Mathematics

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20 pages, 569 KiB

Open AccessArticle

Optimal Control Theory and Calculus of Variations in Mathematical Models of Chemotherapy of Malignant Tumors

by Nina Subbotina, Natalia Novoselova and Evgenii Krupennikov

Mathematics 2023, 11(20), 4301; https://doi.org/10.3390/math11204301 - 16 Oct 2023

Viewed by 1042

This paper is devoted to the analysis of mathematical models of chemotherapy for malignant tumors growing according to the Gompertz law or the generalized logistic law. The influence of the therapeutic agent on the tumor dynamics is determined by a therapy function depending [...] Read more.

This paper is devoted to the analysis of mathematical models of chemotherapy for malignant tumors growing according to the Gompertz law or the generalized logistic law. The influence of the therapeutic agent on the tumor dynamics is determined by a therapy function depending on the time-varying concentration of the drug in the patient’s body. The case of a non-monotonic therapy function with two maxima is studied. It reflects the use of two different therapeutic agents. The state variables of the dynamics are the tumor volume and the amount of the therapeutic agent able to suppress malignant cells (concentration of the drug in the body). The treatment protocol (the rate of administration of the therapeutic agent) is the control in the dynamics. The optimal control problem for this models is considered. It is the problem of the construction of treatment protocols that provide the minimal tumor volume at the end of the treatment. The solution of this problem was obtained by the authors in previous works via the optimal control theory. The form of the considered therapy functions provides a specific structure for the optimal controls. The managerial insights of this structure are discussed. In this paper, the structure of the viability set is described for the model according to the generalized logistic law. It is the set of the initial states of the model for which one can find a treatment protocol that guarantees that the tumor volume remains within the prescribed limits throughout the treatment. The description of the viability set’s structure is based on the optimal control theory and the theory of Hamilton–Jacobi equations. An inverse problem of therapy is also considered, namely the problem of reconstruction of the treatment protocol and identification of the unknown parameter of the intensity of the tumor growth. Reconstruction is carried out by processing information about the observations of the tumor volume dynamics and the measurements of the drug concentration in the body. A solution to this problem is obtained through the use of a method based on the calculus of variations. The results of the numerical simulations are presented herein. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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26 pages, 756 KiB

Open AccessArticle

Optimal Control of a Two-Patch Dengue Epidemic under Limited Resources

by Edwin Barrios-Rivera, Olga Vasilieva and Mikhail Svinin

Mathematics 2023, 11(18), 3921; https://doi.org/10.3390/math11183921 - 15 Sep 2023

Cited by 1 | Viewed by 1199

Abstract

Despite the ongoing preventive measures of vector control, dengue fever still presents outbreaks, and daily commuting of people also facilitates its propagation. To contain the disease spread after an outbreak has already occurred, the local healthcare authorities are compelled to perform insecticide spraying [...] Read more.

Despite the ongoing preventive measures of vector control, dengue fever still presents outbreaks, and daily commuting of people also facilitates its propagation. To contain the disease spread after an outbreak has already occurred, the local healthcare authorities are compelled to perform insecticide spraying as a corrective measure of vector control, thus trying to avoid massive human infections. Several issues concerned with the practical implementation of such corrective measures can be solved from a mathematical standpoint, and the purpose of this study is to contribute to this strand of research. Using as a basis a two-patch dengue transmission ODE model, we designed the patch-dependent optimal strategies for the insecticide spraying with the optimal control approach. We also analyzed the response of the optimal strategies to three alternative modes of budget cuts under different intensities of daily commuting. Our approach illustrated that trying “to save money” by reducing the budget for corrective control is completely unwise, and the anticipated “savings” will actually turn into considerable additional public spending for treating human infections, which could have been averted by a timely corrective intervention. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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15 pages, 334 KiB

Open AccessArticle

Necessary Conditions for the Optimality and Sustainability of Solutions in Infinite-Horizon Optimal Control Problems

by Sergey M. Aseev

Mathematics 2023, 11(18), 3851; https://doi.org/10.3390/math11183851 - 8 Sep 2023

Viewed by 757

Abstract

The paper deals with an infinite-horizon optimal control problem with general asymptotic endpoint constraints. The fulfillment of constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions. A new version of the Pontryagin maximum principle with [...] Read more.

