mathematical models of dengue fever epidemiology: multi-strain dynamics, immunological aspects associated to disease severity and vaccines. Commun Biomath Sci 1(1):1\u201312. https:\/\/doi.org\/10.5614\/cbms.2017.1.1.1","journal-title":"Commun Biomath Sci"},{"issue":"12","key":"2341_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pntd.0005179","volume":"10","author":"M Aguiar","year":"2016","unstructured":"Aguiar M, Stollenwerk N, Halstead SB (2016) The impact of the newly licensed dengue vaccine in endemic countries. PLoS Negl Trop Dis 10(12):1\u201323. https:\/\/doi.org\/10.1371\/journal.pntd.0005179","journal-title":"PLoS Negl Trop Dis"},{"key":"2341_CR16","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.amc.2018.12.022","volume":"349","author":"N Anggriani","year":"2019","unstructured":"Anggriani N, Tasman H, Ndii MZ, Supriatna AK, Soewono E, Siregar E (2019) The effect of reinfection with the same serotype on dengue transmission dynamics. Appl Math Comput 349:62\u201380. https:\/\/doi.org\/10.1016\/j.amc.2018.12.022","journal-title":"Appl Math Comput"},{"key":"2341_CR17","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1007\/s00285-006-0015-0","volume":"53","author":"N Baca\u00eb","year":"2006","unstructured":"Baca\u00eb N, Guernaoui S (2006) The case of cutaneous leihmananiasis in chichaoua, morocco. J Math Biol 53:421\u2013436. https:\/\/doi.org\/10.1007\/s00285-006-0015-0","journal-title":"J Math Biol"},{"key":"2341_CR18","doi-asserted-by":"publisher","unstructured":"Bhatt S, Gething P, Brady O, al.: (2013) The global distribution and burden of dengue. Nature 496, 504\u2013507 https:\/\/doi.org\/10.1038\/nature12060","DOI":"10.1038\/nature12060"},{"issue":"1","key":"2341_CR19","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s00284-020-02284-w448","volume":"78","author":"P Bhatt","year":"2011","unstructured":"Bhatt P, Sabeena SP, Varma M, Arunkumar G (2011) Current understanding of the pathogenesis of dengue virus infection. Curr Microbiol 78(1):17\u201332. https:\/\/doi.org\/10.1007\/s00284-020-02284-w448","journal-title":"Curr Microbiol"},{"issue":"2","key":"2341_CR20","doi-asserted-by":"publisher","first-page":"361","DOI":"10.3934\/mbe.2004.1.361","volume":"1","author":"C Castillo-Chavez","year":"2004","unstructured":"Castillo-Chavez C, Song B (2004) Dynamical Models of Tuberculosis and Their Applications. Math Biosci Eng 1(2):361\u2013404. https:\/\/doi.org\/10.3934\/mbe.2004.1.361","journal-title":"Math Biosci Eng"},{"key":"2341_CR21","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1016\/j.ecocom.2014.11.005","volume":"21","author":"S Chakraborty","year":"2015","unstructured":"Chakraborty S, Kooi BW, Biswas B, Chattopadhyay J (2015) Revealing the role of predator interference in a predator-prey system with disease in prey population. Ecol Complex 21:100\u2013111. https:\/\/doi.org\/10.1016\/j.ecocom.2014.11.005","journal-title":"Ecol Complex"},{"key":"2341_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mbs.2019.01.010","volume":"310","author":"C Champagne","year":"2019","unstructured":"Champagne C, Cazelles B (2019) Comparison of stochastic and deterministic frameworks in dengue modelling. Math Biosci 310:1\u201312. https:\/\/doi.org\/10.1016\/j.mbs.2019.01.010","journal-title":"Math Biosci"},{"key":"2341_CR23","unstructured":"Doedel EJ, Oldeman B (2012) Auto 07p: Continuation and bifurcation software for ordinary differential equations. Technical report, Concordia University, Montreal, Canada"},{"key":"2341_CR24","doi-asserted-by":"publisher","unstructured":"Driessche P, Watmough J (2008) Further notes on the basic reproduction number. In: Brauer, F., Driessche, P., Wu, J. (eds.) Mathematical Epidemiology, pp. 159\u2013178. Springer, Berlin, Heidelberg . https:\/\/doi.org\/10.1007\/978-3-540-78911-6_6","DOI":"10.1007\/978-3-540-78911-6_6"},{"key":"2341_CR25","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1371\/journal.ppat.1011223","volume":"46","author":"L Esteva","year":"2003","unstructured":"Esteva L, Vargas C (2003) Coexistence of different serotypes