# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

May 2019 , Volume 24 , Issue 5

New horizons in optimal control and biomathematics: A tribute to Urszula Ledzewicz, Helmut Maurer and Heinz Schaettler

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2019, 24(5): ⅰ-ⅱ doi: 10.3934/dcdsb.201905i +[Abstract](932) +[HTML](51) +[PDF](440.81KB)
Abstract:
2019, 24(5): 2017-2038 doi: 10.3934/dcdsb.2019082 +[Abstract](1372) +[HTML](74) +[PDF](1258.63KB)
Abstract:

In this paper, a non-trivial generalization of a mathematical model put forward in [35] to account for the development of resistance by tumors to chemotherapy is presented. A study of the existence and local stability of the solutions, as well as the ultimate dynamics of the model, is addressed. An analysis of different chemotherapeutical protocols using discretization and optimization methods is carried out. A number of objective functionals are considered and the necessary optimality conditions are provided. Since the control variable appears linearly in the associated problem, optimal controls are concatenations of bang-bang and singular arcs. A formula of the singular control in terms of state and adjoint variables is derived analytically. Bang-bang and singular controls from the numerical simulations are obtained where, in particular, singular controls illustrate the metronomic chemotherapy.

2019, 24(5): 2039-2052 doi: 10.3934/dcdsb.2019083 +[Abstract](1043) +[HTML](47) +[PDF](1930.11KB)
Abstract:

This study investigates how optimal control theory may be used to delay the onset of chemotherapy resistance in tumours. An optimal control problem with simple tumour dynamics and an objective functional explicitly penalising drug resistant tumour phenotype is formulated. It is shown that for biologically relevant parameters the system has a single globally attracting positive steady state. The existence of singular arc is then investigated analytically under a very general form of the resistance penalty in the objective functional. It is shown that the singular controls are of order one and that they satisfy Legendre-Clebsch condition in a subset of the domain. A gradient method for solving the proposed optimal control problem is then used to find the control minimising the objective. The optimal control is found to consist of three intervals: full dose, singular and full dose. The singular part of the control is essential in delaying the onset of drug resistance.

2019, 24(5): 2053-2071 doi: 10.3934/dcdsb.2019084 +[Abstract](1220) +[HTML](45) +[PDF](7168.1KB)
Abstract:

Morphogenesis, the shaping of an organism, is a complex biological process accomplished through an well organized interplay between growth, differentiation and cell movement.It is still today one of the major outstanding problems in the biological sciences. Pattern formation has been well-addressed in the literature with the development of many mathematical models including the famous reaction-diffusion ones. We here take a different approach, introducing a controlled cellular automaton in order to model the signal molecules, known as growth factors, that convey information from one cell to another during an organism's development and help maintain the viability of the adult. This control represents extracellular structures that have been associated with the regulation of stem cell proliferation and are called fractones. In this paper we introduce two co-evolving automata, one describing the perturbed diffusion of growth factors and one accounting for the rules of basic cellular functions (proliferation, differentiation, migration and apoptosis). Fractones are introduced as an external input to control the shaping of multi-cellular organisms; we analyze their influence on the emerging shape. We illustrate our theory with 2 and 3 dimensional simulations. This work presents the foundation upon which to develop cellular automata as a tool to simulate the morphodynamics in embryonic development.

2019, 24(5): 2073-2091 doi: 10.3934/dcdsb.2019085 +[Abstract](1431) +[HTML](60) +[PDF](1149.16KB)
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Nonlinear dynamics of a reaction-diffusion equation with delay is studied with numerical simulations in 1D and 2D cases. Homogeneous in space solutions can manifest time oscillations with period doubling bifurcations and transition to chaos. Transition between two regions with homogeneous oscillations is provided by quasi-waves, propagating solutions without regular structure and often with complex aperiodic oscillations. Dynamics of space dependent solutions is described by a combination of various waves, e.g., bistable, monostable, periodic and quasi-waves.

2019, 24(5): 2093-2124 doi: 10.3934/dcdsb.2019086 +[Abstract](1188) +[HTML](46) +[PDF](576.86KB)
Abstract:

Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature of our results is that we allow the unknown parameters belonging to a mere complete separable metric space (not necessarily compact).

