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doi: 10.3934/dcdsb.2018263

Global eradication for spatially structured populations by regional control

1. 

Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iaşi, "Octav Mayer" Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania

2. 

ADAMSS (Centre for Advanced Applied Mathematical and Statistical Sciences), Universitá degli Studi di Milano, 20133 Milano, Italy

3. 

Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iaşi, Iaşi 700506, Romania

* Corresponding author: Vincenzo Capasso

Received  November 2017 Revised  May 2018 Published  October 2018

This paper concerns problems for the eradication of apopulation by acting on a subregion ω. The dynamics isdescribed by a general reaction-diffusion system, including one ormore populations, subject to a vital dynamics with either locallogistic or nonlocal logistic terms. For the one populationcase, a necessary condition and a sufficient condition foreradicability (zero-stabilizability) are obtained, in terms of the sign of the principaleigenvalue of a suitable elliptic operator acting on the domain$Ω \setminus \overline{ω }$. A feedbackharvesting-like control with a large constant harvesting raterealizes eradication of the population. The problem oferadication is then reformulated in a more convenient way, bytaking into account the total cost of the damages produced by apest population and the costs related to the choice of therelevant subregion, and approximated by a regional optimalcontrol problem with a finite horizon. A conceptual iterativealgorithm is formulated for the simulation of the proposedoptimal control problem. Numerical tests are given to illustratethe effectiveness of the results. Relevant regional controlproblems for two populations reaction-diffusion models, such asprey-predator system, and an SIR epidemic system with spatialstructure and local/nonlocal force of infection, have beenanalyzed too.

Citation: Sebastian Aniţa, Vincenzo Capasso, Ana-Maria Moşneagu. Global eradication for spatially structured populations by regional control. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018263
References:
[1]

S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Acad. Publ., Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.

[2]

S. Aniţa, V. Arnăutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics. From Mathematical Models to Numerical Simulation with MATLAB, Birkhäuser, Basel, 2011. doi: 10.1007/978-0-8176-8098-5.

[3]

S. Aniţa and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic model, Nonlin. Anal. Real World Appl., 3 (2002), 453-464. doi: 10.1016/S1468-1218(01)00025-6.

[4]

S. Aniţa and V. Capasso, A stabilization strategy for a reaction-diffusion system modelling a class of spatially structured epidemic systems (think globally, act locally), Nonlin. Anal. Real World Appl, 10 (2009), 2026-2035. doi: 10.1016/j.nonrwa.2008.03.009.

[5]

S. Aniţa and V. Capasso, On the stabilization of reaction-diffusion systems modelling a class of man-environment epidemics: A review, Math. Meth. Appl. Sci., 33 (2010), 1235-1244. doi: 10.1002/mma.1267.

[6]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control, Nonlin. Anal. Real World Appl., 13 (2012), 725-735. doi: 10.1016/j.nonrwa.2011.08.012.

[7]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling malaria transmission, Discrete Cont. Dynam. Syst., Series B, 17 (2012), 1673-1684. doi: 10.3934/dcdsb.2012.17.1673.

[8]

S. AniţaV. Capasso and A.-M. Moşneagu, Regional control in optimal harvesting problems of population dynamics, Nonlin. Anal., 147 (2016), 191-212. doi: 10.1016/j.na.2016.09.008.

[9]

S. AniţaW.-E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a class of predator-prey systems posed on non coincident spatial domains, Discrete Cont. Dyn. Sys., Series B, 11 (2009), 805-822. doi: 10.3934/dcdsb.2009.11.805.

[10]

V. Arnăutu and P. Neittaanmäki, Optimal Control from Theory to Computer Programs, Kluwer Acad. Publ., Dordrecht, 2003. doi: 10.1007/978-94-017-2488-3.

[11]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in Nonlinear Diffusion (W.E. Fitzgibbon, A.F. Walker, Eds.), Pitman, London, 1977.

[12]

V. Barbu, Partial Differention Equations and Boundary Value Problems, Kluwer Acad. Publ., Dordrecht, 1998. doi: 10.1007/978-94-015-9117-1.

[13]

A. O. Belyakov and V. M. Veliov, On optimal harvesting in age-structured populations, in Dynamic Perspectives on Managerial Decision Making (H. Dawid, K.F. Doerner, G. Feichtinger, P.M. Kort, A. Seidl, Eds.), Springer Internat. Publ., 22 (2016), 1186-1202. doi: 10.1007/978-3-319-39120-5_9.

[14]

A. BressanG. M. Coclite and W. Shen, A multidimensional optimal-harvesting problem with measure-valued solutions, SIAM J. Control Optim., 51 (2013), 1186-1202. doi: 10.1137/110853510.

