{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:01:21Z","timestamp":1774627281763,"version":"3.50.1"},"reference-count":66,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T00:00:00Z","timestamp":1771891200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T00:00:00Z","timestamp":1771891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s12190-026-02790-3","type":"journal-article","created":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T12:47:19Z","timestamp":1771937239000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamical analysis of an infectious disease model with diffusion, distributed delays and multiple transmission routes"],"prefix":"10.1007","volume":"72","author":[{"given":"Chong","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhijian","family":"Ye","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yinggao","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,24]]},"reference":[{"issue":"1\u20133","key":"2790_CR1","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1016\/j.cnsns.2014.08.036","volume":"22","author":"K. Manna","year":"2015","unstructured":"Manna, K., Chakrabarty, S.P.: Chronic hepatitis B infection and HBV DNA-containing capsids: modeling and analysis. Commun. Nonlinear Sci. Numer. Simul. 22(1\u20133), 383\u2013395 (2015)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"6","key":"2790_CR2","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s00285-024-02096-7","volume":"88","author":"S. Foko","year":"2024","unstructured":"Foko, S.: Dynamical analysis of a general delayed HBV infection model with capsids and adaptive immune response in presence of exposed infected hepatocytes. J. Math. Biol. 88(6), 75 (2024)","journal-title":"J. Math. Biol."},{"issue":"21\u201322","key":"2790_CR3","doi-asserted-by":"publisher","first-page":"5047","DOI":"10.1016\/j.apm.2014.03.010","volume":"38","author":"N. Bairagi","year":"2014","unstructured":"Bairagi, N., Adak, D.: Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay. Appl. Math. Modell. 38(21\u201322), 5047\u20135066 (2014)","journal-title":"Appl. Math. Modell."},{"key":"2790_CR4","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1016\/j.matcom.2021.11.015","volume":"194","author":"T.K. Gharahasanlou","year":"2022","unstructured":"Gharahasanlou, T.K., Roomi, V., Hemmatzadeh, Z.: Global stability analysis of viral infection model with logistic growth rate, general incidence function and cellular immunity. Math. Comput. Simul. 194, 64\u201379 (2022)","journal-title":"Math. Comput. Simul."},{"key":"2790_CR5","doi-asserted-by":"crossref","unstructured":"Guo, K., Ma, W.: Rich dynamics of a hepatitis C virus infection model with logistic proliferation and time delays. Stud. Appl. Math. 154(1), e12781 (2025)","DOI":"10.1111\/sapm.12781"},{"key":"2790_CR6","doi-asserted-by":"crossref","unstructured":"Nowak, M.A., Bangham, C.R.M.: Population dynamics of immune responses to persistent viruses. Science 272(5258), 74\u00a879 (1996)","DOI":"10.1126\/science.272.5258.74"},{"issue":"14","key":"2790_CR7","doi-asserted-by":"publisher","first-page":"1650234","DOI":"10.1142\/S0218127416502345","volume":"26","author":"T. Guo","year":"2016","unstructured":"Guo, T., Liu, H., Xu, C., Yan, F.: Dynamics of a delayed HIV-1 infection model with saturation incidence rate and CTL immune response. Int. J. Bifurcation Chaos. 26(14), 1650234 (2016)","journal-title":"Int. J. Bifurcation Chaos"},{"key":"2790_CR8","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.nonrwa.2016.08.001","volume":"34","author":"J. Wang","year":"2017","unstructured":"Wang, J., Lang, J., Zou, X.: Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission. Nonlinear Anal.: Real World Appl. 34, 75\u201396 (2017)","journal-title":"Nonlinear Anal.: Real World Appl."