of dengue virus. J Math Biol 46:31\u201347. https:\/\/doi.org\/10.1371\/journal.ppat.1011223","journal-title":"J Math Biol"},{"key":"2341_CR26","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s002850050064","volume":"35","author":"Z Feng","year":"1997","unstructured":"Feng Z, Velasco-Hern\u00e1ndez J (1997) Competitive exclusion in a vector-host model for the dengue fever. J Math Biol 35:523\u2013544. https:\/\/doi.org\/10.1007\/s002850050064","journal-title":"J Math Biol"},{"issue":"2","key":"2341_CR27","doi-asserted-by":"publisher","first-page":"790","DOI":"10.1073\/pnas.96.2.790","volume":"96","author":"N Ferguson","year":"1999","unstructured":"Ferguson N, Anderson R, Gupta S (1999) The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens. Proc Natl Acad Sci 96(2):790\u2013794. https:\/\/doi.org\/10.1073\/pnas.96.2.790","journal-title":"Proc Natl Acad Sci"},{"key":"2341_CR28","unstructured":"Fischer DB, Halstead SB (1970) Observations related to pathogenesis of dengue hemorrhagic fever. V. Examination of agspecific sequential infection rates using a mathematical model. The Yale Journal of Biology and Medicine 42(5), 329\u2013349"},{"key":"2341_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2019.110014","volume":"484","author":"A Glover","year":"2020","unstructured":"Glover A, White A (2020) A vector-host model to assess the impact of superinfection exclusion on vaccination strategies using dengue and yellow fever as case studies. J Theor Biol 484:110014. https:\/\/doi.org\/10.1016\/j.jtbi.2019.110014","journal-title":"J Theor Biol"},{"key":"2341_CR30","doi-asserted-by":"publisher","unstructured":"Guzman MG, Gubler DJ, Izquierdo A, al.: (2016) Dengue infection. Nat Rev Dis Primers. 2(1), 16055 https:\/\/doi.org\/10.1038\/nrdp.2016.55","DOI":"10.1038\/nrdp.2016.55"},{"issue":"12","key":"2341_CR31","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1038\/nrmicro2460","volume":"8","author":"MG Guzman","year":"2010","unstructured":"Guzman MG, Halstead SB, Artsob H, Buchy P, Farrar J, Gubler DJ, Peeling RW (2010) Dengue: a continuing global threat. Nat Rev Microbiol 8(12):7\u201316. https:\/\/doi.org\/10.1038\/nrmicro2460","journal-title":"Nat Rev Microbiol"},{"key":"2341_CR32","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1016\/s0065-3527(03)60011-4","volume":"60","author":"SB Halstead","year":"2003","unstructured":"Halstead SB (2003) Neutralization and antibody-dependent enhancement of dengue viruses advances in virus research. Academic Press 60:421\u2013467. https:\/\/doi.org\/10.1016\/s0065-3527(03)60011-4","journal-title":"Academic Press"},{"key":"2341_CR33","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.jtbi.2012.11.021","volume":"319","author":"K Hu","year":"2013","unstructured":"Hu K, Thoens C, Bianco S, Edlund S, Davis M, Douglas J, Kaufman JH (2013) The effect of antibody-dependent enhancement, cross immunity, and vector population on the dynamics of dengue fever. J Theor Biol 319:62\u201374. https:\/\/doi.org\/10.1016\/j.jtbi.2012.11.021","journal-title":"J Theor Biol"},{"issue":"35","key":"2341_CR34","doi-asserted-by":"publisher","first-page":"5860","DOI":"10.1016\/j.vaccine.2011.06.042","volume":"29","author":"MA Johansson","year":"2011","unstructured":"Johansson MA, Hombach J, Cummings DAT (2011) Models of the impact of dengue vaccines: A review of current research and potential approaches. Vaccine 29(35):5860\u20135868. https:\/\/doi.org\/10.1016\/j.vaccine.2011.06.042","journal-title":"Vaccine"},{"key":"2341_CR35","doi-asserted-by":"publisher","unstructured":"Kaklamanos P, Pugliese A, Sensi M, Sottile S (2024) A geometric analysis of the SIRS model with secondary infections. SIAM Journal on Applied Mathematics 84 https:\/\/doi.org\/10.1137\/23M1565632","DOI":"10.1137\/23M1565632"},{"key":"2341_CR36","doi-asserted-by":"crossref","unstructured":"Kooi BW, Rashkov P, Venturino E (2023) Multi-Strain Host-Vector Dengue Modeling: Dynamics and Control, pp. 110\u2013142. Chapman and Hall\/CRC, New