[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83439-en":3,"doc-seo-83439-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83439,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Fractal Fractional HIV Dynamics with Mittag Leffler Kernel Analysis Stability and Numerical Simulations","A fractal–fractional HIV transmission model is developed using the Atangana–Baleanu–Caputo operator with a Mittag–Leffler kernel to embed memory and hereditary effects in disease progression. The study establishes existence and uniqueness of solutions through appropriate analytical methods and evaluates Hyers–Ulam stability to confirm robustness of the proposed system. Numerical approximations are obtained via Newton polynomial approximation combined with the Atangana–Toufik scheme, supported by sensitivity heatmaps and tornado diagrams to assess parameter influence. Results validate the framework for transient and long-term HIV dynamics.","arXiv :2607 .00061v1 [math .NA] 30 Jun 2026  \nFractal-Fractional HIV Dynamics with Mittag-Leffler Kernel: Analysis, Stability, and Numerical Simulations  \nNiaz Ali Shah 1 , Samad Noeiaghdam2 ,3 ,∗  \n1 Department of Mathematics Abbottabad University of Science and Technology, Abbottabad, 22010 Pakistan.  \n[nuzhatniaz2014@gmail.com](nuzhatniaz2014@gmail.com)  \n2 Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, [China. snoei@hnas.ac.cn](China. snoei@hnas.ac.cn)  \n[3](3 Department of Mathematical Sciences)[ Department of Mathematical Sciences](3 Department of Mathematical Sciences), [Saveetha School of Engineering](Saveetha School of Engineering), [SIMATS](SIMATS), [Chennai](Chennai), [602105](602105) , [India](India).  \nAbstract  \nIn this paper, a fractal–fractional HIV model with the Mittag–Leffler kernel is proposed using the Atangana–Baleanu–Caputo operator to capture the memory and hereditary properties of the disease dynamics. The existence and uniqueness of the solutions are investigated using suitable analytical techniques, and the Hyers–Ulam stability analysis is carried out to verify the stability behavior of the proposed system. For the numerical simulations, the Newton polynomial approximation method together with the Atangana–Toufik numerical scheme is employed to obtain approximate solutions for different parameter settings. Furthermore, several visualization techniques, including sensitivity heatmap representation and tornado diagram analysis, are utilized to study the influence of model parameters on the HIV dynamics. The obtained numerical results demonstrate that the proposed fractal–fractional framework provides an effective and reliable approach for analyzing the transient and long-term behavior of HIV transmission dynamics.  \nkeywords: HIV dynamical model; Fractional order; Atangana-Baleanu-Caputo operator; Sensitivity;  \n1 Introduction  \nDespite major advances in prevention, diagnosis, and antiretroviral therapy, HIV still one of the world’s most important public health isuse. According to recent global estimates, tens of millions of people are affected with HIV, with the highest burden intense in un-developed countries, particularly in sub-Saharan Africa. Over the past two decades, expanded access to treatment has substantially reduced AIDS-related deaths and improved life expectancy, transforming HIV from a fatal disease into a manageable chronic condition for many patients [1] . However, new infections continue to occur each year due to limited healthcare access, social stigma, inequality, lack of education, and insufficient preventive measures in vulnerable populations. International organizations and governments continue to emphasize early diagnosis, universal treatment access, public awareness, vaccination research, and the development of long-acting therapies as essential  \nstrategies for achieving to vanished such dangerous epidemic disease global is still a major problem in the coming decades [2] .  \nHIV is a retrovirus that cannot replicate independently, but it requires the host cells to do so, reproduce independently. It transports single-stranded RNA, which is transformed. the virus reverse transcriptase enzyme after into double-stranded DNA. enters a CD4+ T-cell [3] . The viral DNA becomes incorporated in the host genome, enabling the infected cell to produce viral RNA and proteins. These components gather atthe cell membrane to create immature virions that bud off the cell. and grow up by cleavage by protease, and become complete infectious. particles [5] . HIV infection can be long term without any clinical manifestations. period, and symptoms usually develop when the CD+4 T-cell number is [decreased. to](decreased. to) almost 200cells/mm3 and the viral load grows significantly [4] . Mathematical modeling has played a big role in comprehending HIV. pathogenesis and immunodynamics. Wodarz and Nowak [4] came up with. model systems of viral development and response to therap","cbCailD0VnhATPrY","https://ap.wps.com/l/cbCailD0VnhATPrY","pdf",1946192,4,1,25,"English","en",105,"# Abstract\n# Introduction\n# Dynamical Systems and Fractional Modeling\n# Model Formulation and Analytical Results\n# Numerical Simulations and Parameter Analysis\n# Conclusion","[{\"question\":\"What mathematical operators are used in the proposed HIV model?\",\"answer\":\"The model uses the Atangana–Baleanu–Caputo operator with a Mittag–Leffler kernel to represent memory and hereditary properties of HIV dynamics.\"},{\"question\":\"How does the paper verify stability of the fractal–fractional system?\",\"answer\":\"Hyers–Ulam stability analysis is carried out to assess whether the proposed system remains stable under perturbations.\"},{\"question\":\"Which numerical methods are used to compute approximate solutions?\",\"answer\":\"The Newton polynomial approximation method is combined with the Atangana–Toufik numerical scheme to produce approximate solutions for different parameter settings.\"}]",1784187736,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"fractal-fractional-hiv-dynamics-with-mittag-leffler-kernel-analysis-stability-and-numerical-simulations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/fractal-fractional-hiv-dynamics-with-mittag-leffler-kernel-analysis-stability-and-numerical-simulations/83439/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What mathematical operators are used in the proposed HIV model?","Question",{"text":75,"@type":76},"The model uses the Atangana–Baleanu–Caputo operator with a Mittag–Leffler kernel to represent memory and hereditary properties of HIV dynamics.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper verify stability of the fractal–fractional system?",{"text":80,"@type":76},"Hyers–Ulam stability analysis is carried out to assess whether the proposed system remains stable under perturbations.",{"name":82,"@type":73,"acceptedAnswer":83},"Which numerical methods are used to compute approximate solutions?",{"text":84,"@type":76},"The Newton polynomial approximation method is combined with the Atangana–Toufik numerical scheme to produce approximate solutions for different parameter 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