},{"issue":"2","key":"2790_CR9","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s11424-017-6038-3","volume":"31","author":"Y. Liu","year":"2018","unstructured":"Liu, Y., Wu, C.: Global dynamics for an HIV infection model with Crowley-Martin functional response and two distributed delays. J. Syst. Sci. Complexity. 31(2), 385\u2013395 (2018)","journal-title":"J. Syst. Sci. Complexity"},{"key":"2790_CR10","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1016\/j.physa.2018.09.087","volume":"514","author":"H. Liu","year":"2019","unstructured":"Liu, H., Zhang, J.F.: Dynamics of two time delays differential equation model to HIV latent infection. Physica A: Stat. Mech. its Appl. 514, 384\u2013395 (2019)","journal-title":"Physica A: Stat. Mech. Its Appl."},{"issue":"1","key":"2790_CR11","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/s12190-018-1213-9","volume":"60","author":"H. Miao","year":"2019","unstructured":"Miao, H., Kang, C.: Stability and Hopf bifurcation analysis for an HIV infection model with beddington-DeAngelis incidence and two delays. J. Appl. Math. Comput. 60(1), 265\u2013290 (2019)","journal-title":"J. Appl. Math. Comput."},{"key":"2790_CR12","doi-asserted-by":"crossref","unstructured":"Yang, Y., Dong, Y., Takeuchi, Y.: Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete Contin. Dyn. Syst.-B 24(2) (2019)","DOI":"10.3934\/dcdsb.2018207"},{"issue":"1","key":"2790_CR13","doi-asserted-by":"publisher","first-page":"57","DOI":"10.3934\/mbe.2021003","volume":"18","author":"J. Ren","year":"2021","unstructured":"Ren, J., Xu, R., Li, L.: Global stability of an HIV infection model with saturated CTL immune response and intracellular delay. Math. Biosci. Eng. 18(1), 57\u201368 (2021)","journal-title":"Math. Biosci. Eng."},{"issue":"2","key":"2790_CR14","doi-asserted-by":"publisher","first-page":"411","DOI":"10.11650\/tjm\/211102","volume":"26","author":"N. Akbari","year":"2022","unstructured":"Akbari, N., Asheghi, R., Nasirian, M.: Stability and dynamic of HIV-1 mathematical model with logistic target cell growth, treatment rate, cure rate and cell-to-cell spread. Taiwan. J. Math. 26(2), 411\u2013441 (2022)","journal-title":"Taiwan. J. Math."},{"key":"2790_CR15","doi-asserted-by":"publisher","first-page":"103618","DOI":"10.1016\/j.nonrwa.2022.103618","volume":"67","author":"R. Xu","year":"2022","unstructured":"Xu, R., Song, C.: Dynamics of an HIV infection model with virus diffusion and latently infected cell activation. Nonlinear Anal.: Real World Appl. 67, 103618 (2022)","journal-title":"Nonlinear Anal.: Real World Appl."},{"issue":"1","key":"2790_CR16","first-page":"1","volume":"28","author":"L. Zhang","year":"2023","unstructured":"Zhang, L., Xu, R.: Dynamics analysis of an HIV infection model with latent reservoir, delayed CTL immune response and immune impairment. Nonlinear Anal.: Modell. Control. 28(1), 1\u201319 (2023)","journal-title":"Nonlinear Anal.: Modell. Control."},{"issue":"9","key":"2790_CR17","doi-asserted-by":"publisher","first-page":"2356","DOI":"10.3934\/dcdss.2024066","volume":"18","author":"P. Murugan","year":"2025","unstructured":"Murugan, P., Mani, P.: Threshold dynamics of time-delay in HIV infection model with immune impairment. Discrete Contin. Dyn. Syst.-S. 18(9), 2356\u20132378 (2025)","journal-title":"Discrete Contin. Dyn. Syst.