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Preface
Global stability of the steady states of an epidemic model incorporating intervention strategies
1. | School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China |
2. | Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA |
In this paper, we investigate the global stability of the steady states of a general reaction-diffusion epidemiological model with infection force under intervention strategies in a spatially heterogeneous environment. We prove that the reproduction number $\mathcal{R}_0$ can be played an essential role in determining whether the disease will extinct or persist: if $\mathcal{R}_0<1$, there is a unique disease-free equilibrium which is globally asymptotically stable; and if $\mathcal{R}_0>1$, there exists a unique endemic equilibrium which is globally asymptotically stable. Furthermore, we study the relation between $\mathcal{R}_0$ with the diffusion and spatial heterogeneity and find that, it seems very necessary to create a low-risk habitat for the population to effectively control the spread of the epidemic disease. This may provide some potential applications in disease control.
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Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.
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Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems-A, 21 (2008), 1-20.
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Health communication during SARS, Emerging Infectious Diseases, 10 (2004), 377-380.
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M. P. Brinn, K. V. Carson, A. J. Esterman, A. B. Chang and B. J. Smith,
Cochrane review: Mass media interventions for preventing smoking in young people, Evidence-Based Child Health: A Cochrane Review Journal, 7 (2012), 86-144.
doi: 10.1002/ebch.1808. |
[5] |
Y. Cai, Y. Kang, M. Banerjee and W. Wang,
A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502.
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[6] |
Y. Cai and W. M. Wang,
Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 989-1013.
doi: 10.3934/dcdsb.2015.20.989. |
[7] |
Y. Cai and W. M. Wang,
Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Analysis: Real World Applications, 30 (2016), 99-125.
doi: 10.1016/j.nonrwa.2015.12.002. |
[8] |
Y. Cai, Z. Wang and W. M. Wang,
Endemic dynamics in a host-parasite epidemiological model within spatially heterogeneous environment, Applied Mathematics Letters, 61 (2016), 129-136.
doi: 10.1016/j.aml.2016.05.011. |
[9] |
R. S. Cantrell and C. Cosner,
Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., 2003.
doi: 10.1002/0470871296. |
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J. Cui, X. Tao and H. Zhu,
An SIS infection model incorporating media coverage, Journal of Mathematics, 38 (2008), 1323-1334.
doi: 10.1216/RMJ-2008-38-5-1323. |
[11] |
J. Cui, Y. Sun and H. Zhu,
The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.
doi: 10.1007/s10884-007-9075-0. |
[12] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz,
On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.
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[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 Journal on Mathematical Analysis, 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, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893-910.
doi: 10.3934/dcdsb.2004.4.893. |
[15] |
J. Ge, K. I. Kim, Z. Lin and H. Zhu,
A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, Journal of Differential Equations, 259 (2015), 5486-5509.
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Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66.
doi: 10.3934/mbe.2010.7.51. |
[19] |
P. A. Khanam, B. Khuda, T. T. Khane and A. Ashraf,
Awareness of sexually transmitted disease among women and service providers in rural bangladesh, International Journal of STD & AIDS, 8 (1997), 688-696.
doi: 10.1258/0956462971919066. |
[20] |
T. Kuniya and J. Wang,
Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Applicable Analysis, (2016), 1-26.
doi: 10.1080/00036811.2016.1199796. |
[21] |
A. K. Misra, A. Sharma and J. B. Shukla,
Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Mathematical and Computer Modelling, 53 (2011), 1221-1228.
doi: 10.1016/j.mcm.2010.12.005. |
[22] |
C. Neuhauser,
Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314.
|
[23] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 198. Springer New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[24] |
R. Peng,
Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, Journal of Differential Equations, 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002. |
[25] |
R. Peng and S. Liu,
Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 239-247.
doi: 10.1016/j.na.2008.10.043. |
[26] |
R. Peng and X. Zhao,
A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
doi: 10.1088/0951-7715/25/5/1451. |
[27] |
R. Peng and F. Yi,
Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D: Nonlinear Phenomena, 259 (2013), 8-25.
doi: 10.1016/j.physd.2013.05.006. |
[28] |
M. H. Protter and H. F. Weinberger,
Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, 1967. |
[29] |
M. Robinson, N. I. Stilianakis and Y. Drossinos,
Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012), 116-126.
doi: 10.1016/j.jtbi.2011.12.015. |
[30] |
J. Shi, Z. Xie and K. Little,
Cross-diffusion induced instability and stability in reaction-diffusion systems, Journal of Applied Analysis and Computation, 1 (2011), 95-119.
