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June  2019, 24(6): 2511-2533. 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, 2019, 24 (6) : 2511-2533. doi: 10.3934/dcdsb.2018263
References:
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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

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[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. Google Scholar

[16]

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

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[30]

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

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[37]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[30]

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

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[37]

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

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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