On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies

Ledzewicz, Urszula; Schättler, Heinz

doi:10.1007/s10957-020-01754-2

Cited by 16 publications

(22 citation statements)

References 47 publications

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“…This controversy leads to a question regarding the extent to which results obtained from a specific objective functional are valid and can be trusted. Unfortunately, despite its obvious practical relevance, so far this issue has attracted rather small attention: so far the authors were able to find only two publications [13,14] that address this issue. Both publications are of the same authors and are dealing with within-a-patient disease dynamics.…”

Section: Discussionmentioning

confidence: 99%

“…Another similar related problem is a form of the functional dependency of the cost on the state variables and the impact of this on the optimal control. However, despite the apparent practical relevance of this problem, very little mathematical work was so far done in this direction: so far, the authors were able to find only two publications [13,14] that address this issue.…”

Section: Introductionmentioning

confidence: 99%

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Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

Saldaña¹,

Camacho-Gutíerrez²,

Korobeinikov³

2020

Preprint

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In applications of the optimal control theory to problems in medicine and biology, the dependency of the objective functional on the control itself is often a matter of controversy. In this paper, we explore the impact of the dependency using reasonably simple SIR and SEIRS epidemic models. To qualitatively compare the outcomes for different objective functionals, we apply the cost-effectiveness analysis. Our result shows that, at least for the comparatively inexpensive controls, the variation of the power at the controls in a biologically feasible range does not significantly affect the forms of the optimal controls and the corresponding optimal state solutions. Moreover, the costs and effectiveness are affected even less. At the same time, the dependency of the cost on the state variables can be very significant.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

Saldaña¹,

Camacho-Gutíerrez²,

Korobeinikov³

2020

Preprint

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show abstract

“…The model that we investigate here is a modified version of the Stepanova model [28] that has been extensively used in the literature, we cite for example [8], [17] and [18] where optimal control approaches were proposed to schedule chemo-and immunotherapy injection profiles. Furthermore, [26] proposed a multiple model predictive control scheme to design chemo-and immunotherapy injection schedules.…”

Section: Dynamical Modelmentioning

confidence: 99%

Robust domain of attraction estimation for a tumor growth model

Moussa

Fiacchini

Alamir

2021

Applied Mathematics and Computation

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This paper deals with the estimation of regions of attraction (RoAs) for a cancer dynamical model. The estimation of this type of sets is very interesting in the field of control for cancer dynamics, since it provides the set of possible initial health indicators, for which there exists a treatment protocol allowing to heal the patient. Therefore, we propose a methodology to estimate the region of attraction of a nonlinear dynamical system describing the interaction between a tumor, the immune system and combined therapies of cancer. Then, we use the characterization of the RoA for a given model parameters vector, in order to derive an estimation of an outer approximation of the robust RoA under parametric uncertainties.

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“…The quadratic cost function that we use in this paper qualitatively reflects this situation and, at the same time, is mathematically convenient. In addition, there are works 6–8 showing that in epidemiological problems, the dependence of the optimal control on the power of control

u (t)

in the objective function is not significant. Hence, the optimal controls obtained in the problems with the payoff objective function () can be considered as a good opportunity for implementation in real life.…”

Section: Optimal Control Problemsmentioning

confidence: 99%

Optimal quarantine‐related strategies for COVID‐19 control models

2021

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At the time when this paper was written, quarantine‐related strategies (from full lockdown to some relaxed preventive measures) were the only available measure to control coronavirus disease 2019 (COVID‐19) epidemic. However, long‐term quarantine and especially full lockdown is an extremely expensive measure. To explore the possibility of controlling and suppressing the COVID‐19 epidemic at the lowest possible cost, we apply optimal control theory. In this paper, we create two controlled Susceptible‐Exposed‐Infectious‐Removed (SEIR) type models describing the spread of COVID‐19 in a human population. For each model, we solve an optimal control problem and find the optimal quarantine strategy that ensures the minimal level of the infected population at the lowest possible cost. The properties of the corresponding optimal controls are established analytically using the Pontryagin maximum principle. The optimal solutions, obtained numerically, validate our analytical results. Additionally, for both controlled models, we find explicit formulas for the basic reproductive ratios in the presence of a constant control and show that while the epidemic can be eventually stopped under long‐term quarantine measures of maximum strength (full lockdown), the strength of quarantine can be reduced under the optimal quarantine policies. The behavior of the appropriate optimal solutions and their dependence on the basic reproductive ratio, population density, and the duration of quarantine are discussed, and practically relevant conclusions are made.

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On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies

Cited by 16 publications

References 47 publications

Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

Impact of a cost functional on the optimal control and the cost-effectiveness: control of a spreading infection as a case study

Robust domain of attraction estimation for a tumor growth model

Optimal quarantine‐related strategies for COVID‐19 control models

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