[15]

D. Bucur and G. Buttazzo, Variational Methods in Some Shape Optimization Problems, Notes of Courses Given by the Teachers at the School, Scuola Normale Superiore, Pisa, 2002.

[16]

V. Capasso, Mathematical Structures of Epidemic Systems(corrected 2nd printing), Lecture Notes in Biomathematics, Vol. 97, Springer-Verlag, Heidelberg, 2008.

[17]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Process, 10 (2001), 266-277. doi: 10.1109/83.902291.

[18]

G. M. Coclite and M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim, 55 (2017), 913-935. doi: 10.1137/16M1061886.

[19]

G. M. CocliteM. Garavello and L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 865-900. doi: 10.3934/dcdss.2018053.

[20]

M.C. Delfour and J.-P. Zolesio, Shapes and Geometries. Metrics, Analysis, Differential Calculus and Optimization. Second Edition, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719826.

[21]

K. R. Fister and S. Lenhart, Optimal harvesting in an age-structured predator-prey model, Appl. Math. Optim., 54 (2006), 1-15. doi: 10.1007/s00245-005-0847-9.

[22]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.

[23]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004.

[24]

P. Getreuer, Chan-Vese Segmentation, IPOL J. Image Process, Online, 2 (2012), 214-224. doi: 10.5201/ipol.2012.g-cv.

[25]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Mathématiques et Applications, Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[26]

F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, Philadelphia, 1975.

[27]

N. Hritonenko and Y. Yatsenko, Optimization of harvesting age in integral age-dependent model of population dynamics, Math. Biosci., 195 (2005), 154-167. doi: 10.1016/j.mbs.2005.03.001.

[28]

D. G. Kendall, Mathematical models of the spread of infection, in Mathematics and Computer Science in Biology and Medicine, H.M.S.O., London, (1965), 213-225.

[29]

S. Lenhart, Using optimal control of parabolic PDEs to investigate population questions, NIMBioS, April 9-11, 2014; https://www.fields.utoronto.ca/programs/scientific/13-14/ BIOMAT/presentations/lenhartToronto3.pdf.

[30]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall, 2007.

[31]

Z. Luo, Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model, Appl. Math. Comput., 186 (2007), 1742-1752. doi: 10.1016/j.amc.2006.08.168.

[32]

Z. LuoW. T. Li and M. Wang, Optimal harvesting control problem for linear periodic age-dependent population dynamics, Appl. Math. Comput., 151 (2004), 789-800. doi: 10.1016/S0096-3003(03)00536-8.

[33]

J. Ohser and F. Mücklich, Statistical Analysis of Microstructures in Materials Science. Statistics in Practice, John Wiley & Sons, New York, 2000.

[34]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, New York, 2003. doi: 10.1007/b98879.

[35]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[36]

J. A. Sethian, Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Univ. Press, Cambridge, 1999.

[37]

J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.

show all references

References:
[1]

S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Acad. Publ., Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.

[2]

S. Aniţa, V. Arnăutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics. From Mathematical Models to Numerical Simulation with MATLAB, Birkhäuser, Basel, 2011. doi: 10.1007/978-0-8176-8098-5.

[3]

S. Aniţa and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic model, Nonlin. Anal. Real World Appl., 3 (2002), 453-464. doi: 10.1016/S1468-1218(01)00025-6.

[4]

S. Aniţa and V. Capasso, A stabilization strategy for a reaction-diffusion system modelling a class of spatially structured epidemic systems (think globally, act locally), Nonlin. Anal. Real World Appl, 10 (2009), 2026-2035. doi: 10.1016/j.nonrwa.2008.03.009.

[5]

S. Aniţa and V. Capasso, On the stabilization of reaction-diffusion systems modelling a class of man-environment epidemics: A review, Math. Meth. Appl. Sci., 33 (2010), 1235-1244. doi: 10.1002/mma.1267.

[6]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control, Nonlin. Anal. Real World Appl., 13 (2012), 725-735. doi: 10.1016/j.nonrwa.2011.08.012.

[7]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling malaria transmission, Discrete Cont. Dynam. Syst., Series B, 17 (2012), 1673-1684. doi: 10.3934/dcdsb.2012.17.1673.

[8]

S. AniţaV. Capasso and A.-M. Moşneagu, Regional control in optimal harvesting problems of population dynamics, Nonlin. Anal., 147 (2016), 191-212. doi: 10.1016/j.na.2016.09.008.

[9]

S. AniţaW.-E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a class of predator-prey systems posed on non coincident spatial domains, Discrete Cont. Dyn. Sys., Series B, 11 (2009), 805-822. doi: 10.3934/dcdsb.2009.11.805.