York","DOI":"10.1201\/9781003035992-6"},{"key":"2341_CR37","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1016\/j.cam.2012.08.008","volume":"252","author":"BW Kooi","year":"2013","unstructured":"Kooi BW, Aguiar M, Stollenwerk N (2013) Bifurcation analysis of a family of multi-strain epidemiology models. J Comput Appl Math 252:148\u2013158. https:\/\/doi.org\/10.1016\/j.cam.2012.08.008","journal-title":"J Comput Appl Math"},{"key":"2341_CR38","unstructured":"Kuznetsov Y (1998) Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Springer, New York . https:\/\/books.google.es\/books?id=y0uq4qEa2ZsC"},{"issue":"2","key":"2341_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pntd.0010631","volume":"17","author":"XY Leung","year":"2023","unstructured":"Leung XY, Islam RM, Adhami M, Ilic D, McDonald L, Palawaththa S, Diug B, Munshi SU, Karim MN (2023) A systematic review of dengue outbreak prediction models: Current scenario and future directions. PLoS Negl Trop Dis 17(2):1\u201321. https:\/\/doi.org\/10.1371\/journal.pntd.0010631","journal-title":"PLoS Negl Trop Dis"},{"key":"2341_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.actatropica.2024.107159","volume":"253","author":"K Liu","year":"2024","unstructured":"Liu K, Fang S, Li Q, Lou Y (2024) Effectiveness evaluation of mosquito suppression strategies on dengue transmission under changing temperature and precipitation. Acta Trop 253:107159. https:\/\/doi.org\/10.1016\/j.actatropica.2024.107159","journal-title":"Acta Trop"},{"issue":"10","key":"2341_CR41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pcbi.1003308","volume":"9","author":"J Louren\u00e7o","year":"2013","unstructured":"Louren\u00e7o J, Recker M (2013) Natural, persistent oscillations in a spatial multi-strain disease system with application to dengue. PLoS Comput Biol 9(10):1\u201311. https:\/\/doi.org\/10.1371\/journal.pcbi.1003308","journal-title":"PLoS Comput Biol"},{"key":"2341_CR42","doi-asserted-by":"crossref","unstructured":"Martcheva M (2015) An Introduction to Mathematical Epidemiology. Texts in Applied Mathematics. Springer, New York . https:\/\/books.google.es\/books?id=r4MOrgEACAAJ","DOI":"10.1007\/978-1-4899-7612-3"},{"key":"2341_CR43","doi-asserted-by":"publisher","unstructured":"Messina JP, Brady OJ, Pigott DM, Brownstein JS, Hoen AG, Hay SI (2014) Da global compendium of human dengue virus occurrence. Sci Data 1: 140004 https:\/\/doi.org\/10.1038\/sdata.2014.4","DOI":"10.1038\/sdata.2014.4"},{"issue":"6","key":"2341_CR44","doi-asserted-by":"publisher","first-page":"2238","DOI":"10.1073\/pnas.0709029105","volume":"105","author":"Y Nagao","year":"2008","unstructured":"Nagao Y, Koelle K (2008) Decreases in dengue transmission may act to increase the incidence of dengue hemorrhagic fever. Proc Natl Acad Sci 105(6):2238\u20132243. https:\/\/doi.org\/10.1073\/pnas.0709029105","journal-title":"Proc Natl Acad Sci"},{"key":"2341_CR45","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.mbs.2014.12.011","volume":"262","author":"MZ Ndii","year":"2015","unstructured":"Ndii MZ, Hickson RI, Allingham D, Mercer GN (2015) Modelling the transmission dynamics of dengue in the presence of wolbachia. Math Biosci 262:157\u2013166. https:\/\/doi.org\/10.1016\/j.mbs.2014.12.011","journal-title":"Math Biosci"},{"key":"2341_CR46","unstructured":"Nishiura H (2008) Duration of short-lived cross-protective immunity against a clinical attack of dengue: A preliminary estimate. WHO Regional Office for South-East Asia https:\/\/iris.who.int\/handle\/10665\/170468"},{"key":"2341_CR47","doi-asserted-by":"publisher","unstructured":"Pakaya R, Daniel D, Widayani P, Utarini A (2023) Spatial model of dengue hemorrhagic fever (dhf) risk: scoping review. BMC Public Health 23(2448) https:\/\/doi.org\/10.1186\/s12889-023-17185-3","DOI":"10.1186\/s12889-023-17185-3"},{"issue":"2","key":"2341_CR48","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1016\/j.mbs.2013.10.007","volume":"246","author":"A Pandey","year":"2013","unstructured":"Pandey A, Mubayi A, Medlock J (2013) Comparing vector\u2013host and sir models for dengue transmission. Math Biosci 