American Institute of Mathematical Sciences

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July  2016, 21(5): 1469-1481. doi: 10.3934/dcdsb.2016007

Environmental risks in a diffusive SIS model incorporating use efficiency of the medical resource

Jing Ge 1, , Zhigui Lin 1, and  Huaiping Zhu 2,

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2. 

LAboratory of Mathematical Parallel Systems (LAMPS), Centre for Disease Modeling, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3

Received  February 2016 Revised  March 2016 Published  April 2016

To capture the impact of spatial heterogeneity of environment and available resource of the public health system on the persistence and extinction of the infectious disease, a simplified spatial SIS reaction-diffusion model with allocation and use efficiency of the medical resource is proposed. A nonlinear space dependent recovery rate is introduced to model impact of available public health resource on the transmission dynamics of the disease. The basic reproduction numbers associated with the diseases in the spatial setting are defined, and then the low, moderate and high risks of the environment are classified. Our results show that the complicated dynamical behaviors of the system are induced by the variation of the use efficiency of medical resources, which suggests that maintaining appropriate number of public health resources and well management are important to control and prevent the temporal-spatial spreading of the infectious disease. The numerical simulations are presented to illustrate the impact of the use efficiency of medical resources on the control of the spreading of infectious disease.
Keywords: environmental risk, heterogeneous environment, allocation of medical resource., nonlinear recovery rate, Diffusive SIS model.
Mathematics Subject Classification: Primary: 35R35; Secondary: 35K6.
Citation: Jing Ge, Zhigui Lin, Huaiping Zhu. Environmental risks in a diffusive SIS model incorporating use efficiency of the medical resource. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1469-1481. doi: 10.3934/dcdsb.2016007
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 reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1.

[2]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1991.

[3]

R. Boaden, N. Proudlove and M. Wilson, An exploratory study of bed management, J. Manag. Med., 13 (2006), 234-250. doi: 10.1108/02689239910292945.

[4]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2011. doi: 10.1007/978-1-4614-1686-9.

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003. doi: 10.1002/0470871296.

[6]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, West Sussex, England, 2000.

[7]

J. Ge, K. I. Kim, Z. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035.

[8]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics, vol. 247. Longman Sci. Tech., Harlow, 1991.

[9]

W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math Biosci Eng., 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51.

[10]

S. B. Jiang, L. Lu and L. Y. Du, Development of SARS vaccines and therapeutics is still needed, Future Virology, 8 (2013), 1-2. doi: 10.2217/fvl.12.126.

[11]

K. I. Kim and Z. G. Lin, Asymptotic behavior of an SEI epidemic model with diffusion, Math. Comput. Modelling, 47 (2008), 1314-1322. doi: 10.1016/j.mcm.2007.08.004.

[12]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer.Math. Soc, Providence, RI, 1968.

[13]

C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.05.003.

[14]

R. Peng, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model, I, J. Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002.

[15]

R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006.

[16]

R. Peng and X. Q. 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.

[17]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[18]

C. H. Shan and H. P. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688. doi: 10.1016/j.jde.2014.05.030.

[19]

C. H. Shan, Y. F. Yi and H. P. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. of Differential Equations, 260 (2016), 4339-4365. doi: 10.1016/j.jde.2015.11.009.

[20]

Q. L. Tang, J. Ge and Z. G. Lin, An SEI-SI avian-human influenza model with diffusion and nonlocal delay, Appl. Math. Comput., 247 (2014), 753-761. doi: 10.1016/j.amc.2014.09.042.

[21]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[22]

H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 227 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006.

[23]

J. Zhou and H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Bio., 32 (1994), 809-834. doi: 10.1007/BF00168799.

[24]

, World Health Organization, World Health Statistics 2005-2011.

[25]

"Summary of probable SARS cases with onset of illness from 1 November 2002 to 31 July 2003"., World Health Organization (WHO). http://www.who.int/csr/sars/country/table2004_04_21/en/

[26]

, http://www.who.int/management/facility/hospital/en/index6.html.

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 reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1.

[2]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1991.

[3]

R. Boaden, N. Proudlove and M. Wilson, An exploratory study of bed management, J. Manag. Med., 13 (2006), 234-250. doi: 10.1108/02689239910292945.

[4]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2011. doi: 10.1007/978-1-4614-1686-9.

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003. doi: 10.1002/0470871296.

[6]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, West Sussex, England, 2000.

[7]

J. Ge, K. I. Kim, Z. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035.

[8]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics, vol. 247. Longman Sci. Tech., Harlow, 1991.

[9]

W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math Biosci Eng., 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51.

[10]

S. B. Jiang, L. Lu and L. Y. Du, Development of SARS vaccines and therapeutics is still needed, Future Virology, 8 (2013), 1-2. doi: 10.2217/fvl.12.126.

[11]

K. I. Kim and Z. G. Lin, Asymptotic behavior of an SEI epidemic model with diffusion, Math. Comput. Modelling, 47 (2008), 1314-1322. doi: 10.1016/j.mcm.2007.08.004.

[12]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer.Math. Soc, Providence, RI, 1968.

[13]

C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.05.003.

[14]

R. Peng, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model, I, J. Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002.

[15]

R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006.

[16]

R. Peng and X. Q. 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.

[17]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

[18]

C. H. Shan and H. P. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688. doi: 10.1016/j.jde.2014.05.030.

[19]

C. H. Shan, Y. F. Yi and H. P. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. of Differential Equations, 260 (2016), 4339-4365. doi: 10.1016/j.jde.2015.11.009.

[20]

Q. L. Tang, J. Ge and Z. G. Lin, An SEI-SI avian-human influenza model with diffusion and nonlocal delay, Appl. Math. Comput., 247 (2014), 753-761. doi: 10.1016/j.amc.2014.09.042.

[21]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[22]

H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 227 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006.

[23]

J. Zhou and H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Bio., 32 (1994), 809-834. doi: 10.1007/BF00168799.

[24]

, World Health Organization, World Health Statistics 2005-2011.

[25]

"Summary of probable SARS cases with onset of illness from 1 November 2002 to 31 July 2003"., World Health Organization (WHO). http://www.who.int/csr/sars/country/table2004_04_21/en/

[26]

, http://www.who.int/management/facility/hospital/en/index6.html.

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