{"id":5007,"date":"2026-04-30T08:44:17","date_gmt":"2026-04-30T08:44:17","guid":{"rendered":"https:\/\/journal.ziu-university.net\/?p=5007"},"modified":"2026-04-30T08:44:18","modified_gmt":"2026-04-30T08:44:18","slug":"analyzing-the-mathematical-model-of-breast-cancer-and-identifying-the-most-important-factors-affecting-it","status":"publish","type":"post","link":"https:\/\/journal.ziu.edu.sy\/?p=5007","title":{"rendered":"Analyzing the Mathematical Model of Breast Cancer and Identifying the Most Important Factors Affecting it"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>Abstract:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This study develops and analyzes an advanced mathematical framework for modeling breast cancer dynamics based on a singularly perturbed logistic delay differential equation with multiple discrete time delays. The proposed model captures essential biological processes governing tumor progression, including delayed cellular proliferation, immune response activation, and treatment effects, thereby improving the realism of classical tumor growth models.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using singular perturbation theory, the system is reduced to a quasi-steady-state discrete form, enabling a clear separation between fast transient dynamics and long-term tumor evolution. The analytical investigation focuses on equilibrium points and their local stability through characteristic equations derived from linearization. Stability conditions are rigorously established, and parameter thresholds governing tumor-free and positive equilibrium states are identified.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Furthermore, bifurcation analysis reveals the presence of Hopf bifurcation, indicating the emergence of oscillatory tumor behavior under specific parameter regimes. This highlights the critical role of time delays and treatment timing in shaping tumor dynamics. The discretized version of the model is also examined, confirming consistency between continuous and discrete representations.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To validate the theoretical findings, numerical simulations are conducted, including bifurcation diagrams and Lyapunov characteristic exponent analysis. Results demonstrate complex dynamical behaviors such as stability switching, oscillations, and potential chaotic responses, reflecting clinically observed tumor variability.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Importantly, the model emphasizes that effective cancer control depends not only on treatment intensity but also on its timing relative to tumor growth kinetics. These findings provide valuable insights for optimizing therapeutic strategies and underscore the importance of early intervention. Overall, the proposed framework offers a robust and biologically interpretable tool for studying multiscale tumor dynamics in breast cancer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key word: Breast cancer, Mathematical model, Delay difference equation, Stability analysis, Hopf bifurcation.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u0627\u0644\u0645\u0644\u062e\u0635<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u062a\u064f\u0637\u0648\u0651\u0631 \u0647\u0630\u0647 \u0627\u0644\u062f\u0631\u0627\u0633\u0629 \u0625\u0637\u0627\u0631\u064b\u0627 \u0631\u064a\u0627\u0636\u064a\u064b\u0651\u0627 \u0645\u062a\u0642\u062f\u0645\u064b\u0627 \u0644\u0646\u0645\u0630\u062c\u0629 \u062f\u064a\u0646\u0627\u0645\u064a\u0643\u064a\u0627\u062a \u0633\u0631\u0637\u0627\u0646 \u0627\u0644\u062b\u062f\u064a \u0648\u062a\u062d\u0644\u0651\u0644\u0647\u060c \u0648\u0630\u0644\u0643 \u0627\u0633\u062a\u0646\u0627\u062f\u064b\u0627 \u0625\u0644\u0649 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\u0630\u0644\u0643 \u0627\u0644\u062a\u0643\u0627\u062b\u0631 \u0627\u0644\u062e\u0644\u0648\u064a \u0627\u0644\u0645\u062a\u0623\u062e\u0631\u060c \u0648\u062a\u0641\u0639\u064a\u0644 \u0627\u0644\u0627\u0633\u062a\u062c\u0627\u0628\u0629 \u0627\u0644\u0645\u0646\u0627\u0639\u064a\u0629\u060c \u0648\u062a\u0623\u062b\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0644\u0627\u062c\u060c \u0645\u0645\u0627 \u064a\u062d\u0633\u0651\u0646 \u0648\u0627\u0642\u0639\u064a\u0629 \u0627\u0644\u0646\u0645\u0627\u0630\u062c \u0627\u0644\u0643\u0644\u0627\u0633\u064a\u0643\u064a\u0629 \u0644\u0646\u0645\u0648 \u0627\u0644\u0648\u0631\u0645.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0627\u0636\u0637\u0631\u0627\u0628 \u0627\u0644\u0645\u0641\u0631\u062f\u060c \u064a\u062a\u0645 \u0627\u062e\u062a\u0632\u0627\u0644 \u0627\u0644\u0646\u0638\u0627\u0645 \u0625\u0644\u0649 \u0634\u0643\u0644 \u0645\u062a\u0642\u0637\u0639 \u0634\u0628\u0647 \u062b\u0627\u0628\u062a \u0627\u0644\u062d\u0627\u0644\u0629\u060c \u0645\u0645\u0627 \u064a\u062a\u064a\u062d \u0641\u0635\u0644\u064b\u0627 \u0648\u0627\u0636\u062d\u064b\u0627 \u0628\u064a\u0646 \u0627\u0644\u062f\u064a\u0646\u0627\u0645\u064a\u0643\u064a\u0627\u062a \u0627\u0644\u0639\u0627\u0628\u0631\u0629 \u0627\u0644\u0633\u0631\u064a\u0639\u0629 \u0648\u0627\u0644\u062a\u0637\u0648\u0631 \u0627\u0644\u0637\u0648\u064a\u0644 \u0627\u0644\u0623\u062c\u0644 \u0644\u0644\u0648\u0631\u0645. \u064a\u0631\u0643\u0632 \u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0646\u0638\u0631\u064a \u0639\u0644\u0649 \u0646\u0642\u0627\u0637 \u0627\u0644\u062a\u0648\u0627\u0632\u0646 \u0648\u0627\u0633\u062a\u0642\u0631\u0627\u0631\u0647\u0627 \u0627\u0644\u0645\u062d\u0644\u064a \u0645\u0646 \u062e\u0644\u0627\u0644 \u0645\u0639\u0627\u062f\u0644\u0627\u062a \u0645\u0645\u064a\u0632\u0629 \u0645\u0633\u062a\u0645\u062f\u0629 \u0645\u0646 \u0627\u0644\u0627\u0633\u062a\u062e\u0637\u0627\u0637 (\u0627\u0644\u062e\u0637\u064a\u0629). \u064a\u062a\u0645 \u062a\u062d\u062f\u064a\u062f \u0634\u0631\u0648\u0637 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0631 \u0628\u0634\u0643\u0644 \u062f\u0642\u064a\u0642\u060c \u0648\u062a\u0639\u064a\u064a\u0646 \u0627\u0644\u0639\u062a\u0628\u0627\u062a \u0627\u0644\u0645\u064f\u0639\u0627\u0645\u0644\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u064f\u062d\u0643\u0645 \u062d\u0627\u0644\u0627\u062a \u0627\u0644\u062a\u0648\u0627\u0632\u0646 \u0627\u0644\u062e\u0627\u0644\u064a \u0645\u0646 \u0627\u0644\u0648\u0631\u0645 \u0648\u0627\u0644\u062a\u0648\u0627\u0632\u0646 \u0627\u0644\u0625\u064a\u062c\u0627\u0628\u064a.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0639\u0644\u0627\u0648\u0629 \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u064a\u0643\u0634\u0641 \u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u062a\u0634\u0639\u0628 \u0639\u0646 \u0648\u062c\u0648\u062f \u062a\u0634\u0639\u0628 \u0647\u0648\u0628\u0641\u060c \u0645\u0645\u0627 \u064a\u0634\u064a\u0631 \u0625\u0644\u0649 \u0638\u0647\u0648\u0631 \u0633\u0644\u0648\u0643 \u0648\u0631\u0645\u064a \u0645\u062a\u0630\u0628\u0630\u0628 \u0641\u064a \u0638\u0644 \u0623\u0646\u0638\u0645\u0629 \u0645\u062d\u062f\u062f\u0629 \u0644\u0644\u0645\u064f\u0639\u0627\u0645\u0650\u0644\u0627\u062a. \u0648\u0647\u0630\u0627 \u064a\u0633\u0644\u0637 \u0627\u0644\u0636\u0648\u0621 \u0639\u0644\u0649 \u0627\u0644\u062f\u0648\u0631 \u0627\u0644\u062d\u0627\u0633\u0645 \u0644\u0644\u062a\u0623\u062e\u064a\u0631\u0627\u062a \u0627\u0644\u0632\u0645\u0646\u064a\u0629 \u0648\u062a\u0648\u0642\u064a\u062a \u0627\u0644\u0639\u0644\u0627\u062c \u0641\u064a \u062a\u0634\u0643\u064a\u0644 \u062f\u064a\u0646\u0627\u0645\u064a\u0643\u064a\u0627\u062a \u0627\u0644\u0648\u0631\u0645. \u0643\u0645\u0627 \u064a\u062a\u0645 \u0641\u062d\u0635 \u0627\u0644\u0646\u0633\u062e\u0629 \u0627\u0644\u0645\u062a\u0642\u0637\u0639\u0629 (\u0627\u0644\u0645\u0646\u0641\u0635\u0644\u0629) \u0645\u0646 \u0627\u0644\u0646\u0645\u0648\u0630\u062c\u060c \u0645\u0645\u0627 \u064a\u0624\u0643\u062f \u0627\u0644\u0627\u062a\u0633\u0627\u0642 \u0628\u064a\u0646 \u0627\u0644\u062a\u0645\u062b\u064a\u0644\u064a\u0646 \u0627\u0644\u0645\u0633\u062a\u0645\u0631 \u0648\u0627\u0644\u0645\u062a\u0642\u0637\u0639.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0644\u0644\u062a\u062d\u0642\u0642 \u0645\u0646 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0627\u0644\u0646\u0638\u0631\u064a\u0629\u060c \u062a\u064f\u062c\u0631\u0649 \u0645\u062d\u0627\u0643\u0627\u0629 \u0639\u062f\u062f\u064a\u0629\u060c \u062a\u062a\u0636\u0645\u0646 \u0645\u062e\u0637\u0637\u0627\u062a \u0627\u0644\u062a\u0634\u0639\u0628 \u0648\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0623\u0633 \u0627\u0644\u0644\u064a\u0627\u0628\u0648\u0646\u0648\u0641\u064a \u0627\u0644\u0645\u0645\u064a\u0632. \u062a\u064f\u0638\u0647\u0631 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0633\u0644\u0648\u0643\u064a\u0627\u062a \u062f\u064a\u0646\u0627\u0645\u064a\u0643\u064a\u0629 \u0645\u0639\u0642\u062f\u0629 \u0645\u062b\u0644 \u062a\u0628\u062f\u064a\u0644 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0631\u060c \u0648\u0627\u0644\u062a\u0630\u0628\u0630\u0628\u0627\u062a\u060c \u0648\u0627\u0633\u062a\u062c\u0627\u0628\u0627\u062a \u0641\u0648\u0636\u0648\u064a\u0629 \u0645\u062d\u062a\u0645\u0644\u0629\u060c \u0645\u0645\u0627 \u064a\u0639\u0643\u0633 \u0627\u0644\u062a\u0628\u0627\u064a\u0646 \u0627\u0644\u0648\u0631\u0645\u064a\u0651 \u0627\u0644\u0645\u0644\u0627\u062d\u0638 \u0633\u0631\u064a\u0631\u064a\u064b\u0627.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0627\u0644\u0623\u0647\u0645 \u0645\u0646 \u0630\u0644\u0643\u060c \u064a\u0624\u0643\u062f \u0627\u0644\u0646\u0645\u0648\u0630\u062c \u0623\u0646 \u0627\u0644\u0633\u064a\u0637\u0631\u0629 \u0627\u0644\u0641\u0639\u0627\u0644\u0629 \u0639\u0644\u0649 \u0627\u0644\u0633\u0631\u0637\u0627\u0646 \u0644\u0627 \u062a\u0639\u062a\u0645\u062f \u0641\u0642\u0637 \u0639\u0644\u0649 \u0643\u062b\u0627\u0641\u0629 \u0627\u0644\u0639\u0644\u0627\u062c\u060c \u0628\u0644 \u0623\u064a\u0636\u064b\u0627 \u0639\u0644\u0649 \u062a\u0648\u0642\u064a\u062a\u0647 \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u062d\u0631\u0643\u064a\u0629 \u0646\u0645\u0648 \u0627\u0644\u0648\u0631\u0645. \u062a\u0648\u0641\u0651\u0631 \u0647\u0630\u0647 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0631\u0624\u0649 \u0642\u064a\u0651\u0645\u0629 \u0644\u062a\u062d\u0633\u064a\u0646 \u0627\u0644\u0627\u0633\u062a\u0631\u0627\u062a\u064a\u062c\u064a\u0627\u062a \u0627\u0644\u0639\u0644\u0627\u062c\u064a\u0629\u060c \u0648\u062a\u0624\u0643\u062f \u0639\u0644\u0649 \u0623\u0647\u0645\u064a\u0629 \u0627\u0644\u062a\u062f\u062e\u0644 \u0627\u0644\u0645\u0628\u0643\u0631. \u0628\u0634\u0643\u0644 \u0639\u0627\u0645\u060c \u064a\u064f\u0642\u062f\u0651\u0645 \u0627\u0644\u0625\u0637\u0627\u0631 \u0627\u0644\u0645\u0642\u062a\u0631\u062d \u0623\u062f\u0627\u0629 \u0642\u0648\u064a\u0629 \u0648\u0642\u0627\u0628\u0644\u0629 \u0644\u0644\u062a\u0641\u0633\u064a\u0631 \u0628\u064a\u0648\u0644\u0648\u062c\u064a\u064b\u0627 \u0644\u062f\u0631\u0627\u0633\u0629 \u062f\u064a\u0646\u0627\u0645\u064a\u0643\u064a\u0627\u062a \u0627\u0644\u0648\u0631\u0645 \u0645\u062a\u0639\u062f\u062f\u0629 \u0627\u0644\u0645\u0642\u0627\u064a\u064a\u0633 \u0641\u064a \u0633\u0631\u0637\u0627\u0646 \u0627\u0644\u062b\u062f\u064a.