-S"},{"issue":"2261","key":"2790_CR18","doi-asserted-by":"publisher","first-page":"20220059","DOI":"10.1098\/rspa.2022.0059","volume":"478","author":"J. Fan","year":"2022","unstructured":"Fan, J., Yin, Q., Xia, C., Perc, M.: Epidemics on multilayer simplicial complexes. Proc. the R. Soc. A. 478(2261), 20220059 (2022)","journal-title":"Proc. The R. Soc. A"},{"issue":"3","key":"2790_CR19","doi-asserted-by":"publisher","first-page":"4267","DOI":"10.1109\/TCSS.2024.3351173","volume":"11","author":"J. Chen","year":"2024","unstructured":"Chen, J., Xia, C., Perc, M.: The SIQRS propagation model with quarantine on simplicial complexes. IEEE Trans. Comput. Soc. Syst. 11(3), 4267\u20134278 (2024). https:\/\/www.who.int\/news-room\/fact-sheets\/detail\/hiv-aids","journal-title":"IEEE Trans. Comput. Soc. Syst."},{"key":"2790_CR20","unstructured":"https:\/\/www.who.int\/news-room\/fact-sheets\/detail\/hiv-aids"},{"issue":"5","key":"2790_CR21","doi-asserted-by":"publisher","first-page":"2767","DOI":"10.3934\/dcdsb.2021159","volume":"27","author":"Z. Liu","year":"2022","unstructured":"Liu, Z., Wang, L., Tan, R.: Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response. Discrete Contin. Dyn. Syst.-B. 27(5), 2767\u20132790 (2022)","journal-title":"Discrete Contin. Dyn. Syst.-B"},{"issue":"4","key":"2790_CR22","doi-asserted-by":"publisher","first-page":"2365","DOI":"10.1007\/s12190-021-01621-x","volume":"68","author":"Y. Yang","year":"2022","unstructured":"Yang, Y., Xu, R.: Mathematical analysis of a delayed HIV infection model with saturated CTL immune response and immune impairment. J. Appl. Math. Comput. 68(4), 2365\u20132380 (2022)","journal-title":"J. Appl. Math. Comput."},{"key":"2790_CR23","doi-asserted-by":"publisher","first-page":"111733","DOI":"10.1016\/j.chaos.2021.111733","volume":"155","author":"X. Jia","year":"2022","unstructured":"Jia, X., Xu, R.: Global dynamics of a delayed HTLV-I infection model with beddington-DeAngelis incidence and immune impairment. Chaos, Solitons Fractals. 155, 111733 (2022)","journal-title":"Chaos, Solitons Fractals"},{"issue":"7484","key":"2790_CR24","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1038\/505492a","volume":"505","author":"A.L. Cox","year":"2014","unstructured":"Cox, A.L., Siliciano, R.F.: Not-so-innocent bystanders. Nature. 505(7484), 492\u2013493 (2014)","journal-title":"Nature"},{"issue":"7484","key":"2790_CR25","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1038\/nature12940","volume":"505","author":"G. Doitsh","year":"2014","unstructured":"Doitsh, G., Galloway, N.L., Geng, X., Yang, Z., Monroe, K.M., Zepeda, O., Greene, W.C.: Cell death by pyroptosis drives CD4 T-cell depletion in HIV-1 infection. Nature. 505(7484), 509\u2013514 (2014)","journal-title":"Nature"},{"key":"2790_CR26","doi-asserted-by":"crossref","unstructured":"Wang, S., Hottz, P., Schechter, M., Rong, L.: Modeling the slow CD4+ T cell decline in HIV-infected individuals. PLoS Comput. Biol. 11(12), e1004665 (2015)","DOI":"10.1371\/journal.pcbi.1004665"},{"key":"2790_CR27","doi-asserted-by":"crossref","unstructured":"Wang, W., Wu, G., Fan, X.: Global dynamics of a novel viral infection model mediated by pattern recognition receptors. Appl. Math. Lett. 109757 (2025)","DOI":"10.1016\/j.aml.2025.109757"},{"key":"2790_CR28","doi-asserted-by":"publisher","first-page":"110695","DOI":"10.1016\/j.chaos.2021.110695","volume":"144","author":"M.M. Manyombe","year":"2021","unstructured":"Manyombe, M.M., Mbang, J., Chendjou, G.: Stability and Hopf bifurcation of a CTL-inclusive HIV-1 infection model with both viral and cellular infections, and three delays. Chaos, Solitons Fractals. 