2019, 24(5): 2125-2147 doi: 10.3934/dcdsb.2019087 +[Abstract](1132) +[HTML](37) +[PDF](794.32KB)
Abstract:

In the Hes1 gene expression system the protein (present as dimers) bounds to the promoter of its own DNA blocking transcription of its mRNA. This negative feedback leads to an oscillatory behavior, which is observed experimentally. Classical mathematical model of this system consists of two ordinary differential equations with discrete time delay in the term reflecting transcription. However, transcription takes place in the nucleus while translation occurs in the cytoplasm. This means that the delay present in the system is larger than transcription time. Moreover, in reality it is not discrete but distributed around some mean value. In this paper we present the model of the Hes1 gene expression system and discuss similarities and differences between the model with discrete and distributed delays. It turns out that in the case of distributed delays the region of stability of the steady state is larger than in the case of discrete delay. We also derive conditions that guarantee stability of the steady state for particular delay distributions.

2019, 24(5): 2149-2167 doi: 10.3934/dcdsb.2019088 +[Abstract](1075) +[HTML](54) +[PDF](718.99KB)
Abstract:

We study mathematical properties of a model describing growth of primary brain tumours called low-grade gliomas (LGGs) and their response to chemotherapy. The motivation for considering this particular type of cancer is its large impact on society. LGGs affect mainly young adults and eventually result in death, despite the tumour growth rate being slow. The model studied consists of two non-autonomous ordinary differential equations and is a generalised version of the model proposed by Bogdańska et al. (Math. Biosci. 2017). We discuss the stability of stationary states, prove global stability of tumour-free steady state and, in some cases, justify the existence of periodic solutions. Assuming that chemotherapy effectiveness remains constant in time, we provide analytical estimates and calculate minimal doses of the drug that should eliminate the tumour for particular patients with LGGs.

2019, 24(5): 2169-2188 doi: 10.3934/dcdsb.2019089 +[Abstract](1157) +[HTML](41) +[PDF](10420.22KB)
Abstract:

In this paper we propose a model of tuberculosis (TB) transmission in a heterogeneous population consisting of two different subpopulations, like homeless and non-homeless people. We use the criss-cross model to describe the illness dynamics. This criss-cross model is based on the simple SIS model with constant inflow into both subpopulations and bilinear transmission function. We find conditions for the existence and local stability of stationary states (disease-free and endemic) and fit the model to epidemic data from Warmian-Masurian Province of Poland. Basic reproduction number \begin{document}$\mathcal{R}_0$\end{document} is considered as a threshold parameter for the general model. Applying local center manifold theory we show that when \begin{document}$\mathcal{R}_0 = 1$\end{document} a supercritical bifurcation occurs, and with \begin{document}$\mathcal{R}_0$\end{document} increasing above this threshold the disease-free stationary state loses stability and locally asymptotically stable endemic stationary state appears. Our analysis confirms the hypothesis that homeless individuals may be a specific reservoir of the pathogen and the disease may be transmitted from this subpopulation to the general population.

2019, 24(5): 2189-2204 doi: 10.3934/dcdsb.2019090 +[Abstract](1123) +[HTML](45) +[PDF](402.13KB)
Abstract:

We give a new proof of the maximum principle for optimal control problems with running state constraints. The proof uses the so-called method of \begin{document}$v-$\end{document}change of the time variable introduced by Dubovitskii and Milyutin. In this method, the time \begin{document}$t$\end{document} is considered as a new state variable satisfying the equation \begin{document}${\rm d} t/ {\rm d} \tau = v,$\end{document} where \begin{document}$v(\tau)\ge0$\end{document} is a new control and \begin{document}$\tau$\end{document} a new time. Unlike the general \begin{document}$v-$\end{document}change with an arbitrary \begin{document}$v(\tau),$\end{document} we use a piecewise constant \begin{document}$v.$\end{document} Every such \begin{document}$v-$\end{document}change reduces the original problem to a problem in a finite dimensional space, with a continuum number of inequality constrains corresponding to the state constraints. The stationarity conditions in every new problem, being written in terms of the original time \begin{document}$t,$\end{document} give a weak* compact set of normalized tuples of Lagrange multipliers. The family of these compacta is centered and thus has a nonempty intersection. An arbitrary tuple of Lagrange multipliers belonging to the latter ensures the maximum principle.

