Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination
We show a simple model of the dynamics of a viral process based, on the determination of the Kaplan-Meier curve <i>P</i> of the virus. Together with the function of the newly infected individuals <i>I</i>, this model allows us to predict the evolution of the resulting epidemi...
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MDPI AG
2020-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/8/1260 |
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| author | Jose M. Calabuig Luis M. García-Raffi Albert García-Valiente Enrique A. Sánchez-Pérez |
| author_facet | Jose M. Calabuig Luis M. García-Raffi Albert García-Valiente Enrique A. Sánchez-Pérez |
| author_sort | Jose M. Calabuig |
| collection | DOAJ |
| description | We show a simple model of the dynamics of a viral process based, on the determination of the Kaplan-Meier curve <i>P</i> of the virus. Together with the function of the newly infected individuals <i>I</i>, this model allows us to predict the evolution of the resulting epidemic process in terms of the number <i>E</i> of the death patients plus individuals who have overcome the disease. Our model has as a starting point the representation of <i>E</i> as the convolution of <i>I</i> and <i>P</i>. It allows introducing information about latent patients—patients who have already been cured but are still potentially infectious, and re-infected individuals. We also provide three methods for the estimation of <i>P</i> using real data, all of them based on the minimization of the quadratic error: the exact solution using the associated Lagrangian function and Karush-Kuhn-Tucker conditions, a Monte Carlo computational scheme acting on the total set of local minima, and a genetic algorithm for the approximation of the global minima. Although the calculation of the exact solutions of all the linear systems provided by the use of the Lagrangian naturally gives the best optimization result, the huge number of such systems that appear when the time variable increases makes it necessary to use numerical methods. We have chosen the genetic algorithms. Indeed, we show that the results obtained in this way provide good solutions for the model. |
| first_indexed | 2024-03-10T18:02:00Z |
| format | Article |
| id | doaj.art-404d5affbf09433cb0cfb05a858202a4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:02:00Z |
| publishDate | 2020-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-404d5affbf09433cb0cfb05a858202a42023-11-20T08:45:57ZengMDPI AGMathematics2227-73902020-08-0188126010.3390/math8081260Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve DeterminationJose M. Calabuig0Luis M. García-Raffi1Albert García-Valiente2Enrique A. Sánchez-Pérez3Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 València, SpainInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 València, SpainUniversitat de València, Doctor Moliner, 10, 46100 Burjassot (València), SpainInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 València, SpainWe show a simple model of the dynamics of a viral process based, on the determination of the Kaplan-Meier curve <i>P</i> of the virus. Together with the function of the newly infected individuals <i>I</i>, this model allows us to predict the evolution of the resulting epidemic process in terms of the number <i>E</i> of the death patients plus individuals who have overcome the disease. Our model has as a starting point the representation of <i>E</i> as the convolution of <i>I</i> and <i>P</i>. It allows introducing information about latent patients—patients who have already been cured but are still potentially infectious, and re-infected individuals. We also provide three methods for the estimation of <i>P</i> using real data, all of them based on the minimization of the quadratic error: the exact solution using the associated Lagrangian function and Karush-Kuhn-Tucker conditions, a Monte Carlo computational scheme acting on the total set of local minima, and a genetic algorithm for the approximation of the global minima. Although the calculation of the exact solutions of all the linear systems provided by the use of the Lagrangian naturally gives the best optimization result, the huge number of such systems that appear when the time variable increases makes it necessary to use numerical methods. We have chosen the genetic algorithms. Indeed, we show that the results obtained in this way provide good solutions for the model.https://www.mdpi.com/2227-7390/8/8/1260Kaplan-Meiersurvivalquadraticoptimizationepidemicmodel |
| spellingShingle | Jose M. Calabuig Luis M. García-Raffi Albert García-Valiente Enrique A. Sánchez-Pérez Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination Mathematics Kaplan-Meier survival quadratic optimization epidemic model |
| title | Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination |
| title_full | Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination |
| title_fullStr | Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination |
| title_full_unstemmed | Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination |
| title_short | Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination |
| title_sort | evolution model for epidemic diseases based on the kaplan meier curve determination |
| topic | Kaplan-Meier survival quadratic optimization epidemic model |
| url | https://www.mdpi.com/2227-7390/8/8/1260 |
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