[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86412-en":3,"doc-seo-86412-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86412,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","A Nine-Compartment Nonlinear Epidemic Model with Spline-Based Identification of Time-Varying Transmission and Vaccination Dynamics","A nine-compartment nonlinear epidemic model is developed to capture two co-circulating COVID-19 strains, a super-spreader subpopulation, partial vaccine-induced immunity with waning, and explicit hospitalization with differentiated mortality. Time-dependent transmission and vaccination rates are treated as control inputs and identified from Italian COVID-19 data (Jan–May 2021) using a PCHIP control-node parameterization that reduces calibration to a constrained SQP problem. A parametric bootstrap quantifies uncertainty, while analytical well-posedness, stability, reproduction number, identifiability, and error bounds establish convergence of the spline approximation.","arXiv :2606 .07413v1 [math .OC] 5 Jun 2026  \nA NINE-COMPARTMENT NONLINEAR EPIDEMIC MODEL WITH SPLINE-BASED IDENTIFICATION OF TIME-VARYING TRANSMISSION AND VACCINATION DYNAMICS: APPLICATION  \nTO THE COVID-19 THIRD WAVE IN ITALY ∗  \nLOKMAN RACHID MELHANI†, ANTONINO SFERLAZZA‡, LARS GRÜNE§ , DOMINIQUE PERSANO ADORNO¶ , FILIPPO D’IPPOLITO†, OMAR ENZO SANTANGELO∥ , IVAN  \nMARCHESE†, ANTONINO LO BURGIO\\#, AND ALBERTO FIRENZE††  \nAbstract. We develop a nine-compartment nonlinear epidemic model incorporating two cocirculating viral strains (ancestral I1 and the Alpha variant B.1.1.7 I2, which is 43–90% more transmissible, c2 = 1 .5), a super-spreader subpopulation, partial vaccine-induced immunity with waning, and explicit hospitalization dynamics with differentiated mortality. Transmission and vaccination rates are treated as time-varying control inputs and identified from Italian COVID-19 data (January–May 2021) via a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) control-node parameterization, reducing calibration to a fourteen-variable Sequential Quadratic Programming (SQP) problem with monotonicity and box constraints. A parametric bootstrap (n = 1000) quantifies parameter uncertainty. The calibrated model achieves R2 = 0 .966 for active hospitalizations, R2 = 0 .987 for cumulative fatalities, and R2 = 0 .999 for cumulative vaccinations. Well-posedness, the basic reproduction number in closed form, and local and global stability of the disease-free equilibrium are established analytically. An L∞ approximation error bound shows that the PCHIP control-node parameterization converges to the true time-varying parameters at rate O (h2 ) as the node spacing vanishes. Local identifiability and a noise stability bound are established via the Fisher information matrix. A sufficient threshold condition proves epidemic decay under time-varying suppression whenever the effective reproduction number remains persistently below one. Sensitivity analyses consistently rank hospital throughput parameters above the transmission rate, providing a mathematical basis for the observation that reactive containment measures cannot prevent a hospitalization peak already driven by the pre-existing latent viral load.  \nKey words. COVID-19, epidemic modeling, nonlinear inverse problem, ODE-constrained optimization, spline approximation, PCHIP, time-varying parameters, parameter identification, SQP, sensitivity analysis, basic reproduction number  \nAMS subject classifications. 92D30, 34A55, 49M37, 65K10, 65L09, 34D20, 93B07  \n1. Introduction.  \n1.1. Motivation and Context. The COVID-19 pandemic caused by SARSCoV-2 has produced the most severe global public health crisis since the influenza pandemic of 1918 . From the first reported cluster in Wuhan, China in December 2019 [35], the virus spread with a speed that outpaced containment efforts in most highincome countries, producing successive epidemic waves separated by periods of partial control [26, 30 , 15] . Compartmental models in the tradition of Kermack and McKendrick [18] provided the conceptual scaffolding for much of the COVID-19 modeling  \n∗ Submitted to the editors June 8, 2026 .  \nFunding: This research was supported by the University of Palermo.  \n†Department of Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy ([melhanilokmanrachid@gmail.com](melhanilokmanrachid@gmail.com)).  \n‡Department of Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy.  \n§ Department of Mathematics, University of Bayreuth, Bayreuth, Germany.  \n¶ Department of Physics and Chemistry “E. Segré”, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy.  \n∥ Regional Health Care and Social Agency of Lodi, Azienda Socio-Sanitaria Territoriale (ASST) Lodi, 26900 Lodi, Italy.  \n\\# InEmbryo S.r.l.s., Via Rosario Riolo 60, 90141 Palermo, Italy.  \n††Department of Internal Medicine “PROMISE”, University of Palermo, 90127 Palermo, Italy.  \neffort [17, 3 , 7], and ","cbCaijl824CKNaYM","https://ap.wps.com/l/cbCaijl824CKNaYM","pdf",1123954,5,1,23,"English","en",105,"# Introduction\n## Motivation and Context\n## Limitations of Classical Models\n## The Calibration Problem as a Nonlinear Inverse Problem","[{\"question\":\"What key features does the nine-compartment model include for COVID-19?\",\"answer\":\"It incorporates two co-circulating viral strains, a super-spreader subpopulation, partial vaccine-induced immunity with waning, and explicit hospitalization dynamics with differentiated mortality.\"},{\"question\":\"How are time-varying transmission and vaccination rates identified from data?\",\"answer\":\"Transmission and vaccination rates are modeled as time-varying control inputs and identified from Italian COVID-19 data using a PCHIP control-node parameterization, converting calibration into a constrained SQP problem.\"},{\"question\":\"What theoretical guarantees are provided for the model and identification approach?\",\"answer\":\"The work establishes well-posedness, derives the basic reproduction number, proves local and global stability of the disease-free equilibrium, and provides approximation convergence and identifiability results via the Fisher information matrix.\"}]",1784211579,58,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-nine-compartment-nonlinear-epidemic-model-with-spline-based-identification-of-time-varying-transmission-and-vaccination-dynamics","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/a-nine-compartment-nonlinear-epidemic-model-with-spline-based-identification-of-time-varying-transmission-and-vaccination-dynamics/86412/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What key features does the nine-compartment model include for COVID-19?","Question",{"text":76,"@type":77},"It incorporates two co-circulating viral strains, a super-spreader subpopulation, partial vaccine-induced immunity with waning, and explicit hospitalization dynamics with differentiated mortality.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are time-varying transmission and vaccination rates identified from data?",{"text":81,"@type":77},"Transmission and vaccination rates are modeled as time-varying control inputs and identified from Italian COVID-19 data using a PCHIP control-node parameterization, converting calibration into a constrained SQP problem.",{"name":83,"@type":74,"acceptedAnswer":84},"What theoretical guarantees are provided for the model and identification approach?",{"text":85,"@type":77},"The work establishes well-posedness, derives the basic reproduction number, proves local and global stability of the disease-free equilibrium, and provides approximation convergence and identifiability results via the Fisher information matrix.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & 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