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

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October 2017, 14(5&6): 1071-1089. doi: 10.3934/mbe.2017056

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

Yongli Cai ^1, , Yun Kang ^2, and Weiming Wang ^1,,

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

* Corresponding author: Weiming Wang

Received July 2016 Accepted October 2016 Published May 2017

Fund Project: The authors would like to thank the anonymous referees for very helpful suggestions and comments which led to improvement of our original manuscript. This research was supported by the National Science Foundation of China (11601179, 61373005 & 61672013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (16KJB110003). The research of YK is partially supported by NSF-DMS (Award Number 1313312) and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472)

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.

Keywords: Basic reproduction number, disease-free equilibrium, endemic, spatial heterogeneity.

Mathematics Subject Classification: Primary: 35B36, 45M10; Secondary: 92C15.

Citation: Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1071-1089. doi: 10.3934/mbe.2017056

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., 2003. doi: 10.1002/0470871296. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	D. Henry and D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag, Berlin, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	C. Neuhauser, Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, 1967. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., 2003. doi: 10.1002/0470871296. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	D. Henry and D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag, Berlin, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	C. Neuhauser, Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, 1967. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. In the low-risk domain of model (4), (a) the influence of the diffusion coefficient $d$ on $\mathcal{R}_0$; (b) the influence of the spatial heterogeneity of environment on $\mathcal{R}_0$. The parameters are taken as (33)

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Figure 2. In the high-risk domain of model (4), (a) the influence of the diffusion coefficient $d$ on $\mathcal{R}_0$; (b) the influence of the spatial heterogeneity of environment on $\mathcal{R}_0$. The parameters are taken as (33)

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2018 Impact Factor: 1.313

American Institute of Mathematical Sciences