scholarly journals CHARACTERIZING AND MANAGING AN EPIDEMIC: A FIRST PRINCIPLES MODEL AND A CLOSED FORM SOLUTION TO THE KERMACK AND MCKENDRICK EQUATIONS

2021 ◽  
Author(s):  
Theodore G Duclos ◽  
Thomas A Reichert
Keyword(s):  
Closed Form ◽  
Complete Solution ◽  
Closed Form Solution ◽  
Prima Facie ◽  
Form Solution ◽  
Public Health Decision ◽  
Mobility Data ◽  
Containment Measures ◽  
Two Parameters ◽  
Analytical Tools

We derived a closed-form solution to the original epidemic equations formulated by Kermack and McKendrick in 1927 (1). The complete solution is validated using independently measured mobility data and accurate predictions of COVID-19 case dynamics in multiple countries. It replicates the observed phenomenology, quantitates pandemic dynamics, and provides simple analytical tools for policy makers. Of particular note, it projects that increased social containment measures shorten an epidemic and reduce the ultimate number of cases and deaths. In contrast, the widely used Susceptible Infectious Recovered (SIR) models, based on an approximation to Kermack and McKendricks original equations, project that strong containment measures delay the peak in daily infections, causing a longer epidemic. These projections contradict both the complete solution and the observed phenomenology in COVID-19 pandemic data. The closed-form solution elucidates that the two parameters classically used as constants in approximate SIR models cannot, in fact, be reasonably assumed to be constant in real epidemics. This prima facie failure forces the conclusion that the approximate SIR models should not be used to characterize or manage epidemics. As a replacement to the SIR models, the closed-form solution and the expressions derived from the solution form a complete set of analytical tools that can accurately diagnose the state of an epidemic and provide proper guidance for public health decision makers.

scholarly journals The Missing Link: A Closed Form Solution to the Kermack and McKendrick Model Equations

2021 ◽  
Author(s):  
Theodore G Duclos ◽  
Thomas A Reichert
Keyword(s):  
Closed Form ◽  
Complete Solution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Policy Makers ◽  
Mobility Data ◽  
Forward Limit ◽  
Containment Measures ◽  
Model Equations ◽  
Analytical Tools

Susceptible infectious recovered (SIR) models are widely used for estimating the dynamics of epidemics. Such models project that containment measures flatten the curve, i.e., reduce but delay the peak in daily infections, cause a longer epidemic, and increase the death toll. These projections have entered common understanding; individuals and governments often advocate lifting containment measures such as social distancing to shift the peak forward, limit societal and economic disruption, and reduce mortality. It was, then, an extraordinary surprise to discover that COVID-19 pandemic data exhibit phenomenology opposite to the projections of SIR models. With the knowledge that the commonly used SIR equations only approximate the original equations developed to describe epidemics, we identified a closed form solution to the original epidemic equations. Unlike the commonly used approximations, the closed form solution replicates the observed phenomenology and quantitates pandemic dynamics with simple analytical tools for policy makers. The complete solution is validated using independently measured mobility data and accurately predicts COVID19 case numbers in multiple countries.

scholarly journals Location and Shape Reconstruction of 2D Dielectric Objects by Means of a Closed-Form Method: Preliminary Experimental Results

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Gian Luigi Gragnani ◽  
Maurizio Diaz Mendez
Keyword(s):  
Closed Form ◽  
Cross Sections ◽  
Complete Solution ◽  
Closed Form Solution ◽  
Computation Time ◽  
Shape Reconstruction ◽  
Form Solution ◽  
Scattering Data ◽  
Equivalent Sources ◽  
Value Decomposition

An analytical approach to location and shape reconstruction of dielectric scatterers, that was recently proposed, is tested against experimental data. Since the cross-sections of the scatterers do not depend on the z coordinate, a 2D problem can be formulated. A closed-form singular value decomposition of the scattering integral operator is derived and is used to determine the radiating components of the equivalent source density. This is a preliminary step toward a more complete solution, which will take into account the incident field inside the investigation domain in order to provide the dielectric features of the scatterer and also the nonradiating sources. Reconstructions of the equivalent sources, performed on some scattering data belonging to the Fresnel database, show the capabilities of the method and, thanks to the closed-form solution, results are obtained in a very short computation time.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

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