The paper deals with an infinite-horizon optimal control problem with general asymptotic endpoint constraints. The fulfillment of constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions. A new version of the Pontryagin maximum principle with an explicitly specified adjoint variable is developed. The proof of the main results is based on the fact that the restriction of the optimal process to any finite time interval is a solution to the corresponding finite-horizon problem containing the conditional cost of the phase vector as a terminal term. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

15 pages, 1001 KiB

Open AccessFeature PaperArticle

Dynamics of Persistent Epidemic and Optimal Control of Vaccination

by Masoud Saade, Sebastian Aniţa and Vitaly Volpert

Mathematics 2023, 11(17), 3770; https://doi.org/10.3390/math11173770 - 2 Sep 2023

Cited by 3 | Viewed by 1172

Abstract

This paper is devoted to a model of epidemic progression, taking into account vaccination and immunity waning. The model consists of a system of delay differential equations with time delays determined by the disease duration and immunity loss. Periodic epidemic outbreaks emerge as [...] Read more.

This paper is devoted to a model of epidemic progression, taking into account vaccination and immunity waning. The model consists of a system of delay differential equations with time delays determined by the disease duration and immunity loss. Periodic epidemic outbreaks emerge as a result of the instability of a positive stationary solution if the basic reproduction number exceeds some critical value. Vaccination can change epidemic dynamics, resulting in more complex aperiodic oscillations confirmed by some data on Influenza A in Norway. Furthermore, the measures of social distancing during the COVID-19 pandemic weakened seasonal influenza in 2021, but increased it during the next year. Optimal control allows for the minimization of epidemic cost by vaccination. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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14 pages, 257 KiB

Open AccessArticle

An Optimal Control Problem Related to the RSS Model

by Alexander J. Zaslavski

Mathematics 2023, 11(17), 3762; https://doi.org/10.3390/math11173762 - 1 Sep 2023

Viewed by 741

Abstract

In this paper, we consider a discrete-time optimal control problem related to the model of Robinson, Solow and Srinivasan. We analyze this optimal control problem without concavity assumptions on a non-concave utility function which represents the preferences of the planner and establish the [...] Read more.

In this paper, we consider a discrete-time optimal control problem related to the model of Robinson, Solow and Srinivasan. We analyze this optimal control problem without concavity assumptions on a non-concave utility function which represents the preferences of the planner and establish the existence of good programs and optimal programs which are Stiglitz production programs. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

26 pages, 672 KiB

Open AccessArticle

Application of an Optimal Control Therapeutic Approach for the Memory-Regulated Infection Mechanism of Leprosy through Caputo–Fabrizio Fractional Derivative

by Xianbing Cao, Salil Ghosh, Sourav Rana, Homagnic Bose and Priti Kumar Roy

Mathematics 2023, 11(17), 3630; https://doi.org/10.3390/math11173630 - 22 Aug 2023

Cited by 2 | Viewed by 1100

Abstract

Leprosy (Hansen’s disease) is an infectious, neglected tropical skin disease caused by the bacterium Mycobacterium leprae (M. leprae). It is crucial to note that the dynamic behavior of any living microorganism such as M. leprae not only depends on the conditions [...] Read more.

Leprosy (Hansen’s disease) is an infectious, neglected tropical skin disease caused by the bacterium Mycobacterium leprae (M. leprae). It is crucial to note that the dynamic behavior of any living microorganism such as M. leprae not only depends on the conditions of its current state (e.g., substrate concentration, medium condition, etc.) but also on those of its previous states. In this article, we have developed a three-dimensional mathematical model involving concentrations of healthy Schwann cells, infected Schwann cells, and M. leprae bacteria in order to predict the dynamic changes in the cells during the disease dissemination process; additionally, we investigated the effect of memory on system cell populations, especially on the M. leprae bacterial population, by analyzing the Caputo–Fabrizio fractionalized version of the model. Most importantly, we developed and investigated a fractionalized optimal-control-induced system comprising the combined drug dose therapy of Ofloxacin and Dapsone intended to achieve a more realistic treatment regime for leprosy. The main goal of our research article is to compare this fractional-order system with the corresponding integer-order model and also to distinguish the rich dynamics exhibited by the optimal-control-induced system based on different values of the fractional order

ζ \in (0, 1)

. All of the analytical results are validated through proper numerical simulations and are compared with some real clinical data. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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29 pages, 621 KiB

Open AccessArticle

Optimal Melanoma Treatment Protocols for a Bilinear Control Model

by Evgenii Khailov and Ellina Grigorieva

Mathematics 2023, 11(15), 3289; https://doi.org/10.3390/math11153289 - 26 Jul 2023

Cited by 1 | Viewed by 1124

Abstract

In this research, for a given time interval, which is the general period of melanoma treatment, a bilinear control model is considered, given by a system of differential equations, which describes the interaction between drug-sensitive and drug-resistant cancer cells both during drug therapy [...] Read more.