|
[31] |
H. L. Smith,
Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[32] |
C. Sun, W. Yang, J. Arino and K. Khan,
Effect of media-induced social distancing on disease transmission in a two patch setting, Mathematical Biosciences, 230 (2011), 87-95.
doi: 10.1016/j.mbs.2011.01.005. |
[33] |
S. Tang, Y. Xiao, L. Yuan, R. A. Cheke and J. Wu,
Campus quarantine (FengXiao) for curbing emergent infectious diseases: lessons from mitigating a/H1N1 in Xi'an, China, Journal of Theoretical Biology, 295 (2012), 47-58.
doi: 10.1016/j.jtbi.2011.10.035. |
[34] |
J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission ISRN Biomathematics, 2012 (2012), Article ID 581274, 10 pages.
doi: 10.5402/2012/581274. |
[35] |
J. M. Tchuenche, N. Dube, C. P. Bhunu and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza,
BMC Public Health, 11 (2011), S5. |
[36] |
N. Tuncer and M. Martcheva,
Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, Journal of Biological Dynamics, 6 (2012), 406-439.
doi: 10.1080/17513758.2011.614697. |
[37] |
P. Vanden Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[38] |
J. Wang, R. Zhang and T. Kuniya,
The dynamics of an SVIR epidemiological model with infection age, IMA Journal of Applied Mathematics, 81 (2016), 321-343.
doi: 10.1093/imamat/hxv039. |
[39] |
W. D. Wang,
Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006), 267-279.
doi: 10.3934/mbe.2006.3.267. |
[40] |
W. D. Wang and X. Zhao,
Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
[41] |
D. Xiao and S. Ruan,
Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
[42] |
Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak Scientific Reports, 5 (2015), 7838.
doi: 10.1038/srep07838. |
[43] |
Y. Xiao, T. Zhao and S. Tang,
Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Mathematical Biosciences and Engineering, 10 (2013), 445-461.
doi: 10.3934/mbe.2013.10.445. |
[44] |
M. E. Young, G. R. Norman and K. R. Humphreys, Medicine in the popular press: The influence of the media on perceptions of disease PLoS One, 3 (2008), e3552.
doi: 10.1371/journal.pone.0003552. |
show all references
References:
[1] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai,
Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.
doi: 10.1137/060672522. |
[2] |
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, Discrete and Continuous Dynamical Systems-A, 21 (2008), 1-20.
doi: 10.3934/dcds.2008.21.1. |
[3] |
P. M. Arguin, A. W. Navin, S. F. Steele, L. H. Weld and P. E. Kozarsky,
Health communication during SARS, Emerging Infectious Diseases, 10 (2004), 377-380.
doi: 10.3201/eid1002.030812. |
[4] |
M. P. Brinn, K. V. Carson, A. J. Esterman, A. B. Chang and B. J. Smith,
Cochrane review: Mass media interventions for preventing smoking in young people, Evidence-Based Child Health: A Cochrane Review Journal, 7 (2012), 86-144.
doi: 10.1002/ebch.1808. |
[5] |
Y. Cai, Y. Kang, M. Banerjee and W. Wang,
A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502.
doi: 10.1016/j.jde.2015.08.024. |
[6] |
Y. Cai and W. M. Wang,
Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 989-1013.
doi: 10.3934/dcdsb.2015.20.989. |
[7] |
Y. Cai and W. M. Wang,
Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Analysis: Real World Applications, 30 (2016), 99-125.
doi: 10.1016/j.nonrwa.2015.12.002. |
[8] |
Y. Cai, Z. Wang and W. M. Wang,
Endemic dynamics in a host-parasite epidemiological model within spatially heterogeneous environment, Applied Mathematics Letters, 61 (2016), 129-136.
doi: 10.1016/j.aml.2016.05.011. |
[9] |
R. S. Cantrell and C. Cosner,
Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., 2003.
doi: 10.1002/0470871296. |
[10] |
J. Cui, X. Tao and H. Zhu,
An SIS infection model incorporating media coverage, Journal of Mathematics, 38 (2008), 1323-1334.
doi: 10.1216/RMJ-2008-38-5-1323. |
[11] |
J. Cui, Y. Sun and H. Zhu,
The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.
doi: 10.1007/s10884-007-9075-0. |
[12] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz,
On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[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 Journal on Mathematical Analysis, 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, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893-910.
doi: 10.3934/dcdsb.2004.4.893. |
[15] |
J. Ge, K. I. Kim, Z. Lin and H. Zhu,
A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, Journal of Differential Equations, 259 (2015), 5486-5509.
