{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T20:04:07Z","timestamp":1774728247419,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T00:00:00Z","timestamp":1706572800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Migration or dispersal of population plays an important role in disease transmission during an outbreak. In this work, we have proposed an SIRS compartmental epidemic model in order to analyze the system dynamics in a two-patch environment. Both the deterministic and fractional order systems have been considered in order to observe the impact of population dispersal. The following analysis has shown that we can have an infected system even if the basic reproduction number in one patch becomes less than unity. Moreover, higher dispersal towards a patch controls the infection level in the other patch to a greater extent. In the optimal control problem (both integer order and fractional), it is assumed that people\u2019s dispersal rate will depend on the disease prevalence, and as such will be treated as a time-dependent control intervention. The numerical results reveal that there is a higher amount of recovery cases in both patches in the presence of optimal dispersal (both integer order and fractional). Not only that, implementation of people\u2019s awareness reduces the infection level significantly even if people disperse at a comparatively higher rate. In a fractional system, it is observed that there will be a higher amount of recovery cases if the order of derivative is less than unity. The effect of fractional order is omnipotent in achieving a stable situation.<\/jats:p>","DOI":"10.3390\/axioms13020094","type":"journal-article","created":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T12:06:58Z","timestamp":1706616418000},"page":"94","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Analysis of an SIRS Model in Two-Patch Environment in Presence of Optimal Dispersal Strategy"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0690-9058","authenticated-orcid":false,"given":"Sangeeta","family":"Saha","sequence":"first","affiliation":[{"name":"MS2 Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6512-9700","authenticated-orcid":false,"given":"Meghadri","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2956-6265","authenticated-orcid":false,"given":"Guruprasad","family":"Samanta","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1098\/rspa.1927.0118","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_2","unstructured":"Wang, W. (2007). Mathematics for Life Science and Medicine, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1007\/s11538-015-0113-5","article-title":"SIS and SIR epidemic models under virtual dispersal","volume":"77","author":"Bichara","year":"2015","journal-title":"Bull. Math. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.chaos.2015.11.018","article-title":"Optimal control and stability analysis of an epidemic model with population dispersal","volume":"83","author":"Jana","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hu, L., Wang, S., Zheng, T., Hu, H., Kang, Y., Nie, L.F., and Teng, Z. (2022). The Effects of Migration and Limited Medical Resources of the Transmission of SARS-CoV-2 Model with Two Patches. Bull. Math. Biol., 84.","DOI":"10.1007\/s11538-022-01010-w"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.jmaa.2005.01.034","article-title":"The effect of population dispersal on the spread of a disease","volume":"308","author":"Jin","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Abhishek, V., and Srivastava, V. (2023). SIS Epidemic Spreading under Multi-layer Population Dispersal in Patchy Environments. IEEE Trans. Control. Netw. Syst., 1\u201312.","DOI":"10.1109\/TCNS.2023.3272839"},{"key":"ref_8","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1142\/S179304802050006X","article-title":"Optimal Control of Fractional Order COVID-19 Epidemic Spreading in Japan and India 2020","volume":"15","author":"Das","year":"2020","journal-title":"Biophys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s10957-012-0233-0","article-title":"The Necessary Conditions of Fractional Optimal Control in the Sense of Caputo","volume":"156","author":"Guo","year":"2013","journal-title":"J. Optim. Theory Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s11071-004-3764-6","article-title":"A general formulation and solution scheme for fractional optimal control problems","volume":"38","author":"Agarwal","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Das, M., Samanta, G., and De la Sen, M. (2021). A Fractional Ordered COVID-19 Model Incorporating Comorbidity and Vaccination. Mathematics, 9.","DOI":"10.3390\/math9212806"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Das, M., Samanta, G., and De la Sen, M. (2022). A Fractional Order Model to Study the Effectiveness of Government Measures and Public Behaviours in COVID-19 Pandemic. Mathematics, 10.","DOI":"10.3390\/math10163020"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jtbi.2012.02.010","article-title":"Assessing the role of spatial heterogeneity and human movement in malaria dynamics and control","volume":"303","author":"Prosper","year":"2012","journal-title":"J. Theor. