{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T10:56:12Z","timestamp":1764413772366,"version":"3.37.3"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T00:00:00Z","timestamp":1591315200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T00:00:00Z","timestamp":1591315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2020,10]]},"DOI":"10.1007\/s12190-020-01371-2","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T06:03:06Z","timestamp":1591336986000},"page":"615-633","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals"],"prefix":"10.1007","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4556-9466","authenticated-orcid":false,"given":"M. H.","family":"Akrami","sequence":"first","affiliation":[]},{"given":"A.","family":"Atabaigi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,6,5]]},"reference":[{"key":"1371_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physleta.2006.04.087","volume":"358","author":"E Ahmed","year":"2006","unstructured":"Ahmed, E., El-Sayed, A., El-Saka, H.A.A.: On some Routh\u2013Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R\u00f6ssler, Chua and Chen systems. Phys. Lett. A 358, 1\u20134 (2006)","journal-title":"Phys. Lett. A"},{"issue":"1","key":"1371_CR2","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1016\/j.jksus.2018.07.007","volume":"32","author":"S Aman","year":"2020","unstructured":"Aman, S., Al-Mdallal, Q., Khan, I.: Heat transfer and second order slip effect on mhd flow of fractional maxwell fluid in a porous medium. J. King Saud Univ. Sci. 32(1), 450\u2013458 (2020)","journal-title":"J. King Saud Univ. Sci."},{"key":"1371_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3516-1","volume-title":"Mathematical Models in Population Biology and Epidemiology","author":"F Brauer","year":"2001","unstructured":"Brauer, F., Castillo-Ch\u00e1vez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, Berlin (2001)"},{"issue":"1\u20132","key":"1371_CR4","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/0025-5564(78)90006-8","volume":"42","author":"V Capasso","year":"1978","unstructured":"Capasso, V., Serio, G.: A generalization of the Kermack\u2013Mckendrick deterministic epidemic model. Math. Biosci. 42(1\u20132), 43\u201361 (1978)","journal-title":"Math. Biosci."},{"key":"1371_CR5","doi-asserted-by":"publisher","DOI":"10.1142\/7709","volume-title":"Fractional Order Systems: Modeling And Control Applications","author":"R Caponetto","year":"2010","unstructured":"Caponetto, R.: Fractional Order Systems: Modeling And Control Applications, vol. 72. World Scientific, Singapore (2010)"},{"key":"1371_CR6","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.nonrwa.2015.11.001","volume":"30","author":"Y Chena","year":"2016","unstructured":"Chena, Y., Zouc, S., Yang, J.: Global analysis of an sir epidemic model with infection age and saturated incidence. Nonlinear Anal. Real World Appl. 30, 16\u201331 (2016)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"2","key":"1371_CR7","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1145\/779359.779362","volume":"29","author":"A Dhooge","year":"2003","unstructured":"Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: Matcont: a matlab package for numerical bifurcation analysis of odes. ACM Trans. Math. Softw. 29(2), 141\u2013164 (2003)","journal-title":"ACM Trans. Math. Softw."},{"key":"1371_CR8","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.joems.2014.02.012","volume":"23","author":"HAA El-Saka","year":"2015","unstructured":"El-Saka, H.A.A.: Backward bifurcations in fractional-order vaccination models. J. Egypt. Math. Soc. 23, 49\u201355 (2015)","journal-title":"J. Egypt. Math. Soc."},{"issue":"1\u20132","key":"1371_CR9","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s11071-008-9383-x","volume":"56","author":"HA El-Saka","year":"2009","unstructured":"El-Saka, H.A., Ahmed, E., Shehata, M.I., El-Sayed, A.M.A.: On stability, persistence, and Hopf bifurcation in fractional order dynamical systems. Nonlinear Dyn. 56(1\u20132), 121 (2009)","journal-title":"Nonlinear Dyn."