<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>1409-2433</journal-id>
<journal-title><![CDATA[Revista de Matemática Teoría y Aplicaciones]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. Mat]]></abbrev-journal-title>
<issn>1409-2433</issn>
<publisher>
<publisher-name><![CDATA[Centro de Investigaciones en Matemática Pura y Aplicada (CIMPA) y Escuela de Matemática, San José, Costa Rica.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1409-24332017000100079</article-id>
<article-id pub-id-type="doi">10.15517/rmta.v24i1.27771</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Optimal control problem for a SEIR type model of ebola epidemics]]></article-title>
<article-title xml:lang="es"><![CDATA[Problema de control óptimo para un modelo del tipo SEIR de la epidemia del ébola]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Grigorieva]]></surname>
<given-names><![CDATA[Ellina V.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Deignan]]></surname>
<given-names><![CDATA[Paul B.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Khailov]]></surname>
<given-names><![CDATA[Evgenii N.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Texas Woman&#8217;s University  ]]></institution>
<addr-line><![CDATA[Denton TX]]></addr-line>
<country>US</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,The University of Texas  ]]></institution>
<addr-line><![CDATA[Dallas TX]]></addr-line>
<country>US</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Moscow State Lomonosov University  ]]></institution>
<addr-line><![CDATA[ Moscow]]></addr-line>
<country>RU</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>06</month>
<year>2017</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>06</month>
<year>2017</year>
</pub-date>
<volume>24</volume>
<numero>1</numero>
<fpage>79</fpage>
<lpage>96</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_arttext&amp;pid=S1409-24332017000100079&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_abstract&amp;pid=S1409-24332017000100079&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_pdf&amp;pid=S1409-24332017000100079&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[AbstractA Susceptible, Exposed, Infectious, and Recovered (SEIR) type control model describing the Ebola epidemic in a population of constant size is considered over a fixed time interval. This model is an extension of the well-known SEIR model and is more suitable to the study of the control mechanism of Ebola epidemics. Along with the traditional SEIR compartments, this model contains an isolated infectious compartment representing the number of infected and exposed individuals that have been isolated from the susceptible individuals. The model has two intervention controls reflecting efforts to protect susceptible individuals from infected and exposed individuals. Additionally, there are two control functions that define efforts for the detection and isolation of infected and exposed individuals. The minimization problem of the sum of total fractions of infected and exposed individuals and total weighted costs of control constraints over a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. Accordingly, the controls are bang-bang functions determined by the corresponding switching functions. In order to estimate the number of zeros of the switching functions, a new approach is proposed based on the analysis of the Cauchy problems for the derivatives of these functions. It is found that the optimal controls of the original problem have at most one switching. This allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of three variables. Results of the numerical solution to this problem and their analysis are provided.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[ResumenSe considera un modelo de tipo Susceptible, Expuesto, Infeccioso y Recuperado (SEIR) que describe la epidemia del ébola en una población de tamaño constante sobre un intervalo de tiempo fijo. Este modelo es una extensión del bien conocido modelo SEIR y es más adecuado para el estudio del mecanismo de control de la epidemia del ébola. Además de los compartimientos tradicionales del SEIR, este modelo contiene un compartimiento aislado infeccioso que representa el número de individuos infectados y expuestos que han sido aislados de los individuos susceptibles. El modelo tiene dos controles de intervención que reflejan los esfuerzos para proteger a los individuos susceptibles de los individuos infectados y expuestos. Adicionalmente, hay dos funciones de control que definen los esfuerzos para la detección y aislamiento de individuos infectados y expuestos. Se plantea el problema de minimización de la suma del total de cocientes de individuos infectados y expuestos y el total de costos ponderados de restricciones de control sobre un intervalo de tiempo. Para el análisis de los correspondientes controles óptimos, se usa el principio del máximo de Pontryanguin. En consecuencia, los controles son funciones bang-bang determinadas por las correspondientes funciones de cambio. Con el fin de estimar el número de ceros de las funciones de cambio, se propone un nuevo enfoque basado en el análisis de los problemas de Cauchy para las derivadas de estas funciones. Se encontró que los controles óptimos del problema original tienen a lo sumo un cambio. Esto permite la reducción del complejo problema original de control óptimo a resolver un problema mucho más simple de minimización condicional de una función de tres variables. Se presentan los resultados y análisis de la solución numérica a este problema.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[SEIR model]]></kwd>
<kwd lng="en"><![CDATA[nonlinear control system]]></kwd>
<kwd lng="en"><![CDATA[optimal control]]></kwd>
<kwd lng="en"><![CDATA[Pontryagin maximum principle]]></kwd>
<kwd lng="en"><![CDATA[switching function]]></kwd>
<kwd lng="es"><![CDATA[modelo SEIR]]></kwd>
<kwd lng="es"><![CDATA[sistema de control no lineal]]></kwd>
<kwd lng="es"><![CDATA[control óptimo]]></kwd>
<kwd lng="es"><![CDATA[principio del máximo de Pontryaguin]]></kwd>
<kwd lng="es"><![CDATA[función de cambio]]></kwd>
</kwd-group>
</article-meta>
</front><back>
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