144, 110695 (2021)","journal-title":"Chaos, Solitons Fractals"},{"issue":"1","key":"2790_CR29","doi-asserted-by":"publisher","first-page":"401","DOI":"10.3934\/dcdsb.2012.17.401","volume":"17","author":"Y. Yan","year":"2011","unstructured":"Yan, Y., Wang, W.: Global stability of a five-dimensional model with immune responsesand delay. Discrete Contin. Dyn. Syst.-B. 17(1), 401\u2013416 (2011)","journal-title":"Discrete Contin. Dyn. Syst.-B"},{"key":"2790_CR30","doi-asserted-by":"publisher","first-page":"108631","DOI":"10.1016\/j.aml.2023.108631","volume":"142","author":"J. Lv","year":"2023","unstructured":"Lv, J., Ma, W.: Global asymptotic stability of a delay differential equation model for SARS-CoV-2 virus infection mediated by ACE2 receptor protein. Appl. Math. Lett. 142, 108631 (2023)","journal-title":"Appl. Math. Lett."},{"key":"2790_CR31","doi-asserted-by":"publisher","first-page":"108110","DOI":"10.1016\/j.aml.2022.108110","volume":"132","author":"Y. Jiang","year":"2022","unstructured":"Jiang, Y., Zhang, T.: Global stability of a cytokine-enhanced viral infection model with nonlinear incidence rate and time delays. Appl. Math. Lett. 132, 108110 (2022)","journal-title":"Appl. Math. Lett."},{"key":"2790_CR32","doi-asserted-by":"publisher","first-page":"113357","DOI":"10.1016\/j.chaos.2023.113357","volume":"170","author":"T. Zhang","year":"2023","unstructured":"Zhang, T., Xu, X., Wang, X.: Dynamic analysis of a cytokine-enhanced viral infection model with time delays and CTL immune response. Chaos, Solitons Fractals. 170, 113357 (2023)","journal-title":"Chaos, Solitons Fractals"},{"key":"2790_CR33","doi-asserted-by":"crossref","unstructured":"Ye, Z., Zhou, Y., Zheng, Z., Chen, C.: Stability and Hopf bifurcation of a cytokine-enhanced HIV infection model with antibody immune response delay. Int. J. Biomath. 2450037 (2024)","DOI":"10.1142\/S1793524524500372"},{"issue":"11","key":"2790_CR34","doi-asserted-by":"publisher","first-page":"2450133","DOI":"10.1142\/S0218127424501335","volume":"34","author":"C. Chen","year":"2024","unstructured":"Chen, C., Ye, Z., Zhou, Y.: Stability and Hopf bifurcation of a cytokine-enhanced HIV infection model with saturation incidence and three delays. Int. J. Bifurcation Chaos. 34(11), 2450133 (2024)","journal-title":"Int. J. Bifurcation Chaos"},{"key":"2790_CR35","doi-asserted-by":"crossref","unstructured":"Baishya, C., Premakumari, R.N., Asharani, J.R., Bonyah, E.: Modeling breast cancer dynamics with fractional derivatives: immunotherapy and circuit-based analysis. Sci. Afr. e02842 (2025)","DOI":"10.1016\/j.sciaf.2025.e02842"},{"issue":"1","key":"2790_CR36","doi-asserted-by":"publisher","first-page":"94","DOI":"10.22436\/jmcs.041.01.07","volume":"41","author":"C. Baishya","year":"2025","unstructured":"Baishya, C., George, R., Premakumari, R.N., Rangappa, A.J., Etemad, S., Alkhafaji, A.T.: A prey-predator approach to tumor-immune and cancer treatment: a circuit-based analysis with non-local derivatives. J. Math. Comput. Sci. 41(1), 94\u2013113 (2025)","journal-title":"J. Math. Comput. Sci."},{"issue":"1","key":"2790_CR37","doi-asserted-by":"publisher","first-page":"6867","DOI":"10.1038\/s41598-023-33961-y","volume":"13","author":"S.W. Teklu","year":"2023","unstructured":"Teklu, S.W.: Analysis of fractional order model on higher institution students\u00a1\u00af anxiety towards mathematics with optimal control theory. Sci. Rep. 13(1), 6867 (2023)","journal-title":"Sci. Rep."},{"issue":"1","key":"2790_CR38","doi-asserted-by":"publisher","first-page":"13550","DOI":"10.1038\/s41598-023-40745-x","volume":"13","author":"B. Liu","year":"2023","unstructured":"Liu, B., Farid, S., Ullah, S., Altanji, M., Nawaz, R., Wondimagegnhu Teklu, S.: Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach. Sci. Rep. 13(1), 13550 (2023)","journal-title":"Sci. Rep."