[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85629-en":3,"doc-seo-85629-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85629,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Extended Kalman Filter-Based State Estimation for a Nine-Compartment Nonlinear Epidemic Model","Real-time epidemic management requires continuous access to the full latent state that surveillance cannot directly measure, including exposed and infectious subpopulations by strain, waning vaccination-derived immunity, and the recovered pool affecting residual susceptibility. This work develops state estimation for a nine-compartment nonlinear epidemic model with two co-circulating viral strains, hospitalization and mortality, using three public observables: hospitalizations, cumulative fatalities, and vaccinated immune stock. Extended Kalman Filter design is supported by Lie-derivative observability and convergence theory, with covariance tuning validated on calibrated in-silico benchmarks.","arXiv :2606 . 16305v1 [math .OC] 15 Jun 2026  \nExtended Kalman Filter-Based State Estimation for a Nine-Compartment  \nNonlinear Epidemic Model–  \nConvergence Analysis and In-Silico Benchmark Calibrated on the COVID-19  \nThird Wave in Italy  \nLokman Rachid Melhania , Antonino Sferlazzaa , Dominique Persano Adornob , Filippo D’Ippolitoa , Antonino Lo  \nBurgiod , Alberto Firenzec  \na Department of Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy b Department of Physics and Chemistry “E. Segr´e”, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy c Department of Internal Medicine “Promise,” University of Palermo, 90127 Palermo, Italy  \ndInEmbryo S.r.l.s., Via Rosario Riolo 60, 90141 Palermo, Italy  \nAbstract  \nThe real-time management of an infectious disease outbreak requires continuous knowledge of the full epidemic state, including quantities that surveillance systems cannot directly measure: the size of the latent exposed population, the number of actively infectious individuals stratified by strain and transmission potential, the vaccinated immune fraction, and the recovered pool that determines residual susceptibility. This paper addresses the state-estimation problem for the nine-compartment nonlinear epidemic model of the companion study [1] that incorporates two co-circulating viral strains with identical transmissibility multiplier c2 = cP = 1.5, a super-spreader subpopulation, partial vaccinederived immunity with waning, explicit hospitalization dynamics, and disease-induced mortality. The time-varying transmission and vaccination rates are treated as known inputs identified from data by a companion spline-based calibration procedure, so that the remaining problem is precisely the reconstruction of all nine biological states from the three observables systematically reported by public health agencies: active hospitalizations H, cumulative fatalities F, and the vaccinated immune stock V. This paper is a methodological contribution: all numerical experiments use synthetic measurements generated from the calibrated model, so the reported RMSE figures are methodology benchmarksand should not be interpreted as real-time predictive accuracy on live surveillance data.  \nThe paper makes four contributions. The Extended Kalman Filter construction itself is standard; the contributions lie in the observability and convergence theory that surround it, and in the principled covariance design, rather than in the filter recursion. First, a nonlinear observability analysis within the Lie-derivative framework computes the analytical observability codistribution. A six-step algebraic derivation (Lemma 1) proves in closed form that at Lie levels 0, 1, 2 the matrix is rank-deficient at the calibrated symmetric parameter values δi = δp , r1 = r2 , yielding | det(O9)| = δw γ2aκρ2w 21(δi − δp )2 |r1 − r2 | (where δw denotes the waning rate of the recovered pool) with a twodimensional kernel consisting of an I2 ↔ P swap direction and an R-anchored direction. Augmentation by the third Lie derivative restores full rank 9; the treatment-recovery rate r2 is identified as the structural symmetry-breaking parameter. Second, on this observability basis an Extended Kalman Filter (EKF) is designed on the Euler-discretized  \ndynamics, with the analytical 9 × 9 state Jacobian in closed form and the Joseph stabilized covariance update. Third, the local exponential boundedness of the estimation error in mean square is established as a full theorem by verifying the four hypotheses of the Reif–G¨unther–Yaz–Unbehauen framework for the specific system at hand; in particular, hypothesis (A1) (uniform covariance bounds) is established via an explicit 8-step observability Gramian argument together with the dual controllability bound supplied by a positive-definite process noise covariance. The proof exploits the fact that all nonlinearities of the vector field are bilinear and that the measurement map is exactly linear, ","cbCaip1zHIdf5APz","https://ap.wps.com/l/cbCaip1zHIdf5APz","pdf",551034,1,50,"English","en",105,"# Abstract\n# Introduction\n## Motivation and Contribution","[{\"question\":\"Which epidemic states are reconstructed by the proposed estimator?\",\"answer\":\"The method reconstructs all nine biological compartments, including latent exposed and actively infectious subpopulations stratified by strain and transmission potential, vaccinated immune fraction with waning, and the recovered pool that shapes residual susceptibility.\"},{\"question\":\"What observable data are used as inputs to the state estimation problem?\",\"answer\":\"Estimation relies on three public-health observables: active hospitalizations H, cumulative fatalities F, and the vaccinated immune stock V, while transmission and vaccination rates are treated as known inputs identified from calibration.\"},{\"question\":\"How is convergence and error behavior established for the Extended Kalman Filter?\",\"answer\":\"Local mean-square exponential boundedness is proven by verifying the four hypotheses in a Reif–Günther–Yaz–Unbehauen framework, with uniform covariance bounds derived via observability Gramian arguments and controlled through positive-definite process noise covariance.\"}]",1784205068,126,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"extended-kalman-filter-based-state-estimation-for-a-nine-compartment-nonlinear-epidemic-model","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/extended-kalman-filter-based-state-estimation-for-a-nine-compartment-nonlinear-epidemic-model/85629/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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},"Which epidemic states are reconstructed by the proposed estimator?","Question",{"text":75,"@type":76},"The method reconstructs all nine biological compartments, including latent exposed and actively infectious subpopulations stratified by strain and transmission potential, vaccinated immune fraction with waning, and the recovered pool that shapes residual susceptibility.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What observable data are used as inputs to the state estimation problem?",{"text":80,"@type":76},"Estimation relies on three public-health observables: active hospitalizations H, cumulative fatalities F, and the vaccinated immune stock V, while transmission and vaccination rates are treated as known inputs identified from calibration.",{"name":82,"@type":73,"acceptedAnswer":83},"How is convergence and error behavior established for the Extended Kalman Filter?",{"text":84,"@type":76},"Local mean-square exponential boundedness is proven by verifying the four hypotheses in a Reif–Günther–Yaz–Unbehauen framework, with uniform covariance bounds derived via observability Gramian arguments and controlled through positive-definite process noise covariance.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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