246(2):252\u2013259. https:\/\/doi.org\/10.1016\/j.mbs.2013.10.007","journal-title":"Math Biosci"},{"key":"2341_CR49","doi-asserted-by":"publisher","unstructured":"Prasad R, Sagar SK, Parveen S, Dohare R (2022) Mathematical modeling in perspective of vector-borne viral infections: a review. Beni-Suef Univ J Basic Appl Sci 11(102) https:\/\/doi.org\/10.1186\/s43088-022-00282-4","DOI":"10.1186\/s43088-022-00282-4"},{"issue":"1","key":"2341_CR50","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1080\/17513758.2020.1864038","volume":"15","author":"P Rashkov","year":"2021","unstructured":"Rashkov P, Kooi BW (2021) Complexity of host-vector dynamics in a two-strain dengue model. J Biol Dyn 15(1):35\u201372. https:\/\/doi.org\/10.1080\/17513758.2020.1864038","journal-title":"J Biol Dyn"},{"issue":"5","key":"2341_CR51","doi-asserted-by":"publisher","first-page":"4314","DOI":"10.3934\/mbe.2019215","volume":"16","author":"P Rashkov","year":"2019","unstructured":"Rashkov P, Venturino E, Aguiar M, Stollenwerk N, Kooi BW (2019) On the role of vector modeling in a minimalistic epidemic model. Math Biosci Eng 16(5):4314\u20134338. https:\/\/doi.org\/10.3934\/mbe.2019215","journal-title":"Math Biosci Eng"},{"issue":"1667","key":"2341_CR52","doi-asserted-by":"publisher","first-page":"2541","DOI":"10.1098\/rspb.2009.0331","volume":"276","author":"M Recker","year":"2009","unstructured":"Recker M, Blyuss KB, Simmons CP, Hien TT, Wills B, Farrar J, Gupta S (2009) Immunological serotype interactions and their effect on the epidemiological pattern of dengue. Proc R Soc B Biol Sci 276(1667):2541\u20132548. https:\/\/doi.org\/10.1098\/rspb.2009.0331","journal-title":"Proc R Soc B Biol Sci"},{"issue":"10","key":"2341_CR53","doi-asserted-by":"publisher","first-page":"2105","DOI":"10.1080\/00207160.2013.783208","volume":"90","author":"F Rocha","year":"2013","unstructured":"Rocha F, Aguiar M, Souza M, Stollenwerk N (2013) Time-scale separation and centre manifold analysis describing vector-borne disease dynamics. Int J Comput Math 90(10):2105\u20132125. https:\/\/doi.org\/10.1080\/00207160.2013.783208","journal-title":"Int J Comput Math"},{"key":"2341_CR54","doi-asserted-by":"publisher","unstructured":"Sangkawibha N, Rojanasuphot S, Ahandrik S, Viriyapongse S, al.: (1984) Risk factors in dengue shock syndrome: a prospective epidemiologic study in rayong, thailand. Am J Epidemiol 120(5), 653\u2013669 https:\/\/doi.org\/10.1093\/oxfordjournals.aje.a113932","DOI":"10.1093\/oxfordjournals.aje.a113932"},{"issue":"6","key":"2341_CR55","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.72.066201","volume":"72","author":"I Schwartz","year":"2005","unstructured":"Schwartz I, Shaw L, Cummings D, Billings L, McCrary M, Burke D (2005) Chaotic desynchronization of multistrain diseases. Phys Rev E 72(6):066201. https:\/\/doi.org\/10.1103\/PhysRevE.72.066201","journal-title":"Phys Rev E"},{"key":"2341_CR56","doi-asserted-by":"publisher","DOI":"10.1016\/j.actatropica.2020.105452","volume":"207","author":"MI Simoy","year":"2020","unstructured":"Simoy MI, Aparicio JP (2020) Ross-macdonald models: Which one should we use? Acta Trop 207:105452. https:\/\/doi.org\/10.1016\/j.actatropica.2020.105452","journal-title":"Acta Trop"},{"key":"2341_CR57","doi-asserted-by":"publisher","first-page":"8064","DOI":"10.3934\/mbe.2021400","volume":"18","author":"B Song","year":"2021","unstructured":"Song B (2021) Basic reinfection number and backward bifurcation. Math Biosci Eng 18:8064\u20138083. https:\/\/doi.org\/10.3934\/mbe.2021400","journal-title":"Math Biosci Eng"},{"key":"2341_CR58","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2023.107663","volume":"128","author":"V Steindorf","year":"2024","unstructured":"Steindorf V, Oliva S, Stollenwerk N, Aguiar M (2024) Symmetry in a multi-strain epidemiological model with distributed delay as a general cross-protection period and disease enhancement factor. Commun Nonlinear Sci Numer Simul 128:107663. https:\/\/doi.org\/10.1016\/j.cnsns.2023.107663","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"3","key":"2341_CR59","doi-asserted-by":"publisher","first-page":"2950","DOI":"10.3934\/mbe.2022136","volume":"19","author":"V