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.mbs.2018.07.004","article-title":"The distribution of the time taken for an epidemic to spread between two communities","volume":"303","author":"Yan","year":"2018","journal-title":"Math. Biosci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s11071-020-05896-w","article-title":"Epidemic model of COVID-19 outbreak by inducing behavioural response in population","volume":"102","author":"Saha","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.matcom.2022.04.025","article-title":"Impact of optimal vaccination and social distancing on COVID-19 pandemic","volume":"200","author":"Saha","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/978-3-031-17778-1_10","article-title":"Analysis of a COVID-19 Model Implementing Social Distancingas an Optimal Control Strategy","volume":"Volume 14","author":"Rezaei","year":"2023","journal-title":"Integrated Science of Global Epidemics. Integrated Science"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1460","DOI":"10.1016\/j.jmaa.2006.07.057","article-title":"Global dynamics of SIS models with transport-related infection","volume":"329","author":"Takeuchi","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","first-page":"8458","article-title":"Global dynamics of a two-patch SIS model with infection during transport","volume":"217","author":"Wang","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/S0022-247X(03)00428-1","article-title":"Threshold of disease transmission in a patch environment","volume":"285","author":"Wang","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.mbs.2002.11.001","article-title":"An epidemic model in a patchy environment","volume":"190","author":"Wang","year":"2004","journal-title":"Math. Biosci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.1137\/S0036139903431245","article-title":"An age-structured epidemic model in a patchy environment","volume":"65","author":"Wang","year":"2005","journal-title":"SIAM J. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S1793048018500017","article-title":"Mathematical analysis of an epidemic system in the presence of optimal control and population dispersal","volume":"13","author":"Nandi","year":"2018","journal-title":"Biophys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/j.jtbi.2007.03.032","article-title":"An SEIS epidemic model with transport-related infection","volume":"247","author":"Wan","year":"2007","journal-title":"J. Theor. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1080\/08898480306720","article-title":"A multi-city epidemic model","volume":"10","author":"Arino","year":"2003","journal-title":"Math. Popul. Stud."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jtbi.2011.06.025","article-title":"Dynamics of an SIQS epidemic model with transport-related infection and exit\u2013entry screenings","volume":"285","author":"Liu","year":"2011","journal-title":"J. Theor. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.apm.2017.03.004","article-title":"Adaptive dispersal effect on the spread of a disease in a patchy environment","volume":"47","author":"Cheng","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_29","first-page":"764278","article-title":"Transmission dynamics of a two-city SIR epidemic model with transport-related infections","volume":"2014","author":"Chen","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Watmough","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_31","unstructured":"Arriola, L., and Hyman, J. (2005). Lecture Notes, Forward and Adjoint Sensitivity Analysis: With Applications in Dynamical Systems, Linear Algebra and Optimisation. Math. Theor. Biol. Inst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2281","DOI":"10.1080\/00207160802624331","article-title":"On linear stability of predictor-corrector algorithms for fractional differential equations","volume":"87","author":"Garrappa","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s00285-014-0761-3","article-title":"The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population","volume":"70","author":"Kassa","year":"2015","journal-title":"J. Math. Biol."},{"key":"ref_34","first-page":"1","article-title":"Optimal control of an epidemic through educational campaigns","volume":"2006","author":"Castilho","year":"2006","journal-title":"Electron. J. Differ. Equations (EJDE)"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"469","DOI":"10.3934\/mbe.2009.6.469","article-title":"Optimal control applied to vaccination and treatment strategies for various epidemiological models","volume":"6","author":"Gaff","year":"2009","journal-title":"Math. Biosci. Eng."},{"key":"ref_36","first-page":"194","article-title":"Optimal control of vaccination and treatment for an SIR epidemiological model","volume":"8","author":"Yusuf","year":"2012","journal-title":"World J. Model. Simul."},{"key":"ref_37","unstructured":"Kirk, D.E. (2012). Optimal Control Theory: An Introduction, Dover Publications. Dover Books on Electrical Engineering."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Kheiri, H., and Jafari, M. (2018). Optimal control of a fractional-order model for the HIV\/AIDS epidemic. Int. J. Biomath., 11.","DOI":"10.1142\/S1793524518500869"},{"key":"ref_39","unstructured":"Coddington, A., and Levinson, N. (1955). Theory of Ordinary Differential Equations, Tata McGraw-Hill Companies."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Fleming, W., and Rishel, R. (1975). Deterministic and Stochastic Optimal Control, Springer.","DOI":"10.1007\/978-1-4612-6380-7"},{"key":"ref_41","unstructured":"Pontryagin, L.S. (1987). Mathematical Theory of Optimal Processes, CRC Press."},{"key":"ref_42","first-page":"1","article-title":"Laplace transform of fractional order differential equations","volume":"2015","author":"Liang","year":"2015","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2433","DOI":"10.1007\/s11071-011-0157-5","article-title":"Stability analysis of Caputo fractional-order non linear system revisited","volume":"67","author":"Delavari","year":"2012","journal-title":"Nonlinear Dyn."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/94\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:51:55Z","timestamp":1760104315000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/94"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,30]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["axioms13020094"],"URL":"https:\/\/doi.org\/10.3390\/axioms13020094","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,30]]}}}