Publication
Optimal control and biomathematics: modeling, control and optimization
| Summary: | In this Ph.D. thesis, we apply optimal control theory to various mathematical models, including a pharmacokinetic/pharmacodynamic (PK/PD) model, a fractional PK/PD model, and an epidemiological SEIR models. First, we focus on the PK/PD model to control the infusion of propofol. We begin by analyzing the mathematical model of anesthesia and provide an analytical solution to the time-optimal control problem for the induction phase of anesthesia. Our approach aligns closely with results obtained using the standard shooting method. Utilizing the Pontryagin minimum principle, we propose a new analytical method for solving the time-optimal control problem. Our findings reveal that the optimal continuous infusion rate of the anesthetic and the minimum time required to transition from an awake state to an anesthetized state are consistent between both methods. Furthermore, we extend our analysis to a PK/PD anesthesia model using psi-caputo fractional derivatives. The second part of the thesis focuses on the development of an SEIR model. Initially, we explore mathematical models for COVID-19 with discrete time delays and vaccination. Specifically, we introduce a time delay to account for the delayed migration of individuals from susceptible to infected states. We establish sufficient conditions for the local stability of both endemic and disease-free equilibrium points in the presence of positive time delays. To address the COVID-19 pandemic, we propose a generalized SEIR-type control model. Additionally, we introduce three time-dependent controls for the SEIR model and analyze the optimal control problem with respect to real data transmission in Italy. Our results demonstrate the effectiveness of the model, particularly concerning the number of quarantined and recovered individuals. By considering Pontryagin controls, we illustrate the potential for significant reductions in susceptible, exposed, infected, quarantined/hospitalized, and deceased individuals through increased population protection. We also present a model for maintaining the efficacy of COVID-19 vaccines during transportation and distribution, emphasizing the importance of vaccination in controlling the pandemic. |
|---|---|
| Main Authors: | Abdelaziz, Zaitri Mohamed |
| Subject: | Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| Year: | 2023 |
| Country: | Portugal |
| Document type: | doctoral thesis |
| Access type: | open access |
| Associated institution: | Universidade de Aveiro |
| Language: | English |
| Origin: | RIA - Repositório Institucional da Universidade de Aveiro |
| _version_ | 1877769355942428672 |
|---|---|
| author | Abdelaziz, Zaitri Mohamed |
| author_facet | Abdelaziz, Zaitri Mohamed |
| author_role | author |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Abdelaziz, Zaitri Mohamed\"}] |
| datacite.creators.creator.creatorName.fl_str_mv | Abdelaziz, Zaitri Mohamed |
| datacite.date.Accepted.fl_str_mv | 2023-09-29T00:00:00Z |
| datacite.date.available.fl_str_mv | 2023-11-23T09:53:54Z |
| datacite.date.embargoed.fl_str_mv | 2023-11-23T09:53:54Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| datacite.titles.title.fl_str_mv | Optimal control and biomathematics: modeling, control and optimization |
| dc.creator.none.fl_str_mv | Abdelaziz, Zaitri Mohamed |
| dc.date.Accepted.fl_str_mv | 2023-09-29T00:00:00Z |
| dc.date.available.fl_str_mv | 2023-11-23T09:53:54Z |
| dc.date.embargoed.fl_str_mv | 2023-11-23T09:53:54Z |
| dc.description.none.fl_str_mv | Nesta tese de doutoramento, aplicamos a teoria do controlo ótimo a um modelo farmacocinético/farmacodinâmico (PK/PD) e a um modelo epidemiológico do tipo SEIR. Primeiro, estudamos as propriedades do modelo PK/PD para controlar a infusão de propofol. Começamos por analisar um modelo matemático para a anestesia e determinamos uma solução analítica para o problema de controlo ótimo de tempo mínimo para a fase de indução da anestesia, mostrando que esta coincide numericamente com a solução obtida usando o método de tiro. Considerando o princípio do mínimo de Pontryagin, resolvemos o problema de controlo ótimo de tempo mínimo