{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T19:37:33Z","timestamp":1776973053789,"version":"3.51.4"},"reference-count":63,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T00:00:00Z","timestamp":1668643200000},"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>In this study, we developed an autonomous non-linear epidemic model for the transmission dynamics of susceptible, vaccinated, infected, and recovered individuals (SVIR model) with non-linear saturation incidence and vaccination rates. The non-linear saturation incidence rate significantly reduces the death ratio of infected individuals by increasing human immunity. We discuss a detailed explanation of the model equilibrium, its basic reproduction number R0, local stability, and global stability. The disease-free equilibrium is observed to be stable if R0&lt;1, while the endemic equilibrium exists and the disease exists permanently in the population if R0&gt;1. To approximate the solution of the model, the well-known Runge\u2013Kutta (RK4) methodology is utilized. The implications of numerous parameters on the population dynamics of susceptible, vaccinated, infected, and recovered individuals are addressed. We discovered that increasing the value of the disease-included death rate \u03c8 has a negative impact on those affected, while it has a positive impact on other populations. Furthermore, the value of interaction between vaccinated and infected \u03bb2 has a decreasing impact on vulnerable and vaccinated people, while increasing in other populations. On the other hand, the model is solved using Euler and Euler-modified techniques, and the results are compared numerically and graphically. The quantitative computations demonstrate that the RK4 method provides very precise solutions compared to the other approaches. The results show that the suggested SVIR model that approximates the solution method is accurate and useful.<\/jats:p>","DOI":"10.3390\/axioms11110651","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T03:27:44Z","timestamp":1668655664000},"page":"651","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2640-2377","authenticated-orcid":false,"given":"Attaullah","family":"Attaullah","sequence":"first","affiliation":[{"name":"Department of Mathematics & Statistics, Bacha Khan University, Charsadda 24461, Pakistan"}]},{"given":"Adil","family":"Khurshaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Swabi, Swabi 23430, Pakistan"}]},{"given":"Zeeshan","family":"Zeeshan","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, Bacha Khan University, Charsadda 24461, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6364-3690","authenticated-orcid":false,"given":"Sultan","family":"Alyobi","sequence":"additional","affiliation":[{"name":"King Abdulaziz University, College of Science & Arts, Department of Mathematics, Rabigh, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3952-4341","authenticated-orcid":false,"given":"Mansour F.","family":"Yassen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Aflaj 11912, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Damietta, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5485-8956","authenticated-orcid":false,"given":"Din","family":"Prathumwan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1097\/QCO.0b013e32814a5729","article-title":"Leptospirosis: Pathogenesis, immunity and diagnosis","volume":"20","author":"Palaniappan","year":"2007","journal-title":"J. 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