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The model incorporates a Beddington\u2013DeAngelis functional response to describe the infection incidence, accounting for saturation effects in both target cells and free virus particles. It further includes a cure rate for infected cells, representing the efficacy of antiretroviral therapy or intrinsic immune clearance, and logistic growth for CD4\n                    <jats:inline-formula>\n                      <jats:tex-math>$$^+$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    T-cell populations. A novel contribution is the explicit inclusion of both cellular (cytotoxic T-lymphocytes, CTLs) and humoral (antibody) immune responses. We perform a complete dynamical analysis of the continuous-time system, deriving the basic reproduction number\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal {R}_0$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    as a sharp threshold. We establish the existence and uniqueness of the disease-free and endemic equilibria and analyze their local stability. Furthermore, we prove the global asymptotic stability of the endemic equilibrium when\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal {R}_0&gt; 1$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    using a Lyapunov function. To facilitate numerical investigation, we construct a dynamically consistent nonstandard finite difference (NSFD) discretization that preserves the positivity and stability properties of the continuous model. Numerical simulations validate the theoretical findings and illustrate the distinct roles of the saturation parameters\n                    <jats:inline-formula>\n                      <jats:tex-math>$$m_1$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$m_2$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , as well as the immune response, in modulating infection outcomes. The results highlight how the interplay between viral kinetics and immune effectors can determine disease progression or clearance, providing theoretical insights that could inform therapeutic strategies.\n                  <\/jats:p>","DOI":"10.1038\/s41598-026-47946-0","type":"journal-article","created":{"date-parts":[[2026,5,26]],"date-time":"2026-05-26T09:56:24Z","timestamp":1779789384000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A modified HIV model with Beddington\u2013DeAngelis incidence and cure rate"],"prefix":"10.1038","volume":"16","author":[{"given":"Sarah","family":"Ramadan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sanaa","family":"Salman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ahmed","family":"EL-Sayed","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,26]]},"reference":[{"key":"47946_CR1","unstructured":"UNAIDS. 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