},{"key":"2790_CR39","doi-asserted-by":"crossref","unstructured":"Teklu, S.W.: Insight into the optimal control strategies on corruption dynamics using fractional order derivatives. Sci. Afr. 23, e02069 (2024)","DOI":"10.1016\/j.sciaf.2024.e02069"},{"issue":"4","key":"2790_CR40","doi-asserted-by":"publisher","first-page":"207","DOI":"10.3390\/fractalfract9040207","volume":"9","author":"M. G\u00fcm\u00fcs","year":"2025","unstructured":"G\u00fcm\u00fcs, M., Teklu, S.W.: Cost-benefit and dynamical Investigation of a fractional-order corruption population dynamical system. Fractal Fract. 9(4), 207 (2025)","journal-title":"Fractal Fract."},{"issue":"5","key":"2790_CR41","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1088\/0951-7715\/25\/5\/1451","volume":"25","author":"R. Peng","year":"2012","unstructured":"Peng, R., Zhao, X.Q.: A reaction-diffusion SIS epidemic model in a time-periodic environment. Nonlinearity. 25(5), 1451 (2012)","journal-title":"Nonlinearity"},{"key":"2790_CR42","doi-asserted-by":"publisher","first-page":"103116","DOI":"10.1016\/j.nonrwa.2020.103116","volume":"55","author":"A.M. Elaiw","year":"2020","unstructured":"Elaiw, A.M., Al Agha, A.D.: Analysis of a delayed and diffusive oncolytic M1 virotherapy model with immune response. Nonlinear Anal.: Real World Appl. 55, 103116 (2020)","journal-title":"Nonlinear Anal.: Real World Appl."},{"key":"2790_CR43","doi-asserted-by":"publisher","first-page":"103763","DOI":"10.1016\/j.nonrwa.2022.103763","volume":"72","author":"W. Wang","year":"2023","unstructured":"Wang, W., Feng, Z.: Global dynamics of a diffusive viral infection model with spatial heterogeneity. Nonlinear Anal.: Real World Appl. 72, 103763 (2023)","journal-title":"Nonlinear Anal.: Real World Appl."},{"issue":"17","key":"2790_CR44","doi-asserted-by":"publisher","first-page":"15656","DOI":"10.1002\/mma.70040","volume":"48","author":"W. Wang","year":"2025","unstructured":"Wang, W., Wang, X., Fan, X.: On the global attractivity of a diffusive viral infection model with spatial heterogeneity. Math. Methods The Appl. Sci. 48(17), 15656\u201315660 (2025)","journal-title":"Math. Methods The Appl. Sci."},{"issue":"3","key":"2790_CR45","doi-asserted-by":"publisher","first-page":"540","DOI":"10.1007\/s11538-017-0389-8","volume":"80","author":"W. Wang","year":"2018","unstructured":"Wang, W., Zhang, T.: Caspase-1-mediated pyroptosis of the predominance for driving CD4+ T cells death: a nonlocal spatial mathematical model. Bull. Math. Biol. 80(3), 540\u2013582 (2018)","journal-title":"Bull. Math. Biol."},{"key":"2790_CR46","doi-asserted-by":"publisher","first-page":"112950","DOI":"10.1016\/j.chaos.2022.112950","volume":"166","author":"X. Yang","year":"2023","unstructured":"Yang, X., Su, Y., Zhuo, X., Gao, T.: Global analysis for a delayed HCV model with saturation incidence and two target cells. Chaos, Solitons Fractals. 166, 112950 (2023)","journal-title":"Chaos, Solitons Fractals"},{"issue":"9","key":"2790_CR47","doi-asserted-by":"publisher","first-page":"2799","DOI":"10.1016\/j.camwa.2011.03.050","volume":"61","author":"R. Xu","year":"2011","unstructured":"Xu, R.: Global dynamics of an HIV-1 infection model with distributed intracellular delays. Comput. Math. Appl. 61(9), 2799\u20132805 (2011)","journal-title":"Comput. Math. Appl."},{"key":"2790_CR48","doi-asserted-by":"publisher","first-page":"128604","DOI":"10.1016\/j.physa.2023.128604","volume":"616","author":"A.M. Elaiw","year":"2023","unstructured":"Elaiw, A.M., Alsaedi, A.J., Hobiny, A.D., Aly, S.: Stability of a delayed SARS-CoV-2 reactivation model with logistic growth and adaptive immune response. Physica A: Stat. Mech. its Appl. 616, 128604 (2023)","journal-title":"Physica A: Stat. Mech. Its Appl."