através de um novo método analítico e mostramos que a taxa de infusão contínua ótima do anestésico e o tempo mínimo requerido para passar do estado de vigília para o estado de anestesia são semelhantes usando os dois métodos. Além disso, analisamos um modelo fracionário de Anestesia PK/PD via derivadas fracionais de psi-Caputo. A segunda parte da tese é dedicada ao desenvolvimento de um modelo do tipo SEIR. Primeiramente, analisamos modelos matemáticos para a COVID-19 com tempos de atrasos discretos e vacinação. Mas precisamente, introduzimos um tempo de atraso que representa, matematicamente, o fato de a migração de indivíduos suscetíveis para infetados estar sujeita a tempos de atraso. Um dos resultados mais importantes em sistemas dinâmicos é a estabilidade. Nesta tese demonstramos condições suficientes para a estabilidade local dos pontos de equilíbrio endémico e livre de doença, para qualquer tempo de atraso positive. Para combater a propagação da COVID-19, propomos um modelo com controlo, generalizando o modelo do tipo SEIR. Além disso, introduzimos três controlos ao modelo SEIR e analisamos o problema de controlo ótimo da transmissão da doença usando dados reais de Itália. Os nossos resultados mostram o ajuste do modelo aos dados reais, em particular no que diz respeito ao número de indivíduos em quarentena e recuperados. Considerando os controlos de Pontryagin, mostramos como num mundo perfeito seria possível diminuir drasticamente o número de indivíduos suscetíveis, expostos, infetados, em quarentena/hospitalizados e óbitos, aumentando a população de protegidos. Além disso, apresentamos um modelo para manter a eficácia da vacina para a COVID-19 desde o transporte da área de armazenamento na fábrica até ao destino desejado e introduzir a vacina na população suscetível, a fim de controlar a disseminação da COVID-19. Mostramos a importância da vacina para o controlo da propagação da COVID-19 e também na melhoria do resultado que poderia ser obtido se o número de vacinas disponíveis satisfizesse as necessidades da população e fossem distribuídas de acordo com a teoria do controlo ótimo. |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | http://hdl.handle.net/10773/39707 |
| dc.language.none.fl_str_mv | eng |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| dc.title.fl_str_mv | Optimal control and biomathematics: modeling, control and optimization |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_db06 |
| description | In this Ph.D. thesis, we apply optimal control theory to various mathematical models, including a pharmacokinetic/pharmacodynamic (PK/PD) model, a fractional PK/PD model, and an epidemiological SEIR models. First, we focus on the PK/PD model to control the infusion of propofol. We begin by analyzing the mathematical model of anesthesia and provide an analytical solution to the time-optimal control problem for the induction phase of anesthesia. Our approach aligns closely with results obtained using the standard shooting method. Utilizing the Pontryagin minimum principle, we propose a new analytical method for solving the time-optimal control problem. Our findings reveal that the optimal continuous infusion rate of the anesthetic and the minimum time required to transition from an awake state to an anesthetized state are consistent between both methods. Furthermore, we extend our analysis to a PK/PD anesthesia model using psi-caputo fractional derivatives. The second part of the thesis focuses on the development of an SEIR model. Initially, we explore mathematical models for COVID-19 with discrete time delays and vaccination. Specifically, we introduce a time delay to account for the delayed migration of individuals from susceptible to infected states. We establish sufficient conditions for the local stability of both endemic and disease-free equilibrium points in the presence of positive time delays. To address the COVID-19 pandemic, we propose a generalized SEIR-type control model. Additionally, we introduce three time-dependent controls for the SEIR model and analyze the optimal control problem with respect to real data transmission in Italy. Our results demonstrate the effectiveness of the model, particularly concerning the number of quarantined and recovered individuals. By considering Pontryagin controls, we illustrate the potential for significant reductions in susceptible, exposed, infected, quarantined/hospitalized, and deceased individuals through increased population protection. We also present a model for maintaining the efficacy of COVID-19 vaccines during transportation and distribution, emphasizing the importance of vaccination in controlling the pandemic. |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | doctoralThesis |