Steindorf","year":"2022","unstructured":"Steindorf V, Oliva S, Wu J (2022) Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model. Math Biosci Eng 19(3):2950\u20132984. https:\/\/doi.org\/10.3934\/mbe.2022136","journal-title":"Math Biosci Eng"},{"key":"2341_CR60","doi-asserted-by":"publisher","unstructured":"Steindorf V, Srivastav AK, Stollenwerk N, Kooi BW, Aguiar M (2022) Modeling secondary infections with temporary immunity and disease enhancement factor: Mechanisms for complex dynamics in simple epidemiological models. Chaos, Solitons & Fractals 164, 112709https:\/\/doi.org\/10.1016\/j.chaos.2022.112709","DOI":"10.1016\/j.chaos.2022.112709"},{"issue":"1","key":"2341_CR61","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s12879-024-09995-6","volume":"24","author":"V Steindorf","year":"2024","unstructured":"Steindorf V, Srivastav A, Stollenwerk N, Kooi B, Aguiar M (2024) Beyond the biting - limited impact of explicit mosquito dynamics in dengue models. BMC Infect Dis 24(1):1\u201311. https:\/\/doi.org\/10.1186\/s12879-024-09995-6","journal-title":"BMC Infect Dis"},{"key":"2341_CR62","unstructured":"Stollenwerk N, Aguiar M, Kooi BW (2018) Chaos via torus destruction in population biology and its implications for data analysis. In: Book of Abstract Symposium on Biomathematics 2018, p. 3"},{"key":"2341_CR63","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/j.ecocom.2016.12.009","volume":"30","author":"N Stollenwerk","year":"2017","unstructured":"Stollenwerk N, Sommer PF, Kooi B, Mateus L, Ghaffari P, Aguiar M (2017) Hopf and torus bifurcations, torus destruction and chaos in population biology. Ecol Complex 30:91\u201399. https:\/\/doi.org\/10.1016\/j.ecocom.2016.12.009","journal-title":"Ecol Complex"},{"key":"2341_CR64","unstructured":"Sympy: https:\/\/colab.research.google.com\/drive\/1lfvw0-z-pu199o-fnwltrfsuwvqgw3qx?usp=sharing (2002)"},{"issue":"1\u20132","key":"2341_CR65","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","volume":"180","author":"P Van Den Driessche","year":"2002","unstructured":"Van Den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1\u20132):29\u201348. https:\/\/doi.org\/10.1016\/S0025-5564(02)00108-6","journal-title":"Math Biosci"},{"key":"2341_CR66","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1007\/s10884-008-9111-8","volume":"520","author":"W Wang","year":"2008","unstructured":"Wang W, Zhoa X-Q (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Differential Equations 520:699\u2013717. https:\/\/doi.org\/10.1007\/s10884-008-9111-8","journal-title":"J Differential Equations"},{"issue":"31","key":"2341_CR67","doi-asserted-by":"publisher","first-page":"11802","DOI":"10.1073\/pnas.0602960103","volume":"103","author":"HJ Wearing","year":"2006","unstructured":"Wearing HJ, Rohani P (2006) Ecological and immunological determinants of dengue epidemics. Proc Natl Acad Sci 103(31):11802\u201311807. https:\/\/doi.org\/10.1073\/pnas.0602960103","journal-title":"Proc Natl Acad Sci"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02341-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-026-02341-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02341-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T06:27:33Z","timestamp":1773037653000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-026-02341-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,10]]},"references-count":67,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["2341"],"URL":"https:\/\/doi.org\/10.1007\/s00285-026-02341-1","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,10]]},"assertion":[{"value":"20 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 November 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 January 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare they have no known competing financial interests that could have appeared to influence the results reported in this work.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"26"}}