2019, 24(5): 2205-2217 doi: 10.3934/dcdsb.2019091 +[Abstract](1088) +[HTML](43) +[PDF](1337.2KB)
Abstract:

In the present paper we propose and study a simple model of collagen remodeling occurring in latter stage of tendon healing process. The model is an integro-differential equation describing the possibility of an alignment of collagen fibers in a finite time. We show that the solutions may either exist globally in time or blow-up in a finite time depending on initial data. The latter behavior can be related to the healing of injury without the scar formation in a finite time: a full alignment of collagen fibers. We believe that the present model is an essential ingredient of the full description of collagen remodeling.

2019, 24(5): 2219-2235 doi: 10.3934/dcdsb.2019092 +[Abstract](1422) +[HTML](55) +[PDF](495.98KB)
Abstract:

We discuss and compare numerical methods to solve singular optimal control problems by the direct method. Our discussion is illustrated by an Autonomous Underwater Vehicle (AUV) problem with state constraints. For this problem, we test four different approaches to solve numerically our problem via the direct method. After discretizing the optimal control problem we solve the resulting optimization problem with (ⅰ) A Mathematical Programming Language (\begin{document}$\text{AMPL}$\end{document}), (ⅱ) the Imperial College London Optimal Control Software (\begin{document}$\text{ICLOCS}$\end{document}), (ⅲ) the Gauss Pseudospectral Optimization Software (\begin{document}$\text{GPOPS}$\end{document}) as well as with (ⅳ) a new algorithm based on mixed-binary non-linear programming reported in [7]. This algorithm consists on converting the optimal control problem to a Mixed Binary Optimal Control (MBOC) problem which is then transcribed to a mixed binary non-linear programming problem (\begin{document}$\text{MBNLP}$\end{document}) problem using Legendre-Radau pseudospectral method. Our case study shows that, in contrast with the first three approaches we test (all relying on \begin{document}$\text{IPOPT}$\end{document} or other numerical optimization software packages like \begin{document}$\text{KNITRO}$\end{document}), the \begin{document}$\text{MBOC}$\end{document} approach detects the structure of the AUV's problem without a priori information of optimal control and computes the switching times accurately.

2019, 24(5): 2237-2250 doi: 10.3934/dcdsb.2019093 +[Abstract](1224) +[HTML](37) +[PDF](914.18KB)
Abstract:

Most clinical trials with combination therapy fail. One of the reasons is that not enough forethought is given to the interaction between the different agents, as well as the potential negative side-effects that may arise in the combined therapy. In the present paper we consider a generic cancer model with combination therapy consisting of chemotherapy agent \begin{document}$X$\end{document} and checkpoint inhibitor \begin{document}$A$\end{document}. We use a mathematical model to investigate the results of injecting different amounts \begin{document}$\gamma_X$\end{document} of \begin{document}$X$\end{document} and \begin{document}$\gamma_A$\end{document} of \begin{document}$A$\end{document}. We show that there are some regions in the \begin{document}$(\gamma_A,\gamma_X)$\end{document}-plane where as increase in \begin{document}$\gamma_X$\end{document} or \begin{document}$\gamma_A$\end{document} actually decreases the tumor volume; such 'regions of antagonism' should be avoided in clinical trials. We also show how to achieve the same level of tumor volume reduction with least negative-side effects, where the side-effects are represented by the level of inflammation of the tumor microenvironment.

2019, 24(5): 2251-2280 doi: 10.3934/dcdsb.2019094 +[Abstract](1061) +[HTML](41) +[PDF](557.99KB)
Abstract:

