Publicação
Optimal control and biomathematics: modeling, control and optimization
| Resumo: | 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. |
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| Autores principais: | Abdelaziz, Zaitri Mohamed |
| Assunto: | 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 |
| Ano: | 2023 |
| País: | Portugal |
| Tipo de documento: | tese de doutoramento |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Aveiro |
| Idioma: | inglês |
| Origem: | RIA - Repositório Institucional da Universidade de Aveiro |
| Resumo: | 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. |
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Atividades financiadas
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