A Nine-Compartment Nonlinear Epidemic Model with Spline-Based Identification of Time-Varying Transmission and Vaccination Dynamics

Jul 16, 2026

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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.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages23

Reading Time58 min

Introduction

Motivation and Context

Limitations of Classical Models

The Calibration Problem as a Nonlinear Inverse Problem

arXiv :2606 .07413v1 [math .OC] 5 Jun 2026 A NINE-COMPARTMENT NONLINEAR EPIDEMIC MODEL WITH SPLINE-BASED IDENTIFICATION OF TIME-VARYING TRANSMISSION AND VACCINATION DYNAMICS: APPLICATION TO THE COVID-19 THIRD WAVE IN ITALY ∗ LOKMAN RACHID MELHANI†, ANTONINO SFERLAZZA‡, LARS GRÜNE§ , DOMINIQUE PERSANO ADORNO¶ , FILIPPO D’IPPOLITO†, OMAR ENZO SANTANGELO∥ , IVAN MARCHESE†, ANTONINO LO BURGIO\#, AND ALBERTO FIRENZE†† Abstract. 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. Key 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 AMS subject classifications. 92D30, 34A55, 49M37, 65K10, 65L09, 34D20, 93B07 1. Introduction. 1.1. 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 ∗ Submitted to the editors June 8, 2026 . Funding: This research was supported by the University of Palermo. †Department of Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy ([melhanilokmanrachid@gmail.com](melhanilokmanrachid@gmail.com)). ‡Department of Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy. § Department of Mathematics, University of Bayreuth, Bayreuth, Germany. ¶ Department of Physics and Chemistry “E. Segré”, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy. ∥ Regional Health Care and Social Agency of Lodi, Azienda Socio-Sanitaria Territoriale (ASST) Lodi, 26900 Lodi, Italy. \# InEmbryo S.r.l.s., Via Rosario Riolo 60, 90141 Palermo, Italy. ††Department of Internal Medicine “PROMISE”, University of Palermo, 90127 Palermo, Italy. effort [17, 3 , 7], and

FAQs

What key features does the nine-compartment model include for COVID-19?

It incorporates two co-circulating viral strains, a super-spreader subpopulation, partial vaccine-induced immunity with waning, and explicit hospitalization dynamics with differentiated mortality.

How are time-varying transmission and vaccination rates identified from data?

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.

What theoretical guarantees are provided for the model and identification approach?

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.

Jul 16, 2026

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