We consider a nonlinear system of differential equations describing a process of the psoriasis treatment. Its phase variables are the concentrations of T-lymphocytes, keratinocytes and dendritic cells. A scalar bounded control is introduced into this system to reflect the medication dosage. For such a control system, on a given time interval the minimization problem of the Bolza type functional is stated. Its terminal term is the concentration of keratinocytes at the terminal time, and its integral term is the product of the non-negative weighting coefficient with the total cost of the psoriasis treatment. This cost is linear in the control and proportional to the concentration of keratinocytes. For the analysis of such a problem, the Pontryagin maximum principle is used. As a result of this analysis, it is shown that if the weighting coefficient is zero, then the corresponding optimal control can contain a singular arc. We establish that it is a chattering control, and therefore does not make much sense as a type of a medical treatment. If the weighting coefficient is positive, then the corresponding optimal control is bang-bang, and it can be presented as a type of psoriasis treatment. In addition, when this coefficient tends to zero, such optimal controls can be considered as chattering approximations. Therefore, the convergence of these optimal controls, the corresponding optimal solutions of the original system, and the minimum values of the functional are studied. The obtained theoretical results are illustrated by numerical calculations and the corresponding conclusions are made.

2019, 24(5): 2281-2292 doi: 10.3934/dcdsb.2019095 +[Abstract](1023) +[HTML](38) +[PDF](369.44KB)
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In the paper, we derive a maximum principle for a Bolza problem described by an integro-differential equation of Volterra type. We use the Dubovitskii-Milyutin approach.

2019, 24(5): 2293-2313 doi: 10.3934/dcdsb.2019096 +[Abstract](1193) +[HTML](278) +[PDF](471.3KB)
Abstract:

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.

2019, 24(5): 2315-2334 doi: 10.3934/dcdsb.2019097 +[Abstract](1213) +[HTML](46) +[PDF](519.38KB)
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We analyze the structure of optimal protocols for a mathematical model of tumor anti-angiogenic treatment. The control represents the concentration of the agent and we consider the problem to administer an a priori given total amount of agents in order to achieve a minimum tumor volume/maximum tumor reduction. In earlier work, this problem was studied with a log-kill type pharmacodynamic model for drug effects which does not account for saturation of the drug concentration. Here we study the effect of incorporating a Michaelis-Menten (MM) or \begin{document}$E_{\max}$\end{document}-type pharmacodynamic model, the most commonly used model in the field of pharmacometrics. We compare the formulations of both problems and the resulting solutions. The reformulated problem with \begin{document}$E_{\max}$\end{document} pharmacodynamics is no longer linear in the control. This results in qualitative changes in the structure of optimal controls which, in line with an interpretation as concentrations, now are continuous while discontinuities exist if the log-kill model is used which is more in line with an interpretation of the control as dose rates. In spite of these qualitative differences, similarities in the structures of solutions can be observed. Both aspects are discussed theoretically and illustrated numerically.

2019, 24(5): 2335-2364 doi: 10.3934/dcdsb.2019098 +[Abstract](1312) +[HTML](48) +[PDF](657.45KB)
Abstract:

This article addresses the problem of controlling a constrained, continuous–time, nonlinear system through Model Predictive Control (MPC). In particular, we focus on methods to efficiently and accurately solve the underlying optimal control problem (OCP). In the numerical solution of a nonlinear OCP, some form of discretization must be used at some stage. There are, however, benefits in postponing the discretization process and maintain a continuous-time model until a later stage. This is because that way we can exploit additional freedom to select the number and the location of the discretization node points.We propose an adaptive time–mesh refinement (AMR) algorithm that iteratively finds an adequate time–mesh satisfying a pre–defined bound on the local error estimate of the obtained trajectories. The algorithm provides a time–dependent stopping criterion, enabling us to impose higher accuracy in the initial parts of the receding horizon, which are more relevant to MPC. Additionally, we analyze the conditions to guarantee closed–loop stability of the MPC framework using the AMR algorithm. The numerical results show that the proposed AMR strategy can obtain solutions as fast as methods using a coarse equidistant–spaced mesh and, on the other hand, as accurate as methods using a fine equidistant–spaced mesh. Therefore, the OCP can be solved, and the MPC law obtained, faster and/or more accurately than with discrete-time MPC schemes using equidistant–spaced meshes.

2019, 24(5): 2365-2381 doi: 10.3934/dcdsb.2019099 +[Abstract](1184) +[HTML](268) +[PDF](440.63KB)
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We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

2019, 24(5): 2383-2405 doi: 10.3934/dcdsb.2019100 +[Abstract](1176) +[HTML](39) +[PDF](6258.57KB)
Abstract:

We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.

2018  Impact Factor: 1.008