| id | ria_945cd3d261b033ffa773264f91d9ea73 |
| identifier.url.fl_str_mv | http://hdl.handle.net/10773/39707 |
| inst_facet_str | urn:organizationAcronym:ua{{{_:::_}}}Universidade de Aveiro |
| instacron_str | ua |
| institution | Universidade de Aveiro |
| instname_str | Universidade de Aveiro |
| language | eng |
| network_acronym_str | ria |
| network_name_str | RIA - Repositório Institucional da Universidade de Aveiro |
| oai_identifier_str | oai:ria.ua.pt:10773/39707 |
| organization_str_mv | urn:organizationAcronym:ua |
| person_str_mv | Abdelaziz, Zaitri Mohamed |
| publishDate | 2023 |
| repo_facet_str | urn:repositoryAcronym:ria{{{_:::_}}}RIA - Repositório Institucional da Universidade de Aveiro |
| reponame_str | RIA - Repositório Institucional da Universidade de Aveiro |
| repository_id_str | urn:repositoryAcronym:ria |
| service_str_mv | urn:repositoryAcronym:ria |
| spelling | pt_PTIn this Ph.D. thesis, we apply optimal control theory to various mathematical models, including a pharmacokinetic/pharmacodynamic (PK/PD) model, a fractional PK/PD model, and an epidemiological SEIR models. First, we focus on the PK/PD model to control the infusion of propofol. We begin by analyzing the mathematical model of anesthesia and provide an analytical solution to the time-optimal control problem for the induction phase of anesthesia. Our approach aligns closely with results obtained using the standard shooting method. Utilizing the Pontryagin minimum principle, we propose a new analytical method for solving the time-optimal control problem. Our findings reveal that the optimal continuous infusion rate of the anesthetic and the minimum time required to transition from an awake state to an anesthetized state are consistent between both methods. Furthermore, we extend our analysis to a PK/PD anesthesia model using psi-caputo fractional derivatives. The second part of the thesis focuses on the development of an SEIR model. Initially, we explore mathematical models for COVID-19 with discrete time delays and vaccination. Specifically, we introduce a time delay to account for the delayed migration of individuals from susceptible to infected states. We establish sufficient conditions for the local stability of both endemic and disease-free equilibrium points in the presence of positive time delays. To address the COVID-19 pandemic, we propose a generalized SEIR-type control model. Additionally, we introduce three time-dependent controls for the SEIR model and analyze the optimal control problem with respect to real data transmission in Italy. Our results demonstrate the effectiveness of the model, particularly concerning the number of quarantined and recovered individuals. By considering Pontryagin controls, we illustrate the potential for significant reductions in susceptible, exposed, infected, quarantined/hospitalized, and deceased individuals through increased population protection. We also present a model for maintaining the efficacy of COVID-19 vaccines during transportation and distribution, emphasizing the importance of vaccination in controlling the pandemic.pt_PTNesta tese de doutoramento, aplicamos a teoria do controlo ótimo a um modelo farmacocinético/farmacodinâmico (PK/PD) e a um modelo epidemiológico do tipo SEIR. Primeiro, estudamos as propriedades do modelo PK/PD para controlar a infusão de propofol. Começamos por analisar um modelo matemático para a anestesia e determinamos uma solução analítica para o problema de controlo ótimo de tempo mínimo para a fase de indução da anestesia, mostrando que esta coincide numericamente com a solução obtida usando o método de tiro. Considerando o princípio do mínimo de Pontryagin, resolvemos