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Discrete and Continuous Dynamical Systems - S

March 2022 , Volume 15 , Issue 3

Issue on optimization and differential equations

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Catherine Choquet, M. Rchid Sidi Ammi, Mouhcine Tilioua and Delfim F. M. Torres
2022, 15(3): i-ii doi: 10.3934/dcdss.2022015 +[Abstract](5409) +[HTML](232) +[PDF](97.84KB)
Optimal control of an HIV model with a trilinear antibody growth function
Karam Allali, Sanaa Harroudi and Delfim F. M. Torres
2022, 15(3): 501-518 doi: 10.3934/dcdss.2021148 +[Abstract](530) +[HTML](137) +[PDF](526.48KB)

We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.

Optimality conditions involving the Mittag–Leffler tempered fractional derivative
Ricardo Almeida and M. Luísa Morgado
2022, 15(3): 519-534 doi: 10.3934/dcdss.2021149 +[Abstract](529) +[HTML](132) +[PDF](322.7KB)

In this work we study problems of the calculus of the variations, where the differential operator is a generalization of the tempered fractional derivative. Different types of necessary conditions required to determine the optimal curves are proved. Problems with additional constraints are also studied. A numerical method is presented, based on discretization of the variational problem. Through several examples, we show the efficiency of the method.

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays
Şeyma Bılazeroğlu, Huseyin Merdan and Luca Guerrini
2022, 15(3): 535-554 doi: 10.3934/dcdss.2021150 +[Abstract](456) +[HTML](188) +[PDF](537.45KB)

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.

Mathematical modeling of algal blooms due to swine CAFOs in Eastern North Carolina
Amy Henderson, Emek Kose, Allison Lewis and Ellen R. Swanson
2022, 15(3): 555-572 doi: 10.3934/dcdss.2021151 +[Abstract](524) +[HTML](135) +[PDF](879.54KB)

Dramatic strides have been made in treating human waste to remove pathogens and excess nutrients before discharge into the environment, to the benefit of ground and surface water quality. Yet these advances have been undermined by the dramatic growth of Confined Animal Feeding Operations (CAFOs) which produce voluminous quantities of untreated waste. Industrial swine routinely produce waste streams similar to that of a municipality, yet these wastes are held in open-pit "lagoons" which are at risk of rupture or overflow. Eastern North Carolina is a coastal plain with productive estuaries which are imperiled by more than 2000 permitted swine facilities housing over 9 million hogs; the associated 3,500 permitted manure lagoons pose a risk to sensitive estuarine ecosystems, as breaches or overflows send large plumes of nutrient and pathogen-rich waste into surface waters. Understanding the relationship between nutrient pulses and surface water quality in coastal environments is essential to effective CAFO policy formation. In this work, we develop a system of ODEs to model algae growth in a coastal estuary due to a manure lagoon breach and investigate nutrient thresholds above which algal blooms are unresolvable.

A non-standard class of variational problems of Herglotz type
Natália Martins
2022, 15(3): 573-586 doi: 10.3934/dcdss.2021152 +[Abstract](408) +[HTML](132) +[PDF](297.75KB)

In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function \begin{document}$ x $\end{document} and its derivative and an unknown functional \begin{document}$ z $\end{document}, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.

An optimal control problem applied to a wastewater treatment plant
M. Teresa T. Monteiro, Isabel Espírito Santo and Helena Sofia Rodrigues
2022, 15(3): 587-601 doi: 10.3934/dcdss.2021153 +[Abstract](466) +[HTML](133) +[PDF](603.61KB)

This paper aims to present a mathematical model that describes the operation of an activated sludge system during one day. Such system is used in the majority of wastewater treatment plants and depends strongly on the dissolved oxygen, since it is a biological treatment. To guarantee the appropriate amount of dissolved oxygen, expensive aeration strategies are demanded, leading to high costs in terms of energy consumption. It was considered a typical domestic effluent as the wastewater to test the mathematical model and it was used the ASM1 to describe the activated sludge behaviour. An optimal control problem was formulated whose cost functional considers the trade-off between the minimization of the control variable herein considered (the dissolved oxygen) and the quality index that is the amount of pollution. The optimal control problem is treated as a nonlinear optimization problem after discretization by direct methods. The problem was then coded in the AMPL programming language in order to carry out numerical simulations using the NLP solver IPOPT from NEOS Server.

Set-valued problems under bounded variation assumptions involving the Hausdorff excess
Bianca Satco
2022, 15(3): 603-620 doi: 10.3934/dcdss.2021154 +[Abstract](377) +[HTML](117) +[PDF](368.28KB)

In the very general framework of a (possibly infinite dimensional) Banach space \begin{document}$ X $\end{document}, we are concerned with the existence of bounded variation solutions for measure differential inclusions

where \begin{document}$ dg $\end{document} is the Stieltjes measure generated by a nondecreasing left-continuous function.

This class of differential problems covers a wide variety of problems occuring when studying the behaviour of dynamical systems, such as: differential and difference inclusions, dynamic inclusions on time scales and impulsive differential problems. The connection between the solution set associated to a given measure \begin{document}$ dg $\end{document} and the solution sets associated to some sequence of measures \begin{document}$ dg_n $\end{document} strongly convergent to \begin{document}$ dg $\end{document} is also investigated.

The multifunction \begin{document}$ G : [0,1] \times X \to \mathcal{P}(X) $\end{document} with compact values is assumed to satisfy excess bounded variation conditions, which are less restrictive comparing to bounded variation with respect to the Hausdorff-Pompeiu metric, thus the presented theory generalizes already known existence and continuous dependence results. The generalization is two-fold, since this is the first study in the setting of infinite dimensional spaces.

Next, by using a set-valued selection principle under excess bounded variation hypotheses, we obtain solutions for a functional inclusion

It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.

Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives
Moulay Rchid Sidi Ammi, Mostafa Tahiri and Delfim F. M. Torres
2022, 15(3): 621-637 doi: 10.3934/dcdss.2021155 +[Abstract](473) +[HTML](126) +[PDF](574.89KB)

The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.

Stability and optimal control of a delayed HIV/AIDS-PrEP model
Cristiana J. Silva
2022, 15(3): 639-654 doi: 10.3934/dcdss.2021156 +[Abstract](485) +[HTML](132) +[PDF](4167.21KB)

In this paper, we propose a time-delayed HIV/AIDS-PrEP model which takes into account the delay on pre-exposure prophylaxis (PrEP) distribution and adherence by uninfected persons that are in high risk of HIV infection, and analyze the impact of this delay on the number of individuals with HIV infection. We prove the existence and stability of two equilibrium points, for any positive time delay. After, an optimal control problem with state and control delays is proposed and analyzed, where the aim is to find the optimal strategy for PrEP implementation that minimizes the number of individuals with HIV infection, with minimal costs. Different scenarios are studied, for which the solutions derived from the Minimum Principle for Multiple Delayed Optimal Control Problems change depending on the values of the time delays and the weights constants associated with the number of HIV infected individuals and PrEP. We observe that changes on the weights constants can lead to a passage from bang-singular-bang to bang-bang extremal controls.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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