Global stability of the steady states of an epidemic model incorporating intervention strategies

Yongli Cai; Yun Kang; Weiming Wang; Yongli Cai; Yun Kang; Weiming Wang

doi:10.3934/mbe.2017056

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1071-1089. doi: 10.3934/mbe.2017056

Previous Article Next Article

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

Received: 01 July 2016 Accepted: 01 October 2016 Published: 01 October 2017
MSC : Primary: 35B36, 45M10; Secondary: 92C15

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 $\mathcal{R}_0 \lt 1$ , there is a unique disease-free equilibrium which is globally asymptotically stable; and if $\mathcal{R}_0 \gt 1$ , 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.
- Basic reproduction number,
- disease-free equilibrium,
- endemic,
- spatial heterogeneity
Citation: Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1071-1089. doi: 10.3934/mbe.2017056

Related Papers:

Abstract

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 $\mathcal{R}_0 \lt 1$ , there is a unique disease-free equilibrium which is globally asymptotically stable; and if $\mathcal{R}_0 \gt 1$ , 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.

References

[1]	[ L. J. S. Allen,B. M. Bolker,Y. Lou,A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007): 1283-1309.
[2]	[ L. J. S. Allen,B. M. Bolker,Y. Lou,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.
[3]	[ P. M. Arguin,A. W. Navin,S. F. Steele,L. H. Weld,P. E. Kozarsky, Health communication during SARS, Emerging Infectious Diseases, 10 (2004): 377-380.
[4]	[ M. P. Brinn,K. V. Carson,A. J. Esterman,A. B. Chang,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.
[5]	[ Y. Cai,Y. Kang,M. Banerjee,W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015): 7463-7502.
[6]	[ Y. Cai,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.
[7]	[ Y. Cai,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.
[8]	[ Y. Cai,Z. Wang,W. M. Wang, Endemic dynamics in a host-parasite epidemiological model within spatially heterogeneous environment, Applied Mathematics Letters, 61 (2016): 129-136.
[9]	[ R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., 2003.
[10]	[ J. Cui,X. Tao,H. Zhu, An SIS infection model incorporating media coverage, Journal of Mathematics, 38 (2008): 1323-1334.
[11]	[ J. Cui,Y. Sun,H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008): 31-53.
[12]	[ O. Diekmann,J. A. P. Heesterbeek,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.
[13]	[ W. E. Fitzgibbon,M. Langlais,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.
[14]	[ W. E. Fitzgibbon,M. Langlais,J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Discrete and Continuous Dynamical Systems-B, 4 (2004): 893-910.
[15]	[ J. Ge,K. I. Kim,Z. Lin,H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, Journal of Differential Equations, 259 (2015): 5486-5509.
[16]	[ A. B. Gumel,S. Ruan,T. Day,J. Watmough,F. Brauer, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London B: Biological Sciences, 271 (2004): 2223-2232.
[17]	[ D. Henry and D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag, Berlin, 1981.
[18]	[ W. Huang,M. Han,K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010): 51-66.
[19]	[ P. A. Khanam,B. Khuda,T. T. Khane,A. Ashraf, Awareness of sexually transmitted disease among women and service providers in rural bangladesh, International Journal of STD & AIDS, 8 (1997): 688-696.
[20]	[ T. Kuniya,J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Applicable Analysis, null (2016): 1-26.
[21]	[ A. K. Misra,A. Sharma,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.
[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.
[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.
[25]	[ R. Peng,S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009): 239-247.
[26]	[ R. Peng,X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012): 1451-1471.
[27]	[ R. Peng,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.
[28]	[ M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, 1967.
[29]	[ M. Robinson,N. I. Stilianakis,Y. Drossinos, Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012): 116-126.
[30]	[ J. Shi,Z. Xie,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,K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Mathematical Biosciences, 230 (2011): 87-95.
[33]	[ S. Tang,Y. Xiao,L. Yuan,R. A. Cheke,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.
[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.
[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,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.
[37]	[ P. Vanden Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002): 29-48.
[38]	[ J. Wang,R. Zhang,T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA Journal of Applied Mathematics, 81 (2016): 321-343.
[39]	[ W. D. Wang, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006): 267-279.
[40]	[ W. D. Wang,X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012): 1652-1673.
[41]	[ D. Xiao,S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007): 419-429.
[42]	[ Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak Scientific Reports, 5 (2015), 7838.
[43]	[ Y. Xiao,T. Zhao,S. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Mathematical Biosciences and Engineering, 10 (2013): 445-461.
[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.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)