Evolution Model for Epidemic Diseases Based on the Kaplan-Meier Curve Determination

Abstract

We show a simple model of the dynamics of a viral process based, on the determination of the Kaplan-Meier curve P of the virus. Together with the function of the newly infected individuals I, this model allows us to predict the evolution of the resulting epidemic process in terms of the number E of the death patients plus individuals who have overcome the disease. Our model has as a starting point the representation of E as the convolution of I and P. 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 P 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.

Kaplan-Meier Type Survival Curves for COVID-19: A Health Data Based Decision-Making Tool

“…There is also a technical problem caused by the fact that the data provided by public institutions are not disaggregated, i.e., they give an overall number of new infected people, deaths and cured per day but do not provide the number of days each patient is followed by the health services before being discharged and considered cured. This fact makes it necessary to deconvolute the data to obtain the survival curve using new mathematical procedures, as explained in ( 3 ), which inevitably introduce more errors. Analysis using complete data provided by the health administration would also reduce the error caused by this need for mathematical processing.…”

“…This quantity approximates the number of discharged people Using this equation it is possible to compute the Kaplan-Meier survival curve from the data extracted of the reports of the different countries affected by the pandemia. The complete explanation of how this can be done can be found in ( 3 ).…”

“…Thus, although all these arguments could influence the unusual results of the survival curves these facts do not substantially change the structure of the model (3). Therefore, the problem comes from the data.…”

“…But there is another fact that is even more problematic for the mathematical analysis: the data are aggregated—they are essentially given by the number of new infections, deaths and individuals cured each day, so a disaggregation (deconvolution) procedure is necessary to obtain Kaplan-Meier curves. The mathematical issue of doing this was studied in our work ( 3 ), where several analytical-heuristic methods are considered. Finally, we decided to use Genetic Algorithms for our overall analysis, as it was the method that was shown to be the most efficient.…”

“…To approximate the solution of the equations we use a genetic algorithm approach ( 7 ), which provides estimates of the probability and therefore clear images of the expected infection scenario. A full explanation of the mathematical method we have developed to do this is available in ( 3 ). In this paper we show that our model provides information that can be relevant for the management of health systems for different countries.…”