June  2022, 15(6): 1599-1614. doi: 10.3934/dcdss.2022024

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

* Corresponding author: Taige Wang

Received  August 2021 Revised  December 2021 Published  June 2022 Early access  February 2022

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).

Citation: Taige Wang, Dihong Xu. A quantitative strong unique continuation property of a diffusive SIS model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1599-1614. doi: 10.3934/dcdss.2022024
References:
[1]

M. E. AlexanderC. BowmanS. M. MoghadasR. SummersA. 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. AlexanderS. M. MoghadasP. 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. AllenB. M. BolkerY. 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. ArinoR. 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.

[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.

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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. EscauriazaG. 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. FitzgibbonM. 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. FitzgibbonM. 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. HuangM. 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. AlexanderC. BowmanS. M. MoghadasR. SummersA. 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. AlexanderS. M. MoghadasP. 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. AllenB. M. BolkerY. 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. ArinoR. 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. EscauriazaG. 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. FitzgibbonM. 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. FitzgibbonM. 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. HuangM. 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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