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Local indirect stabilization of same coupled evolution systems through resolvent estimates
A quantitative strong unique continuation property of a diffusive SIS model
| 1. | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA |
| 2. | College of Engineering, Huazhong Agricultural University, Wuhan 430070, China |
This article is concerned with a strong unique continuation property of solutions for a diffusive SIS (Susceptible - Infected - Susceptible, or SI) model, which belongs to a type of observability inequalities in a time interval $ [0, T] $. That is, if one can observe solution on a convex and connected bounded open set $ \omega $ in a bounded domain $ \Omega $ at time $ t = T $, then the norms of solution on $ [0,T) $ on $ \Omega $ are observable. In our discussion, boundary condition is a homogeneous Dirichlet one (hostile boundary condition).
References:
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M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai,
A vaccine model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524.
doi: 10.1137/030600370. |
| [2] |
M. E. Alexander, S. M. Moghadas, P. Rohani and A. R. Summers,
Modelling the effect of a booster vaccine model on disease epidemiology, J. Math. Bio., 52 (2006), 290-306.
doi: 10.1007/s00285-005-0356-0. |
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L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai,
Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Disc. Contin. Dyn. Syst., 21 (2008), 1-20.
doi: 10.3934/dcds.2008.21.1. |
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J. Arino, R. Jordan and P. van der Driessch,
Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60.
doi: 10.1016/j.mbs.2005.09.002. |
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F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second edition. Texts in Applied Mathematics, 40. Springer, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
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K. Deng and Y. Wu,
Dynamics of an SIS epidemic reation-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.
doi: 10.1017/S0308210515000864. |
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O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, 2000. |
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L. Escauriaza,
Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127.
doi: 10.1215/S0012-7094-00-10415-2. |
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L. Escauriaza and F. J. Fernández,
Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.
doi: 10.1007/BF02384566. |
| [10] |
L. Escauriaza, G. Seregin and V. Šverák,
Backward uniqueness for parabolic equations, Arch. Rational Mech. Anal., 169 (2003), 147-157.
doi: 10.1007/s00205-003-0263-8. |
| [11] |
L. Escauriaza and L. Vega,
Carleman inequalities and the heat operator II, Indiana Univ. Math. J., 50 (2001), 1149-1169.
doi: 10.1512/iumj.2001.50.1937. |
| [12] |
F. J. Fernández,
Unique continuation for parabolic operators II, Comm. Partial Differential Equations, 28 (2003), 1597-1604.
doi: 10.1081/PDE-120024523. |
| [13] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan,
A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain, SIAM J. Math. Anal., 33 (2001), 570-588.
doi: 10.1137/S0036141000371757. |
| [14] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan,
A reaction-diffusion system modeling direct and indirect transmission of diseases, Disc. Conti. Dyna. Syst. B, 4 (2004), 893-910.
doi: 10.3934/dcdsb.2004.4.893. |
| [15] |
N. Garofalo and F.-H. Lin,
Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.
doi: 10.1512/iumj.1986.35.35015. |
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G. Giodano,
Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.
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W. Huang, M. Han and K. Liu,
Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Bio. Eng., 7 (2010), 51-66.
doi: 10.3934/mbe.2010.7.51. |
| [19] |
C. E. Kenig,
Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle, Proc. Sympos. Pure Math., 79 (2008), 207-227.
doi: 10.1090/pspum/079/2500494. |
| [20] |
H. Koch and D. Tataru,
Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Comm. PDE, Comm. PDE (2009), 305-366.
doi: 10.1080/03605300902740395. |
| [21] |
I. Kukavica,
Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240.
doi: 10.1215/S0012-7094-98-09111-6. |
| [22] |
E. M. Landis and O. A. Oleinik,
Generalized analyticity and some related properties of solutions of elliptic and parabolic equations, Russian Mathematical Surveys, 29 (1974), 195-212.
|
| [23] |
F.-H. Lin,
A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136.
doi: 10.1002/cpa.3160430105. |
| [24] |
Y. Lou, Several spatial epidemic models in a heterogeneous environment with applications to COVID-19, Canada-China Distinguished Lecture, Mathematics and COVID-19, June 19, 2020. |
| [25] |
R. Peng,
Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, Part I, J. Diff. Eqn., 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002. |
| [26] |
K. D. Phung,
Note on the cost of the approximate controllability for the heat equation with potential, J. Math. Anal. Appl., 295 (2004), 527-238.
doi: 10.1016/j.jmaa.2004.03.059. |
| [27] |
K. D. Phung,
Carleman commutator approach in logarithmic convexity for parabolic equations, Math. Control Rela. Fields, 8 (2018), 899-933.
doi: 10.3934/mcrf.2018040. |
| [28] |
K. D. Phung and G. Wang,
Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.
doi: 10.1016/j.jfa.2010.04.015. |
| [29] |
K. D. Phung and G. Wang,
An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
| [30] |
P. Polacik, Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds, Handbook of Dynamical Systems 2,835–883, Hand. Differ. Equ., North-Holland, Amsterdam, 2002.
doi: 10.1016/S1874-575X(02)80037-6. |
| [31] |
C.-C. Poon,
Unique continuation for parabolic equations, Comm. PDE, 21 (1996), 521-539.