},{"issue":"1","key":"1371_CR10","first-page":"1","volume":"1","author":"AMA El-Sayed","year":"2010","unstructured":"El-Sayed, A.M.A.: On the existence and stability of positive solution for a nonlinear fractional-order differential equation and some applications. Alex J Math 1(1), 1\u201310 (2010)","journal-title":"Alex J Math"},{"key":"1371_CR11","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718195","volume-title":"Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students","author":"B Ermentrout","year":"2002","unstructured":"Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, vol. 14. SIAM, New York (2002)"},{"key":"1371_CR12","doi-asserted-by":"publisher","DOI":"10.1142\/3779","volume-title":"Applications of Fractional Calculus in Physics","author":"R Hilfer","year":"2000","unstructured":"Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)"},{"key":"1371_CR13","first-page":"120","volume":"23","author":"XL Hu","year":"2010","unstructured":"Hu, X.L., Sun, F.G., Wang, C.X.: Global analysis of sir epidemic model with the saturated contact rate and vertical transmission. Basic Sci. J. Textile Univ. 23, 120\u2013122 (2010)","journal-title":"Basic Sci. J. Textile Univ."},{"key":"1371_CR14","doi-asserted-by":"publisher","first-page":"5842","DOI":"10.1016\/j.apm.2011.05.027","volume":"35","author":"Z Huang","year":"2011","unstructured":"Huang, Z., Yang, Q., Cao, J.: Stochastic stability and bifurcation for the chronic state in Marchuk\u2019s model with noise. Appl. Math. Model. 35, 5842\u20135855 (2011)","journal-title":"Appl. Math. Model."},{"key":"1371_CR15","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.physa.2015.12.078","volume":"448","author":"J Huo","year":"2016","unstructured":"Huo, J., Zhao, H.: Dynamical analysis of a fractional sir model with birth and death on heterogeneous complex networks. Phys. A Stat. Mech. Its Appl. 448, 41\u201356 (2016)","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"1371_CR16","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s10441-015-9273-9","volume":"64","author":"S Jana","year":"2016","unstructured":"Jana, S., Kumar\u00a0Nandi, S., Kar, T\u00a0.K.: Complex dynamics of an sir epidemic model with saturated incidence rate and treatment. Acta Biotheor. 64, 65\u201384 (2016)","journal-title":"Acta Biotheor."},{"key":"1371_CR17","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.chaos.2017.03.047","volume":"99","author":"J Li","year":"2017","unstructured":"Li, J., Teng, Z., Wang, G., Zhang, L., Hu, C.: Stability and bifurcation analysis of an sir epidemic model with logistic growth and saturated treatment. Chaos Solitons Fract. 99, 63\u201371 (2017)","journal-title":"Chaos Solitons Fract."},{"key":"1371_CR18","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/BF00277162","volume":"25","author":"W Liu","year":"1987","unstructured":"Liu, W., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biosci. 25, 359\u2013380 (1987)","journal-title":"J. Math. Biosci."},{"key":"1371_CR19","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/BF00276956","volume":"23","author":"W Liu","year":"1986","unstructured":"Liu, W., Levin, S., Iwasa, Y.J.: Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models. J. Math. Biol. 23, 187\u2013204 (1986)","journal-title":"J. Math. Biol."},{"key":"1371_CR20","first-page":"963","volume":"2","author":"D Matignon","year":"1996","unstructured":"Matignon, D.: Stability results for fractional differential equations with applications to control processing. Comput. Eng. Syst. Appl. 2, 963\u2013968 (1996)","journal-title":"Comput. Eng. Syst. Appl."},{"key":"1371_CR21","first-page":"1","volume":"160","author":"A Mouaouine","year":"2018","unstructured":"Mouaouine, A., Boukhouima, A., Hattaf, K., Yousfi, N.: A fractional order sir epidemic model with nonlinear incidence rate. Adv. Differ. Equ. 160, 1\u20139 (2018)","journal-title":"Adv. Differ. Equ."},{"issue":"1","key":"1371_CR22","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1006\/jdeq.1993.1044","volume":"103","author":"L Perko","year":"1993","unstructured":"Perko, L.: Rotated vector fields. J. Differ. Equ. 103(1), 127\u2013145 (1993)","journal-title":"J. Differ. Equ."