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u0627\u0644\u0643\u0644\u0645\u0627\u062a \u0627\u0644\u0645\u0641\u062a\u0627\u062d\u064a\u0629:<\/strong>&nbsp;<strong>\u0633\u0631\u0637\u0627\u0646 \u0627\u0644\u062b\u062f\u064a\u060c \u0646\u0645\u0648\u0630\u062c \u0631\u064a\u0627\u0636\u064a\u060c \u0645\u0639\u0627\u062f\u0644\u0629 \u0641\u0631\u0642 \u062a\u0623\u062e\u064a\u0631\u064a\u0629\u060c \u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0631\u060c \u062a\u0634\u0639\u0628<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">BY : <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>F. M. Atteytallah\u00b9, N. A. Elabd\u00b9*<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>\u00b9 Department of Mathematics, Faculty of Science, University of Darna, Libya<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\r\n\t\t<div class=\"review_wrap\">\r\n\t\t\t<div id=\"review-box\" class=\"review-box review-bottom review-stars\">\r\n\t\t\t\t<div class=\"review-summary\">\r\n\t\t\t\t\t<div class=\"review-final-score\">\r\n\t\t\t\t\t\t<span title=\"\" class=\"post-large-rate stars-large\"><span style=\"width:0%\"><\/span><\/span>\r\n\t\t\t\t\t\t<h4><\/h4>\r\n\t\t\t\t\t<\/div>\r\n\t\t\t\t\r\n\t\t\t\t<div class=\"review-short-summary\"><a href=\"https:\/\/drive.google.com\/file\/d\/1-JX8Y4lLTjFRAoQEziR8XJukIrvjdKho\/view?usp=drive_link\" class=\"taq-button taq-medium taq-square taq-flat\" style=\"background-color:#a0ce4e\" target=\"_blank\"><i class=\"fa fa-download\"><\/i><span class=\"button-text\">Download<\/span><\/a>\r\n\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t\t\r\n\t\t\t<div class=\"user-rate-wrap\">\r\n\t\t\t\t<span class=\"user-rating-text\">\r\n\t\t\t\t\t<strong>\u062a\u0642\u064a\u064a\u0645 \u0627\u0644\u0645\u0633\u062a\u062e\u062f\u0645\u0648\u0646: <\/strong>\r\n\t\t\t\t\t<span class=\"taq-score\"><\/span>\r\n\t\t\t\t\t<small>\u0643\u0646 \u0623\u0648\u0644 \u0627\u0644\u0645\u0635\u0648\u062a\u0648\u0646 !<\/small>\r\n\t\t\t\t<\/span>\r\n\r\n\t\t\t\t<div data-rate=\"0\" data-id=\"5007\" class=\"user-rate taq-user-rate-active\">\r\n\t\t\t\t\t<span class=\"user-rate-image post-large-rate stars-large\">\r\n\t\t\t\t\t\t<span style=\"width:0%\"><\/span>\r\n\t\t\t\t\t<\/span>\r\n\t\t\t\t<\/div>\r\n\r\n\t\t\t\t<div class=\"taq-clear\"><\/div>\r\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Abstract: This study develops and analyzes an advanced mathematical framework for modeling breast cancer dynamics based on a singularly perturbed logistic delay differential equation with multiple discrete time delays. The proposed model captures essential biological processes governing tumor progression, including delayed cellular proliferation, immune response activation, and treatment effects, thereby improving the realism of classical &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[162],"tags":[],"class_list":["post-5007","post","type-post","status-publish","format-standard","","category-162"],"_links":{"self":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts\/5007","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5007"}],"version-history":[{"count":1,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts\/5007\/revisions"}],"predecessor-version":[{"id":5008,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=\/wp\/v2\/posts\/5007\/revisions\/5008"}],"wp:attachment":[{"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/journal.ziu.edu.sy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}