{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T04:08:56Z","timestamp":1789186136006,"version":"build-2803163510"},"reference-count":29,"publisher":"IMR Press","issue":"6","license":[{"start":{"date-parts":[[2023,6,25]],"date-time":"2023-06-25T00:00:00Z","timestamp":1687651200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Front. Biosci. (Landmark Ed)"],"abstract":"<jats:p>Background: Mathematical models reflecting the epidemiological dynamics of dengue infection have been discovered dating back to 1970. The four serotypes (DENV-1 to DENV-4) that cause dengue fever are antigenically related but different viruses that are transmitted by mosquitoes. It is a significant global public health issue since 2.5 billion individuals are at risk of contracting the virus. Methods: The purpose of this study is to carefully examine the transmission of dengue with a time delay. A dengue transmission dynamic model with two delays, the standard incidence, loss of immunity, recovery from infectiousness, and partial protection of the human population was developed. Results: Both endemic equilibrium and illness-free equilibrium were examined in terms of the stability theory of delay differential equations. As long as the basic reproduction number (R0) is less than unity, the illness-free equilibrium is locally asymptotically stable; however, when R0 exceeds unity, the equilibrium becomes unstable. The existence of Hopf bifurcation with delay as a bifurcation parameter and the conditions for endemic equilibrium stability were examined. To validate the theoretical results, numerical simulations were done. Conclusions: The length of the time delay in the dengue transmission epidemic model has no effect on the stability of the illness-free equilibrium. Regardless, Hopf bifurcation may occur depending on how much the delay impacts the stability of the underlying equilibrium. This mathematical modelling is effective for providing qualitative evaluations for the recovery of a huge population of afflicted community members with a time delay.<\/jats:p>","DOI":"10.31083\/j.fbl2806117","type":"journal-article","created":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T01:24:58Z","timestamp":1687915498000},"source":"Crossref","is-referenced-by-count":13,"title":["Analysis of Dengue Transmission Dynamic Model by Stability and Hopf Bifurcation with Two-Time Delays"],"prefix":"10.31083","volume":"28","author":[{"given":"Prakash Raj","family":"Murugadoss","sequence":"first","affiliation":[{"name":"Department of Mathematics, A.V.V.M. Sri Pushpam College (Affiliated to Bharathidasan University, Tiruchirappalli), Poondi, 613 503 Thanjavur, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Venkatesh","family":"Ambalarajan","sequence":"first","affiliation":[{"name":"Department of Mathematics, A.V.V.M. Sri Pushpam College (Affiliated to Bharathidasan University, Tiruchirappalli), Poondi, 613 503 Thanjavur, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vinoth","family":"Sivakumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, V.S.B. Engineering College, 639111 Karur, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Prasantha Bharathi","family":"Dhandapani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sri Eshwar College of Engineering, 641202 Coimbatore, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dumitru","family":"Baleanu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Cankara University, 06530 Ankara, Turkey"},{"name":"Institute of Space Sciences, Laboratory of Theoretical Physics, R 76900, Magurele-Bucharest, Romania"},{"name":"Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, 11022801 Beirut, Lebanon"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16242","published-online":{"date-parts":[[2023,6,25]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"Pragati D and Irrusappan H. Vector-Borne Diseases in India. 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