},{"issue":"1","key":"2790_CR49","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1002\/mma.7775","volume":"45","author":"A.M. Elaiw","year":"2022","unstructured":"Elaiw, A.M., AlShamrani, N.H.: Stability of HIV\/HTLV-I co-infection model with delays. Math. Methods the Appl. Sci. 45(1), 238\u2013300 (2022)","journal-title":"Math. Methods The Appl. Sci."},{"issue":"18","key":"2790_CR50","doi-asserted-by":"publisher","first-page":"6385","DOI":"10.1002\/mma.5747","volume":"42","author":"H. Yan","year":"2019","unstructured":"Yan, H., Xiao, Y., Yan, Q., Liu, X.: Dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, general incidence rate, intracellular delay, and CTL immune responses. Math. Methods the Appl. Sci. 42(18), 6385\u20136406 (2019)","journal-title":"Math. Methods The Appl. Sci."},{"key":"2790_CR51","doi-asserted-by":"publisher","first-page":"107047","DOI":"10.1016\/j.aml.2021.107047","volume":"117","author":"X. Zhou","year":"2021","unstructured":"Zhou, X., Zhang, L., Zheng, T., Li, H.L., Teng, Z.: Global stability for a delayed HIV reactivation model with latent infection and beddington-DeAngelis incidence. Appl. Math. Lett. 117, 107047 (2021)","journal-title":"Appl. Math. Lett."},{"issue":"1","key":"2790_CR52","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.mbs.2007.05.004","volume":"210","author":"K. Wang","year":"2007","unstructured":"Wang, K., Wang, W.: Propagation of HBV with spatial dependence. Math. Biosci. 210(1), 78\u201395 (2007)","journal-title":"Math. Biosci."},{"key":"2790_CR53","unstructured":"Brauner, C.M., Jolly, D., Lorenzi, L., Thiebaut, R.: Heterogeneous viral environment in a HIV spatial model. (2009). arxiv preprint arxiv:0905.2023"},{"issue":"11","key":"2790_CR54","doi-asserted-by":"publisher","first-page":"2312","DOI":"10.1080\/00036811.2014.955797","volume":"93","author":"F.B. Wang","year":"2014","unstructured":"Wang, F.B., Huang, Y., Zou, X.: Global dynamics of a PDE in-host viral model. Applicable Anal. 93(11), 2312\u20132329 (2014)","journal-title":"Applicable Anal."},{"issue":"3","key":"2790_CR55","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s00285-023-01869-w","volume":"86","author":"J. Deng","year":"2023","unstructured":"Deng, J., Shu, H., Wang, L., Wang, X.S.: Viral dynamics with immune responses: effects of distributed delays and Filippov antiretroviral therapy. J. Math. Biol. 86(3), 37 (2023)","journal-title":"J. Math. Biol."},{"key":"2790_CR56","doi-asserted-by":"crossref","unstructured":"Hong, L., Li, J., Rong, L., Wang, X.: Global dynamics of a delayed model with cytokine-enhanced viral infection and cell-to-cell transmission. Aims Math. 9(6) (2024)","DOI":"10.3934\/math.2024788"},{"issue":"10","key":"2790_CR57","first-page":"4223","volume":"23","author":"T. Guo","year":"2018","unstructured":"Guo, T., Liu, H., Xu, C., Yan, F.: Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete Contin. Dyn. Syst.-B. 23(10), 4223\u20134242 (2018)","journal-title":"Discrete Contin. Dyn. Syst.-B"},{"issue":"4","key":"2790_CR58","first-page":"485","volume":"2","author":"S. Ruan","year":"1994","unstructured":"Ruan, S., Wu, J.: Reaction-diffusion equations with infinite delay. Canad. Appl. Math. Quart. 2(4), 485\u2013550 (1994)","journal-title":"Canad. Appl. Math. Quart."