[10]

V. Arnăutu and P. Neittaanmäki, Optimal Control from Theory to Computer Programs, Kluwer Acad. Publ., Dordrecht, 2003. doi: 10.1007/978-94-017-2488-3.

[11]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in Nonlinear Diffusion (W.E. Fitzgibbon, A.F. Walker, Eds.), Pitman, London, 1977.

[12]

V. Barbu, Partial Differention Equations and Boundary Value Problems, Kluwer Acad. Publ., Dordrecht, 1998. doi: 10.1007/978-94-015-9117-1.

[13]

A. O. Belyakov and V. M. Veliov, On optimal harvesting in age-structured populations, in Dynamic Perspectives on Managerial Decision Making (H. Dawid, K.F. Doerner, G. Feichtinger, P.M. Kort, A. Seidl, Eds.), Springer Internat. Publ., 22 (2016), 1186-1202. doi: 10.1007/978-3-319-39120-5_9.

[14]

A. BressanG. M. Coclite and W. Shen, A multidimensional optimal-harvesting problem with measure-valued solutions, SIAM J. Control Optim., 51 (2013), 1186-1202. doi: 10.1137/110853510.

[15]

D. Bucur and G. Buttazzo, Variational Methods in Some Shape Optimization Problems, Notes of Courses Given by the Teachers at the School, Scuola Normale Superiore, Pisa, 2002.

[16]

V. Capasso, Mathematical Structures of Epidemic Systems(corrected 2nd printing), Lecture Notes in Biomathematics, Vol. 97, Springer-Verlag, Heidelberg, 2008.

[17]

T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Process, 10 (2001), 266-277. doi: 10.1109/83.902291.

[18]

G. M. Coclite and M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim, 55 (2017), 913-935. doi: 10.1137/16M1061886.

[19]

G. M. CocliteM. Garavello and L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 865-900. doi: 10.3934/dcdss.2018053.

[20]

M.C. Delfour and J.-P. Zolesio, Shapes and Geometries. Metrics, Analysis, Differential Calculus and Optimization. Second Edition, SIAM, Philadelphia, 2011. doi: 10.1137/1.9780898719826.

[21]

K. R. Fister and S. Lenhart, Optimal harvesting in an age-structured predator-prey model, Appl. Math. Optim., 54 (2006), 1-15. doi: 10.1007/s00245-005-0847-9.

[22]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.

[23]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004.

[24]

P. Getreuer, Chan-Vese Segmentation, IPOL J. Image Process, Online, 2 (2012), 214-224. doi: 10.5201/ipol.2012.g-cv.

[25]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Mathématiques et Applications, Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[26]

F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, Philadelphia, 1975.

[27]

N. Hritonenko and Y. Yatsenko, Optimization of harvesting age in integral age-dependent model of population dynamics, Math. Biosci., 195 (2005), 154-167. doi: 10.1016/j.mbs.2005.03.001.

[28]

D. G. Kendall, Mathematical models of the spread of infection, in Mathematics and Computer Science in Biology and Medicine, H.M.S.O., London, (1965), 213-225.

[29]

S. Lenhart, Using optimal control of parabolic PDEs to investigate population questions, NIMBioS, April 9-11, 2014; https://www.fields.utoronto.ca/programs/scientific/13-14/ BIOMAT/presentations/lenhartToronto3.pdf.

[30]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall, 2007.

[31]

Z. Luo, Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model, Appl. Math. Comput., 186 (2007), 1742-1752. doi: 10.1016/j.amc.2006.08.168.

[32]

Z. LuoW. T. Li and M. Wang, Optimal harvesting control problem for linear periodic age-dependent population dynamics, Appl. Math. Comput., 151 (2004), 789-800. doi: 10.1016/S0096-3003(03)00536-8.

[33]

J. Ohser and F. Mücklich, Statistical Analysis of Microstructures in Materials Science. Statistics in Practice, John Wiley & Sons, New York, 2000.

[34]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, New York, 2003. doi: 10.1007/b98879.

[35]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[36]

J. A. Sethian, Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Univ. Press, Cambridge, 1999.

[37]

J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.

Figure 1.  The representation of final iteration of $\omega$ for $\alpha \in \{5,6,7,8,9,10\}$ and $\beta = 0$
Figure 2.  The representation of final iteration of $\omega$ for $\alpha \in \{0.2, 0.3, 0.4, 0.45, 0.5\}$ and $\beta = 0.3$
Figure 3.  The representation of initial and final iterations of $\omega$ for $\alpha \in \{0.5, 1, 2.5\}$ and $\beta = 0.001$
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