The interplay between models and public health policies: Regional control for a class of spatially structured epidemics <em>(think globally, act locally)</em>

Vincenzo Capasso; Sebastian AniȚa; Vincenzo Capasso; Sebastian AniȚa

doi:10.3934/mbe.2018001

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 1: 1-20. doi: 10.3934/mbe.2018001

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The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally)

Vincenzo Capasso ^{1
,
,},
Sebastian AniȚa ²

1.
ADAMSS, Universitá degli Studi di Milano, 20133 MILANO, Italy
2.
Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iași, "Octav Mayer" Institute of Mathematics of the Romanian Academy, Iași 700506, Romania

Received: 12 January 2016 Accepted: 30 October 2016 Published: 01 February 2018
MSC : Primary: 35K57, 92D30, 93C20, 93D15; Secondary: 00-02, 92-02, 92-03

A review is presented here of the research carried out, by a group including the authors, on the mathematical analysis of epidemic systems. Particular attention is paid to recent analysis of optimal control problems related to spatially structured epidemics driven by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so called zero stabilization. In a series of papers, necessary conditions, and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. The stabilizability with a feedback control of harvesting type is related to the magnitude of the principal eigenvalue of a certain operator. The problem of finding the optimal position (by translation) of the support of the feedback stabilizing control is faced, in order to minimize both the infected population and the pollutant at a certain finite time.
- Epidemic systems,
- nonlinear models,
- man-environment epidemics,
- reaction-diffusion systems,
- stabilization,
- principal eigenvalue,
- feedback control,
- optimal regional control
Citation: Vincenzo Capasso, Sebastian AniȚa. The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally)[J]. Mathematical Biosciences and Engineering, 2018, 15(1): 1-20. doi: 10.3934/mbe.2018001

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Abstract

A review is presented here of the research carried out, by a group including the authors, on the mathematical analysis of epidemic systems. Particular attention is paid to recent analysis of optimal control problems related to spatially structured epidemics driven by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so called zero stabilization. In a series of papers, necessary conditions, and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. The stabilizability with a feedback control of harvesting type is related to the magnitude of the principal eigenvalue of a certain operator. The problem of finding the optimal position (by translation) of the support of the feedback stabilizing control is faced, in order to minimize both the infected population and the pollutant at a certain finite time.

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