},{"key":"2790_CR59","volume-title":"Geometric Theory of Semilinear Parabolic Equations","author":"D. Henry","year":"1993","unstructured":"Henry, D.: Geometric theory of semilinear parabolic equations. Springer-Verlag, New York (1993)"},{"key":"2790_CR60","doi-asserted-by":"crossref","unstructured":"La Salle, J.P.: The stability of dynamical systems. Soc. Ind. Appl. Math. (1976)","DOI":"10.1137\/1.9781611970432"},{"issue":"17","key":"2790_CR61","doi-asserted-by":"publisher","first-page":"11930","DOI":"10.1002\/mma.8547","volume":"45","author":"B. Li","year":"2022","unstructured":"Li, B., Zhang, F., Wang, X.: A delayed diffusive HBV model with nonlinear incidence and CTL immune response. Math. Methods the Appl. Sci. 45(17), 11930\u201311961 (2022)","journal-title":"Math. Methods The Appl. Sci."},{"issue":"7","key":"2790_CR62","doi-asserted-by":"publisher","first-page":"1831","DOI":"10.1007\/s00285-017-1202-x","volume":"76","author":"X. Ren","year":"2018","unstructured":"Ren, X., Tian, Y., Liu, L., Liu, X.: A reaction-diffusion within-host HIV model with cell-to-cell transmission. J. Math. Biol. 76(7), 1831\u20131872 (2018)","journal-title":"J. Math. Biol."},{"key":"2790_CR63","doi-asserted-by":"publisher","first-page":"110422","DOI":"10.1016\/j.chaos.2020.110422","volume":"150","author":"N.H. Alshamrani","year":"2021","unstructured":"Alshamrani, N.H.: Stability of a general adaptive immunity HIV infection model with silent infected cell-to-cell spread. Chaos, Solitons Fractals. 150, 110422 (2021)","journal-title":"Chaos, Solitons Fractals"},{"issue":"1","key":"2790_CR64","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1007\/s40840-018-0699-5","volume":"43","author":"T. Guo","year":"2020","unstructured":"Guo, T., Qiu, Z., Rong, L.: Analysis of an HIV model with immune responses and cell-to-cell transmission. Bull. The Malays. Math. Sci. Soc. 43(1), 581\u2013607 (2020)","journal-title":"Bull. The Malays. Math. Sci. Soc."},{"issue":"2","key":"2790_CR65","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s11538-025-01586-z","volume":"88","author":"Y. Tan","year":"2026","unstructured":"Tan, Y., Liu, S., Cai, Y., Sun, X., Yao, R., He, D., Wang, W.: Stochastic modeling and optimal control of HIV-1 infection dynamics under combination antiretroviral therapy. In: Tan, Y. et al. Bull. Math. Biol. 88(2), 20 (2026)","journal-title":"Tan Et Al. Bull. Math. Biol."},{"issue":"3","key":"2790_CR66","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1007\/s00033-023-02015-8","volume":"74","author":"T. Zheng","year":"2023","unstructured":"Zheng, T., Luo, Y., Teng, Z.: Spatial dynamics of a viral infection model with immune response and nonlinear incidence. Z. f\u00fcr Angew. Math. und Phys. 74(3), 124 (2023)","journal-title":"Z. f\u00fcr Angew. Math. und Phys."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02790-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02790-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02790-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:03:17Z","timestamp":1774623797000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02790-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,24]]},"references-count":66,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2790"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02790-3","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,24]]},"assertion":[{"value":"4 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 February 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 February 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 that theyhave no known competing financial interests or personal relationships thatcould have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interest"}}],"article-number":"139"}}