doi: 10.1080/03605309608821195. |
| [32] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, Springer, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
| [33] |
Y. Wu and X. Zou,
Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Diff. Equ., 261 (2016), 4424-4447.
doi: 10.1016/j.jde.2016.06.028. |
show all references
References:
| [1] |
M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai,
A vaccine model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524.
doi: 10.1137/030600370. |
| [2] |
M. E. Alexander, S. M. Moghadas, P. Rohani and A. R. Summers,
Modelling the effect of a booster vaccine model on disease epidemiology, J. Math. Bio., 52 (2006), 290-306.
doi: 10.1007/s00285-005-0356-0. |
| [3] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai,
Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Disc. Contin. Dyn. Syst., 21 (2008), 1-20.
doi: 10.3934/dcds.2008.21.1. |
| [4] |
J. Arino, R. Jordan and P. van der Driessch,
Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60.
doi: 10.1016/j.mbs.2005.09.002. |
| [5] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second edition. Texts in Applied Mathematics, 40. Springer, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
| [6] |
K. Deng and Y. Wu,
Dynamics of an SIS epidemic reation-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.
doi: 10.1017/S0308210515000864. |
| [7] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, 2000. |
| [8] |
L. Escauriaza,
Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127.
doi: 10.1215/S0012-7094-00-10415-2. |
| [9] |
L. Escauriaza and F. J. Fernández,
Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.
doi: 10.1007/BF02384566. |
| [10] |
L. Escauriaza, G. Seregin and V. Šverák,
Backward uniqueness for parabolic equations, Arch. Rational Mech. Anal., 169 (2003), 147-157.
doi: 10.1007/s00205-003-0263-8. |
| [11] |
L. Escauriaza and L. Vega,
Carleman inequalities and the heat operator II, Indiana Univ. Math. J., 50 (2001), 1149-1169.
doi: 10.1512/iumj.2001.50.1937. |
| [12] |
F. J. Fernández,
Unique continuation for parabolic operators II, Comm. Partial Differential Equations, 28 (2003), 1597-1604.
doi: 10.1081/PDE-120024523. |
| [13] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan,
A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain, SIAM J. Math. Anal., 33 (2001), 570-588.
doi: 10.1137/S0036141000371757. |
| [14] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan,
A reaction-diffusion system modeling direct and indirect transmission of diseases, Disc. Conti. Dyna. Syst. B, 4 (2004), 893-910.
doi: 10.3934/dcdsb.2004.4.893. |
| [15] |
N. Garofalo and F.-H. Lin,
Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.
doi: 10.1512/iumj.1986.35.35015. |
| [16] |
G. Giodano,
Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.
|
| [17] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. |
| [18] |
W. Huang, M. Han and K. Liu,
Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Bio. Eng., 7 (2010), 51-66.
doi: 10.3934/mbe.2010.7.51. |
| [19] |
C. E. Kenig,
Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle, Proc. Sympos. Pure Math., 79 (2008), 207-227.
doi: 10.1090/pspum/079/2500494. |
| [20] |
H. Koch and D. Tataru,
Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Comm. PDE, Comm. PDE (2009), 305-366.
doi: 10.1080/03605300902740395. |
| [21] |
I. Kukavica,
Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240.
doi: 10.1215/S0012-7094-98-09111-6. |
| [22] |
E. M. Landis and O. A. Oleinik,
Generalized analyticity and some related properties of solutions of elliptic and parabolic equations, Russian Mathematical Surveys, 29 (1974), 195-212.
|
| [23] |
F.-H. Lin,
A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136.
doi: 10.1002/cpa.3160430105. |
| [24] |
Y. Lou, Several spatial epidemic models in a heterogeneous environment with applications to COVID-19, Canada-China Distinguished Lecture, Mathematics and COVID-19, June 19, 2020. |
| [25] |
R. Peng,
Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, Part I, J. Diff. Eqn., 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002. |
| [26] |
K. D. Phung,
Note on the cost of the approximate controllability for the heat equation with potential, J. Math. Anal. Appl., 295 (2004), 527-238.
doi: 10.1016/j.jmaa.2004.03.059. |
| [27] |
K. D. Phung,
Carleman commutator approach in logarithmic convexity for parabolic equations, Math. Control Rela. Fields, 8 (2018), 899-933.
doi: 10.3934/mcrf.2018040. |
| [28] |
K. D. Phung and G. Wang,
Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.
doi: 10.1016/j.jfa.2010.04.015. |
| [29] |
K. D. Phung and G. Wang,
An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
| [30] |
P. Polacik, Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds, Handbook of Dynamical Systems 2,835–883, Hand. Differ. Equ., North-Holland, Amsterdam, 2002.
doi: 10.1016/S1874-575X(02)80037-6. |
| [31] |
C.-C. Poon,
Unique continuation for parabolic equations, Comm. PDE, 21 (1996), 521-539.
doi: 10.1080/03605309608821195. |
| [32] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, Springer, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
| [33] |
Y. Wu and X. Zou,
Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Diff. Equ., 261 (2016), 4424-4447.
doi: 10.1016/j.jde.2016.06.028. |
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