},{"key":"1371_CR23","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)"},{"issue":"2","key":"1371_CR24","doi-asserted-by":"publisher","first-page":"663","DOI":"10.1007\/s40995-018-0519-7","volume":"42","author":"D Rostamy","year":"2018","unstructured":"Rostamy, D., Mottaghi, E.: Forward and backward bifurcation in a fractional-order sir epidemic model with vaccination. Iran. J. Sci. Technol. Trans. A Sci. 42(2), 663\u2013671 (2018)","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"issue":"5","key":"1371_CR25","doi-asserted-by":"publisher","first-page":"1662","DOI":"10.1016\/j.jde.2014.05.030","volume":"257","author":"C Shan","year":"2014","unstructured":"Shan, C., Zhu, H.: Bifurcations and complex dynamics of an sir model with the impact of the number of hospital beds. J. Differ. Equ. 257(5), 1662\u20131688 (2014)","journal-title":"J. Differ. Equ."},{"key":"1371_CR26","first-page":"504","volume":"316","author":"J Singh","year":"2018","unstructured":"Singh, J., Kumar, D., Hammouch, Z., Atangana, A.: A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Appl. Math. Comput. 316, 504\u2013515 (2018)","journal-title":"Appl. Math. Comput."},{"issue":"8","key":"1371_CR27","doi-asserted-by":"publisher","first-page":"1886","DOI":"10.1016\/j.automatica.2009.04.001","volume":"45","author":"MS Tavazoei","year":"2009","unstructured":"Tavazoei, M.S., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45(8), 1886\u20131890 (2009)","journal-title":"Automatica"},{"key":"1371_CR28","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.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of diseases transmission. Math. Biosci. 180, 29\u201348 (2002)","journal-title":"Math. Biosci."},{"key":"1371_CR29","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1016\/j.jmaa.2003.11.043","volume":"291","author":"W Wang","year":"2004","unstructured":"Wang, W., Ruan, S.: Bifurcation in an epidemic model with constant removal rate of the infectives. J. Math. Anal. Appl. 291, 775\u2013793 (2004)","journal-title":"J. Math. Anal. Appl."},{"issue":"13","key":"1371_CR30","doi-asserted-by":"publisher","first-page":"1750209","DOI":"10.1142\/S0218127417502091","volume":"27","author":"Z Wang","year":"2017","unstructured":"Wang, Z., Wang, X., Li, Y., Huang, X.: Stability and hopf bifurcation of fractional-order complex-valued single neuron model with time delay. Int. J. Bifurc. Chaos 27(13), 1750209 (2017)","journal-title":"Int. J. Bifurc. Chaos"},{"key":"1371_CR31","volume-title":"Introduction to Applied Nonlinear Dynamical Systems and Chaos","author":"S Wiggins","year":"2003","unstructured":"Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, Berlin (2003)"},{"key":"1371_CR32","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1016\/j.jmaa.2008.07.042","volume":"348","author":"X Zhang","year":"2008","unstructured":"Zhang, X., Liu, X.: Backward bifurcation of an epidemic model with saturated treatment. J. Math. Anal. Appl. 348, 433\u2013443 (2008)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1371_CR33","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.nonrwa.2011.07.036","volume":"13","author":"L Zhou","year":"2012","unstructured":"Zhou, L., Fan, M.: Dynamics of an sir epidemic model with limited medical resources revisited. Nonlinear Anal. Real World Appl. 13(1), 312\u2013324 (2012)","journal-title":"Nonlinear Anal. Real World Appl."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01371-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-020-01371-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-020-01371-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,4]],"date-time":"2021-06-04T23:17:18Z","timestamp":1622848638000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-020-01371-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,5]]},"references-count":33,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["1371"],"URL":"https:\/\/doi.org\/10.1007\/s12190-020-01371-2","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2020,6,5]]},"assertion":[{"value":"21 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2020","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}