<?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>S1409-24332022000200177</article-id>
<article-id pub-id-type="doi">10.15517/rmta.v29i2.50888</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[Control óptimo de orden fraccionario para un modelo de eficacia del tratamiento de la tuberculosis con presencia de VIH/sida y diabetes]]></article-title>
<article-title xml:lang="en"><![CDATA[Fractional-order optimal control for a model of tuberculosis treatment efficacy in the presence of HIV/aids and diabetes]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Delgado-Moya]]></surname>
<given-names><![CDATA[Erick]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Pietrus]]></surname>
<given-names><![CDATA[Alain]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,University of São Paulo IME-Rua do Matão ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Brazil</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,University of Antilles  Department of Mathematics and Computer Sciences]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Francia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2022</year>
</pub-date>
<volume>29</volume>
<numero>2</numero>
<fpage>177</fpage>
<lpage>223</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_arttext&amp;pid=S1409-24332022000200177&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_abstract&amp;pid=S1409-24332022000200177&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_pdf&amp;pid=S1409-24332022000200177&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En este trabajo, aprovechando las ventajas del uso de las derivadas de orden fraccionario en el sentido de Caputo, presentamos un estudio de control óptimo de la eficacia del tratamiento de la tuberculosis (TB) en presencia del VIH/SIDA y la diabetes. El modelo matemático que es controlado se encuentra en (17) y estudia la relación entre la tuberculosis, el VIH/SIDA y la diabetes con respecto a la eficacia del tratamiento y diferencia en tuberculosis multirresistente (TB-MDR), y tuberculosis extremadamente resistente (TB-XDR). La definición de los controles se centra en evitar la reinfección/reactivación, la TB-MDR y la TB-XDR en las diferentes subpoblaciones (TB únicamente, TB-VIH/SIDA y TBDiabetes). El modelo dividido en subpoblaciones nos permite diferenciar los comportamientos de la transmisión y la resistencia y evaluar los diferentes costos en la aplicación de los controles. Realizamos simulaciones computacionales con datos de la literatura para estudiar nuestro problema de control en un escenario específico. Exploramos el comportamiento del número básico de reproducción variando la tasa de contacto efectivo y los parámetros asociados a la resistencia para diferentes valores de &#945; (orden fraccional). Estudiamos diferentes estrategias de control basadas en la activación de los controles y encontramos que la más efectiva es cuando activamos todos los controles. Con esta estrategia, reducimos el número de casos resistentes, principalmente en la TB-XDR en diabéticos que tiene un fuerte impacto en la dinámica de resistencia y transmisión de la tuberculosis. Además, esta estrategia evita el crecimiento futuro del número de casos resistentes.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In this paper, taking advantage of the use of fractional order derivatives in the Caputo sense, we present a study of optimal control of tuberculosis (TB) treatment efficacy in the presence of HIV/AIDS and diabetes. The mathematical model to which control is applied is found in (17) and studies the relationship between TB, HIV/AIDS and diabetes with respect to treatment efficacy and differentiates into multidrug-resistant TB (MDRTB), and extensively drug-resistant TB (XDR-TB). The definition of controls focuses on avoiding reinfection/reactivation, MDR-TB and XDR-TB in the different subpopulations (TB only, TB-HIV/AIDS and TB-Diabetes). The model which is divided into subpopulations allows us to differentiate transmission and resistance behaviors and to evaluate the different costs in the application of controls. We performed computational simulations with literature data to study our control problem in a specific scenario. We explored the behavior of the basic reproduction number by varying the effective contact rate and the parameters associated with the resistance for different values of &#945; (fractional order). We studied different control strategies based on the activation of the controls and found that the most effective is when we activate all the controls. With this strategy, we reduce the number of resistant cases, mainly in XDR-TB in diabetics which has a strong impact on the dynamics of TB resistance and transmission. In addition, this strategy avoids future growth in the number of resistant cases.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[diabetes]]></kwd>
<kwd lng="es"><![CDATA[VIH/SIDA]]></kwd>
<kwd lng="es"><![CDATA[derivadas fraccionarias en el sentido de Caputo]]></kwd>
<kwd lng="es"><![CDATA[problema de control óptimo]]></kwd>
<kwd lng="es"><![CDATA[tuberculosis.]]></kwd>
<kwd lng="en"><![CDATA[diabetes]]></kwd>
<kwd lng="en"><![CDATA[HIV/AIDS]]></kwd>
<kwd lng="en"><![CDATA[fractional derivatives in the Caputo sense]]></kwd>
<kwd lng="en"><![CDATA[optimal control problem]]></kwd>
<kwd lng="en"><![CDATA[tuberculosis.]]></kwd>
</kwd-group>
</article-meta>
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