In this research, for a given time interval, which is the general period of melanoma treatment, a bilinear control model is considered, given by a system of differential equations, which describes the interaction between drug-sensitive and drug-resistant cancer cells both during drug therapy and in the absence of it. This model also contains a control function responsible for the transition from the stage of such therapy to the stage of its absence and vice versa. To find the optimal moments of switching between these stages, the problem of minimizing the cancer cells load both during the entire period of melanoma treatment and at its final moment is stated. Such a minimization problem has a nonconvex control set, which can lead to the absence of an optimal solution to the stated minimization problem in the classes of admissible modes traditional for applications. To avoid this problem, the control set is imposed to be convex. As a result, a relaxed minimization problem arises, in which the optimal solution exists. An analytical study of this minimization problem is carried out using the Pontryagin maximum principle. The corresponding optimal solution is found in the form of synthesis and may contain a singular arc. It shows that there are values of the parameters of the bilinear control model, its initial conditions, and the time interval for which the original minimization problem does not have an optimal solution, because it has a sliding mode. Then for such values it is possible to find an approximate optimal solution to the original minimization problem in the class of piecewise constant controls with a predetermined number of switchings. This research presents the results of the analysis of the connection between such an approximate solution of the original minimization problem and the optimal solution of the relaxed minimization problem based on numerical calculations performed in the Maple environment for the specific values of the parameters of the bilinear control model, its initial conditions, and the time interval. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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25 pages, 626 KiB

Open AccessArticle

Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions

by Fahad Al Basir and Teklebirhan Abraha

Mathematics 2023, 11(7), 1687; https://doi.org/10.3390/math11071687 - 31 Mar 2023

Cited by 10 | Viewed by 6096

Abstract

Malaria is a serious illness caused by a parasite, called Plasmodium, transmitted to humans through the bites of female Anopheles mosquitoes. The parasite infects and destroys the red blood cells in the human body leading to symptoms, such as fever, headache, and flu-like [...] Read more.

Malaria is a serious illness caused by a parasite, called Plasmodium, transmitted to humans through the bites of female Anopheles mosquitoes. The parasite infects and destroys the red blood cells in the human body leading to symptoms, such as fever, headache, and flu-like illness. Awareness campaigns that educate people about malaria prevention and control reduce transmission of the disease. In this research, a mathematical model is proposed to study the impact of awareness-based control measures on the transmission dynamics of malaria. Some basic properties of the proposed model, such as non-negativity and boundedness of the solutions, the existence of the equilibrium points, and their stability properties, have been studied using qualitative theory. Disease-free equilibrium is globally asymptotic when the basic reproduction number,

R_{0}

, is less than the number of current cases. Finally, optimal control theory is applied to minimize the cost of disease control and solve the optimal control problem by applying Pontryagin’s minimum principle. Numerical simulations have been provided for the confirmation of the analytical results. Endemic equilibrium exists for

R_{0} > 1

, and a forward transcritical bifurcation occurs at

R_{0} = 1

. The optimal profiles of the treatment process, organizing awareness campaigns, and insecticide uses are obtained for the cost-effectiveness of malaria management. This research concludes that awareness campaigns through social media with an optimal control approach are best for cost-effective malaria management. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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20 pages, 515 KiB

Open AccessArticle

Numerical Fractional Optimal Control of Respiratory Syncytial Virus Infection in Octave/MATLAB

by Silvério Rosa and Delfim F. M. Torres

Mathematics 2023, 11(6), 1511; https://doi.org/10.3390/math11061511 - 20 Mar 2023

Cited by 9 | Viewed by 2269

Abstract

In this article, we develop a simple mathematical GNU Octave/MATLAB code that is easy to modify for the simulation of mathematical models governed by fractional-order differential equations, and for the resolution of fractional-order optimal control problems through Pontryagin’s maximum principle (indirect approach to [...] Read more.

In this article, we develop a simple mathematical GNU Octave/MATLAB code that is easy to modify for the simulation of mathematical models governed by fractional-order differential equations, and for the resolution of fractional-order optimal control problems through Pontryagin’s maximum principle (indirect approach to optimal control). For this purpose, a fractional-order model for the respiratory syncytial virus (RSV) infection is considered. The model is an improvement of one first proposed by the authors in 2018. The initial value problem associated with the RSV infection fractional model is numerically solved using Garrapa’s fde12 solver and two simple methods coded here in Octave/MATLAB: the fractional forward Euler’s method and the predict-evaluate-correct-evaluate (PECE) method of Adams–Bashforth–Moulton. A fractional optimal control problem is then formulated having treatment as the control. The fractional Pontryagin maximum principle is used to characterize the fractional optimal control and the extremals of the problem are determined numerically through the implementation of the forward-backward PECE method. The implemented algorithms are available on GitHub and, at the end of the paper, in appendixes, both for the uncontrolled initial value problem as well as for the fractional optimal control problem, using the free GNU Octave computing software and assuring compatibility with MATLAB. Full article

(This article belongs to the Special Issue Optimal Control Theory and Its Applications in Medical and Biological Sciences)

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