In this paper, we propose a free final time optimal control approach applied to 4 ordinary differential equations which describe the tumor‐immune interactions after the injection of the bacillus Calmette‐Guérin (BCG) in the bladder of a hypothetical patient. The main goal of this optimal control strategy is to find the optimal dosage amount needed in each instillation of BCG for stimulating the immune‐system and then killing superficial bladder tumors and to determine the optimal duration of treatment, adequate for stopping the intravesical therapy with lesser side‐effects. For this, we introduce into the model of interest, a control function which represents the dose of BCG immunotherapy procedure and we formulate a minimization problem where the final time is considered free (nonfixed). The characterization of the sought optimal control noted u∗ is derived based on Pontryagin's maximum principle, while the formulation of the sought optimal final time noted
tfinal∗ is based on formulae of sensitivity which are obtained conditions from the derivative of the objective function with respect to
tfinal∗. We investigate the resolution of the free final time optimal control problem in 3 possible cases: (a) when
tfinal∗ is quadratic in the final gain function, (b) when the final gain function does not depend on
tfinal∗, and (c) when
tfinal∗ is linear in the final gain function. Finally, we obtain the sought optimal dose of BCG, and we conclude that in case (a), we obtain an optimal duration which is more beneficial regarding the activation of immune cells while cases (b) and (c) both provide an optimal duration which is more adequate for the minimization of the tumoral population.
The purpose of this paper is modelling and controlling the spread of COVID-19 disease in Morocco. A nonlinear mathematical model with two subclasses of infectious individuals is proposed. The population is divided into five classes, namely, susceptible (S), exposed (E), undiagnosed infectious (
I
n
c
), diagnosed patients (
I
c
), and removed individuals. To reflect the real dynamic of the COVID-19 transmission in Morocco, the real reported data are used for estimating model parameters. Two controls representing screening effort and limited treatment are considered. Based on viability theory and set-valued analysis, a Lyapunov function is constructed such that both exposed and infected populations are decreased to zero asymptotically. The corresponding controls are derived via a continuous selection of adequately designed feedback map. Numerical simulations are presented with three scenarios (cases when each control is used alone and the case when two controls are combined). Our results show that when only one control is to be applied, screening is the most effective in decreasing the number of people in the three infected compartments, whereas combining both controls is found to be highly effective and leads to a significant improvement in the epidemiological situation of Morocco. To the best of our knowledge, this work is the first one that applies the set-valued approach to a controlled COVID-19 model which agrees with the observed cases in Morocco.
Optimal control problems are an important mathematic tool used to reduce infectious diseases, most of works in this area considered time constant. In this paper, we present a free terminal optimal time control of Chikungunya epidemic model, which is an arthropod-borne virus (arbovirus) transmitted by mosquitoes of Aedes genus, with the order to give a minimum duration needed to reduce the infected group of both human and vector. We present a control simulating program using Matlab routines. The optimal control and the optimal final time are found using Pontryagin's maximum principle and the additional transversality condition for the terminal time. We solved the optimality system by an iterative method, then we confirm the performance of the optimization strategy by numerical simulations.
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