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<p>In this work, we investigate the calibration of diffusion coefficients in a reaction-diffusion system arising in the modelling of the spatial-temporal dynamics of an epidemic. The model assumes a population partitioned into two compartments: susceptible and infected individuals. It captures a scenario with no lasting immunity, allowing recovered individuals to return to the susceptible class. We consider that the diffusion is given by a diagonal matrix with unknown space-dependent entries that should be determined by the knowledge of susceptible and infected populations at some fixed time. The identification of the diffusion parameter is posed as an optimal control problem, where the objective functional is the comparison between the solution of the reaction-diffusion system and the observed data in the <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{2}$</EquationSource> </InlineEquation>-norm, supplemented by a regularization term in <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$L^{2}$</EquationSource> </InlineEquation>-norm of the diffusion coefficient. The analysis of the optimal control formulation is carried out using the Dubovitskii–Milyutin Methodology. We obtain the following main results: the existence of strictly positive solutions for the state equations, demonstrating the existence of optimal solutions, and the necessary optimality conditions of first order.</p>

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The diffusion determination in a reaction-diffusion system by application of Dubovitskii-Milyutin’s formalism

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月29日

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  • Anibal Coronel,
  • Alex Tello,
  • Camila Isoton,
  • Fernando Huancas

摘要

In this work, we investigate the calibration of diffusion coefficients in a reaction-diffusion system arising in the modelling of the spatial-temporal dynamics of an epidemic. The model assumes a population partitioned into two compartments: susceptible and infected individuals. It captures a scenario with no lasting immunity, allowing recovered individuals to return to the susceptible class. We consider that the diffusion is given by a diagonal matrix with unknown space-dependent entries that should be determined by the knowledge of susceptible and infected populations at some fixed time. The identification of the diffusion parameter is posed as an optimal control problem, where the objective functional is the comparison between the solution of the reaction-diffusion system and the observed data in the L 2 $L^{2}$ -norm, supplemented by a regularization term in L 2 $L^{2}$ -norm of the diffusion coefficient. The analysis of the optimal control formulation is carried out using the Dubovitskii–Milyutin Methodology. We obtain the following main results: the existence of strictly positive solutions for the state equations, demonstrating the existence of optimal solutions, and the necessary optimality conditions of first order.