doi: 10.1016/j.jde.2015.06.035. |
[16] |
A. B. Gumel, S. Ruan, T. Day, J. Watmough and F. Brauer,
Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London B: Biological Sciences, 271 (2004), 2223-2232.
doi: 10.1098/rspb.2004.2800. |
[17] |
D. Henry and D. B. Henry,
Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag, Berlin, 1981. |
[18] |
W. Huang, M. Han and K. Liu,
Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66.
doi: 10.3934/mbe.2010.7.51. |
[19] |
P. A. Khanam, B. Khuda, T. T. Khane and A. Ashraf,
Awareness of sexually transmitted disease among women and service providers in rural bangladesh, International Journal of STD & AIDS, 8 (1997), 688-696.
doi: 10.1258/0956462971919066. |
[20] |
T. Kuniya and J. Wang,
Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Applicable Analysis, (2016), 1-26.
doi: 10.1080/00036811.2016.1199796. |
[21] |
A. K. Misra, A. Sharma and J. B. Shukla,
Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Mathematical and Computer Modelling, 53 (2011), 1221-1228.
doi: 10.1016/j.mcm.2010.12.005. |
[22] |
C. Neuhauser,
Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314.
|
[23] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 198. Springer New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[24] |
R. Peng,
Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, Journal of Differential Equations, 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002. |
[25] |
R. Peng and S. Liu,
Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 239-247.
doi: 10.1016/j.na.2008.10.043. |
[26] |
R. Peng and X. Zhao,
A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
doi: 10.1088/0951-7715/25/5/1451. |
[27] |
R. Peng and F. Yi,
Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D: Nonlinear Phenomena, 259 (2013), 8-25.
doi: 10.1016/j.physd.2013.05.006. |
[28] |
M. H. Protter and H. F. Weinberger,
Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, 1967. |
[29] |
M. Robinson, N. I. Stilianakis and Y. Drossinos,
Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012), 116-126.
doi: 10.1016/j.jtbi.2011.12.015. |
[30] |
J. Shi, Z. Xie and K. Little,
Cross-diffusion induced instability and stability in reaction-diffusion systems, Journal of Applied Analysis and Computation, 1 (2011), 95-119.
|
[31] |
H. L. Smith,
Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[32] |
C. Sun, W. Yang, J. Arino and K. Khan,
Effect of media-induced social distancing on disease transmission in a two patch setting, Mathematical Biosciences, 230 (2011), 87-95.
doi: 10.1016/j.mbs.2011.01.005. |
[33] |
S. Tang, Y. Xiao, L. Yuan, R. A. Cheke and J. Wu,
Campus quarantine (FengXiao) for curbing emergent infectious diseases: lessons from mitigating a/H1N1 in Xi'an, China, Journal of Theoretical Biology, 295 (2012), 47-58.
doi: 10.1016/j.jtbi.2011.10.035. |
[34] |
J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission ISRN Biomathematics, 2012 (2012), Article ID 581274, 10 pages.
doi: 10.5402/2012/581274. |
[35] |
J. M. Tchuenche, N. Dube, C. P. Bhunu and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza,
BMC Public Health, 11 (2011), S5. |
[36] |
N. Tuncer and M. Martcheva,
Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, Journal of Biological Dynamics, 6 (2012), 406-439.
doi: 10.1080/17513758.2011.614697. |
[37] |
P. Vanden Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[38] |
J. Wang, R. Zhang and T. Kuniya,
The dynamics of an SVIR epidemiological model with infection age, IMA Journal of Applied Mathematics, 81 (2016), 321-343.
doi: 10.1093/imamat/hxv039. |
[39] |
W. D. Wang,
Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006), 267-279.
doi: 10.3934/mbe.2006.3.267. |
[40] |
W. D. Wang and X. Zhao,
Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
[41] |
D. Xiao and S. Ruan,
Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
[42] |
Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak Scientific Reports, 5 (2015), 7838.
doi: 10.1038/srep07838. |
[43] |
Y. Xiao, T. Zhao and S. Tang,
Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Mathematical Biosciences and Engineering, 10 (2013), 445-461.
doi: 10.3934/mbe.2013.10.445. |
[44] |
M. E. Young, G. R. Norman and K. R. Humphreys, Medicine in the popular press: The influence of the media on perceptions of disease PLoS One, 3 (2008), e3552.
doi: 10.1371/journal.pone.0003552. |


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2018 Impact Factor: 1.313
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