{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:50:33Z","timestamp":1760143833143,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,4]],"date-time":"2024-03-04T00:00:00Z","timestamp":1709510400000},"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>The homotopy perturbation method (HPM) is one of the recent fundamental methods for solving differential equations. However, checking the accuracy of this method has been ignored by some authors in the literature. This paper reanalyzes the nonlinear system of ordinary differential equations (ODEs) describing the SIR epidemic model, which has been solved in the literature utilizing the HPM. The main objective of this work is to obtain a highly accurate analytical solution for this model via a direct technique. The proposed technique is mainly based on reducing the given system to a single nonlinear ODE that can be easily solved. Numerical results are conducted to compare our approach with the previous HPM, where the Runge\u2013Kutta numerical method is chosen as a reference solution. The obtained results reveal that the current technique exhibits better accuracy over HPM in the literature. Moreover, some physical properties are introduced and discussed in detail regarding the influence of the transmission rate on the behavior of the SIR model.<\/jats:p>","DOI":"10.3390\/axioms13030167","type":"journal-article","created":{"date-parts":[[2024,3,4]],"date-time":"2024-03-04T05:31:12Z","timestamp":1709530272000},"page":"167","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Accurate Approximations for a Nonlinear SIR System via an Efficient Analytical Approach: Comparative Analysis"],"prefix":"10.3390","volume":"13","author":[{"given":"Mona","family":"Aljoufi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,4]]},"reference":[{"key":"ref_1","unstructured":"Graunt, J. (1662). Natural and Political Observations Made Upon the Bills of Mortality, Tho. Roycroft for John Martin, James Allestry, and Tho. Dicas."},{"key":"ref_2","unstructured":"Bernoulli, D. (1760). Essai d\u2019une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l\u2019inoculation pour la prevenir. Mem. Math. Phys. Acad. Roy. Sci. Paris, 1."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1098\/rspa.1927.0118","article-title":"Contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. Roy. Soc. Lond. A"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2002). Mathematical Biology: I. An Introduction, Springer.","DOI":"10.1007\/b98868"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Brauer, F., van den Driessche, P., and Wu, J. (2008). Lecture Notes in Mathematical Epidemiology, Springer.","DOI":"10.1007\/978-3-540-78911-6"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"De Abajo, J.G. (2020). Simple mathematics on COVID-19 expansion. MedRxiv.","DOI":"10.1101\/2020.03.17.20037663"},{"key":"ref_7","first-page":"675","article-title":"Dynamical Behaviors of Nonlinear Coronavirus (COVID-19) Model with Numerical Studies","volume":"67","author":"Gepreel","year":"2021","journal-title":"Comput. Mater. Contin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"132674","DOI":"10.1016\/j.physd.2020.132674","article-title":"Inversion of a SIR-based model: A critical analysis about the application to COVID-19 epidemic","volume":"413","author":"Comunian","year":"2020","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/j.apm.2020.08.057","article-title":"Analytical features of the SIR model and their applications to COVID-19","volume":"90","author":"Kudryashov","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"105103","DOI":"10.1016\/j.rinp.2021.105103","article-title":"Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study","volume":"33","author":"Zhou","year":"2022","journal-title":"Results Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"110049","DOI":"10.1016\/j.chaos.2020.110049","article-title":"Modeling and forecasting the COVID-19 pandemic in India","volume":"139","author":"Sarkar","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"110064","DOI":"10.1016\/j.chaos.2020.110064","article-title":"Modelling the downhill of the SARS-CoV-2 in Italy and a universal forecast of the epidemic in the world","volume":"139","author":"Martelloni","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"104289","DOI":"10.1016\/j.rinp.2021.104289","article-title":"An improved SIR model describing the epidemic dynamics of the COVID-19 in China","volume":"25","author":"Zhu","year":"2021","journal-title":"Results Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s11071-022-07471-x","article-title":"Study of COVID-19 epidemiological evolution in India with a multi-wave SIR model","volume":"109","author":"Ghosh","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_15","first-page":"98","article-title":"Modeling and forecasting of COVID-19 using a hybrid dynamic model based on SEIRD with ARIMA corrections","volume":"6","author":"Majdalawieh","year":"2021","journal-title":"Infect. Dis. Model."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Margenov, S., Popivanov, N., Ugrinova, I., and Hristov, T. (2022). Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination. Mathematics, 10.","DOI":"10.3390\/math10152570"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.cnsns.2010.03.012","article-title":"A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities","volume":"16","author":"Ebaid","year":"2011","journal-title":"Commun. Nonlin. Sci. Numer. Simul."},{"key":"ref_18","unstructured":"Liao, S. (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press."},{"key":"ref_19","first-page":"205","article-title":"Application of homotopy analysis method (HAM) to the non-linear KdV equations Astha Chauhan and Rajan Arora","volume":"31","author":"Chauhan","year":"2023","journal-title":"Commun. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.joems.2014.06.015","article-title":"On the convergence of Homotopy perturbation method","volume":"23","author":"Ayati","year":"2015","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1177\/0954408914533104","article-title":"Approximate analytical solution of nonlinear systems using homotopy perturbation method","volume":"230","author":"Bayat","year":"2016","journal-title":"Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academy.","DOI":"10.1007\/978-94-015-8289-6"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/167\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:08:57Z","timestamp":1760105337000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/167"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,4]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030167"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030167","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,3,4]]}}}