Optimal control to limit the spread of COVID-19 in Italy

We apply optimal control theory to a generalized SEIR-type model. The proposed system has three controls, representing social distancing, preventive means, and treatment measures to combat the spread of the COVID-19 pandemic. We analyze such optimal control problem with respect to real data transmission in Italy. Our results show the appropriateness of the model, in particular with respect to the number of quarantined/hospitalized (confirmed and infected) and recovered individuals. Considering the Pontryagin controls, we show how in a perfect world one could have drastically diminish the number of susceptible, exposed, infected, quarantined/hospitalized, and death individuals, by increasing the population of insusceptible/protected.publishe

Controlo ótimo e biomatemática: modelação, controlo e otimização

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.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.Programa Doutoral em Matemátic

Pharmacokinetic/Pharmacodynamic Anesthesia Model Incorporating psi-Caputo Fractional Derivatives

We present a novel Pharmacokinetic/Pharmacodynamic (PK/PD) model for the induction phase of anesthesia, incorporating the $\psi$-Caputo fractional derivative. By employing the Picard iterative process, we derive a solution for a nonhomogeneous $\psi$-Caputo fractional system to characterize the dynamical behavior of the drugs distribution within a patient's body during the anesthesia process. To explore the dynamics of the fractional anesthesia model, we perform numerical analysis on solutions involving various functions of $\psi$ and fractional orders. All numerical simulations are conducted using the MATLAB computing environment. Our results suggest that the $\psi$ functions and the fractional order of differentiation have an important role in the modeling of individual-specific characteristics, taking into account the complex interplay between drug concentration and its effect on the human body. This innovative model serves to advance the understanding of personalized drug responses during anesthesia, paving the way for more precise and tailored approaches to anesthetic drug administration.This research was developed within the project “Mathematical Modelling of Multi-scale Control Systems: applications to human diseases (CoSysM3)”, 2022.03091.PTDC, financially supported by national funds (OE), through FCT/MCTES. The authors are also supported by FCT and CIDMA via projects UIDB/04106/2020 and UIDP/04106/ 2020.publishe

A hybrid direction algorithm for solving optimal control problems

In this paper, we present an algorithm for finding an approximate numerical solution for linear optimal control problems. This algorithm is based on the hybrid direction algorithm developed by Bibi and Bentobache [A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, vol. 92, no.1, pp. 201–216, 2015]. We define an optimality estimate and give a necessary and sufficient condition to characterize the optimality of a certain admissible control of the discretized problem, then we give a numerical example to illustrate the proposed approach. Finally, we present some numerical results which show the convergence of the proposed algorithm to the optimal solution of the presented continuous optimal control problem

Transport and Optimal Control of Vaccination Dynamics for COVID-19

We develop a mathematical model for transferring the vaccine BNT162b2 based on the heat diffusion equation. Then, we apply optimal control theory to the proposed generalized SEIR model. We introduce vaccination for the susceptible population to control the spread of the COVID-19 epidemic. For this, we use the Pontryagin minimum principle to find the necessary optimality conditions for the optimal control. The optimal control problem and the heat diffusion equation are solved numerically. Finally, several simulations are done to study and predict the spread of the COVID-19 epidemic in Italy. In particular, we compare the model in the presence and absence of vaccination.Comment: This is a preprint whose final form is published by Elsevier in the book 'Mathematical Analysis of Infectious Diseases', 1st Edition - June 1, 2022, ISBN: 978032390504