o problema de controlo ótimo de tempo mínimo através de um novo método analítico e mostramos que a taxa de infusão contínua ótima do anestésico e o tempo mínimo requerido para passar do estado de vigília para o estado de anestesia são semelhantes usando os dois métodos. Além disso, analisamos um modelo fracionário de Anestesia PK/PD via derivadas fracionais de psi-Caputo. A segunda parte da tese é dedicada ao desenvolvimento de um modelo do tipo SEIR. Primeiramente, analisamos modelos matemáticos para a COVID-19 com tempos de atrasos discretos e vacinação. Mas precisamente, introduzimos um tempo de atraso que representa, matematicamente, o fato de a migração de indivíduos suscetíveis para infetados estar sujeita a tempos de atraso. Um dos resultados mais importantes em sistemas dinâmicos é a estabilidade. Nesta tese demonstramos condições suficientes para a estabilidade local dos pontos de equilíbrio endémico e livre de doença, para qualquer tempo de atraso positive. Para combater a propagação da COVID-19, propomos um modelo com controlo, generalizando o modelo do tipo SEIR. Além disso, introduzimos três controlos ao modelo SEIR e analisamos o problema de controlo ótimo da transmissão da doença usando dados reais de Itália. Os nossos resultados mostram o ajuste do modelo aos dados reais, em particular no que diz respeito ao número de indivíduos em quarentena e recuperados. Considerando os controlos de Pontryagin, mostramos como num mundo perfeito seria possível diminuir drasticamente o número de indivíduos suscetíveis, expostos, infetados, em quarentena/hospitalizados e óbitos, aumentando a população de protegidos. Além disso, apresentamos um modelo para manter a eficácia da vacina para a COVID-19 desde o transporte da área de armazenamento na fábrica até ao destino desejado e introduzir a vacina na população suscetível, a fim de controlar a disseminação da COVID-19. Mostramos a importância da vacina para o controlo da propagação da COVID-19 e também na melhoria do resultado que poderia ser obtido se o número de vacinas disponíveis satisfizesse as necessidades da população e fossem distribuídas de acordo com a teoria do controlo ótimo.application/pdfengpt_PTOptimal control and biomathematics: modeling, control and optimizationAbdelaziz, Zaitri MohamedHandlehttp://hdl.handle.net/10773/397072023-11-23T09:53:54Z2023-09-29T00:00:00Z2023-09-29http://purl.org/coar/access_right/c_abf2open accesspt_PTMathematical modelingpt_PTHeat equationpt_PTControl systemspt_PTDelayed systemspt_PTOptimizationpt_PTOptimal controlpt_PTPontryagin extremalspt_PTFractional calculuspt_PTSEIR modelpt_PTPharmacokinetic/pharmacodynamic modelpt_PTEquilibrium pointspt_PTStabilitypt_PTAnalysis of the spread of diseasespt_PTVaccination2709771 byteshttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://ria.ua.pt/bitstream/10773/39707/1/Documento_Mohamed_Zaitri.pdfother research producthttp://purl.org/coar/resource_type/c_db06doctoral thesis |
| spellingShingle | Optimal control and biomathematics: modeling, control and optimization Abdelaziz, Zaitri Mohamed Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| status | SINGLETON |
| subject.fl_str_mv | Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| title | Optimal control and biomathematics: modeling, control and optimization |
| title_full | Optimal control and biomathematics: modeling, control and optimization |
| title_fullStr | Optimal control and biomathematics: modeling, control and optimization |
| title_full_unstemmed | Optimal control and biomathematics: modeling, control and optimization |
| title_short | Optimal control and biomathematics: modeling, control and optimization |
| title_sort | Optimal control and biomathematics: modeling, control and optimization |
| topic | Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| topic_facet | Mathematical modeling Heat equation Control systems Delayed systems Optimization Optimal control Pontryagin extremals Fractional calculus SEIR model Pharmacokinetic/pharmacodynamic model Equilibrium points Stability Analysis of the spread of diseases Vaccination |
| url | http://hdl.handle.net/10773/39707 |
| visible | 1 |
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