November  2019, 24(11): 6239-6259. doi: 10.3934/dcdsb.2019137

Analysis of a reaction diffusion model for a reservoir supported spread of infectious disease

Department of Mathematics, University of Houston, Houston, TX 77204, USA

Received  July 2018 Published  November 2019 Early access  July 2019

Motivated by recent outbreaks of the Ebola Virus, we are concerned with the role that a vector reservoir plays in supporting the spatio-temporal spread of a highly lethal disease through a host population. In our context, the reservoir is a species capable of harboring and sustaining the pathogen. We develop models that describe the horizontal spread of the disease among the host population when the host population is in contact with the reservoir and when it is not in contact with the host population. These models are of reaction diffusion type, and they are analyzed, and their long term asymptotic behavior is determined.

Citation: W. E. Fitzgibbon, J. J. Morgan. Analysis of a reaction diffusion model for a reservoir supported spread of infectious disease. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6239-6259. doi: 10.3934/dcdsb.2019137
References:
[1]

N. T. J. Bailey, The Mathematical Theory of Epidemics, Charles Griffin & Company Limited, London, 1957.

[2]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for eigen value problems with indefinite weight function, Journal of Mathematical Analysis and Applications, 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[3]

S. Cantrell and C. Cosner, Spatial Ecology via Reaction Diffusion Equations, J. Wiley and Sons, Hoboken, N.J., 2003. doi: 10.1002/0470871296.

[4]

J. Evans and A. Shenk, Solutions to nerve axon equations, Biophysical Journal, 10 (1970), 1090-1101. 

[5]

W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, Structured Population Models in Biology and Epidemiology, 115–164, Lecture Notes in Math., 1936, Math. Biosci. Subser., Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5_3.

[6]

W. E. FitzgibbonM. LanglaisF. Marpeau and J. J. Morgan, Modeling the circulation of a disease between two host populations on noncoincident spatial domains, Biological Invasions, 7 (2005), 863-875. 

[7]

W. E. FitzgibbonM. Langlais and J. J. Morgan, A reacton diffusion system modeling the direct and indirect transmission of dieases, Discrete and Continuous Dynamical Systems, Series B, 4 (2004), 893-910.  doi: 10.3934/dcdsb.2004.4.893.

[8]

W. E. FitzgibbonM. Parrott and G. F. Webb, Diffusive epidemic models with crisscross dynamics and incubation, Mathematical Biosciences, 128 (1995), 131-155.  doi: 10.1016/0025-5564(94)00070-G.

[9]

O. A. Ladyshenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations American Mathematical Society, Providence, RI, 1968.

[10]

K. B. Laupland and L. Valiquette, Ebola virus disease, Canadian Journal of Infectious Diseases and Medical Microbiology, 25 (2014), 128-129. 

[11]

M. Marion, Finite dimensional attractors associated with partially dissipative systems, SIAM Journal of Mathematical Analysis, 20 (1989), 815-844.  doi: 10.1137/0520057.

[12]

J. J. Morgan, Boundedness and decay results for for reaction diffusion systems, SIAM J. Math. Anal., 21 (1990), 1172-1189.  doi: 10.1137/0521064.

[13]

R. Nagel (ed.), One Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer, Berlin, 1986.

[14]

M. T. Osterholm, K. A. Moore, N. S. Kelley, L. M. Brosseau, G. Wong, F. A. Murphy, C. J. Peters, J. W. LeDuc, P. K. Russell, M. V. Herp, J. Kapetshi, J. J. T. Muyembe, B. K. Ilunga, J. E. Strong, A. Grolla, A. Wolz, B. Kargbo, D. K. Kargbo, P. Formenty, D. A. Sanders and G. P. LondKobinger, Transmission of the Ebola Viruses: What we know and what we do not know, mBio, American Society for Microbiology, 8 (2017), http://mbio.asm.org/content/6/2/e00137-15.full doi: 10.1128/mBio.00137-15.

[15]

A. Pazy, Semigroups of Operators and Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

J. Richardson, Deadly Ebola Virus Linked to Bush Meat, Food Safety News, 2012, http://www.foodsafetynews.com/2012/09/deadly-african-ebola-virus-linked-to-bushmeat/#.WUrBP_4o47Z

[17]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983.

[18]

R. Swanepool, P. Leman, F. Bart, N. Zachariades, L. Brack, P. Rollins, F, Ksiazek and C. Peters, Experimental inoculation of plants with ebola, Emerging Infectious Diseases, 1996,321–325.

[19]

R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[20]

P. D. WalshK. A. AbernethyM. BermejoR. BeyersP. DeWachterM. E. AkouB. HuijbregtsD. I. MamboungaA. K. TohamA. M. KilbournS. A. LahmS. LatourF. MaiselsC. MbinaY. MihindouS. N. ObiangE. N. EffaM. P. StarkeyP. TelferM. ThibaultC. E. G. TutinL. J. T. White and D. S. Wilkie, Catastrophic ape decline in western equatorial Africa, Nature, 422 (2003), 611-614.  doi: 10.1038/nature01566.

show all references

References:
[1]

N. T. J. Bailey, The Mathematical Theory of Epidemics, Charles Griffin & Company Limited, London, 1957.

[2]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for eigen value problems with indefinite weight function, Journal of Mathematical Analysis and Applications, 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[3]

S. Cantrell and C. Cosner, Spatial Ecology via Reaction Diffusion Equations, J. Wiley and Sons, Hoboken, N.J., 2003. doi: 10.1002/0470871296.

[4]

J. Evans and A. Shenk, Solutions to nerve axon equations, Biophysical Journal, 10 (1970), 1090-1101. 

[5]

W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, Structured Population Models in Biology and Epidemiology, 115–164, Lecture Notes in Math., 1936, Math. Biosci. Subser., Springer, Berlin, 2008. doi: 10.1007/978-3-540-78273-5_3.

[6]

W. E. FitzgibbonM. LanglaisF. Marpeau and J. J. Morgan, Modeling the circulation of a disease between two host populations on noncoincident spatial domains, Biological Invasions, 7 (2005), 863-875. 

[7]

W. E. FitzgibbonM. Langlais and J. J. Morgan, A reacton diffusion system modeling the direct and indirect transmission of dieases, Discrete and Continuous Dynamical Systems, Series B, 4 (2004), 893-910.  doi: 10.3934/dcdsb.2004.4.893.

[8]

W. E. FitzgibbonM. Parrott and G. F. Webb, Diffusive epidemic models with crisscross dynamics and incubation, Mathematical Biosciences, 128 (1995), 131-155.  doi: 10.1016/0025-5564(94)00070-G.

[9]

O. A. Ladyshenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations American Mathematical Society, Providence, RI, 1968.

[10]

K. B. Laupland and L. Valiquette, Ebola virus disease, Canadian Journal of Infectious Diseases and Medical Microbiology, 25 (2014), 128-129. 

[11]

M. Marion, Finite dimensional attractors associated with partially dissipative systems, SIAM Journal of Mathematical Analysis, 20 (1989), 815-844.  doi: 10.1137/0520057.

[12]

J. J. Morgan, Boundedness and decay results for for reaction diffusion systems, SIAM J. Math. Anal., 21 (1990), 1172-1189.  doi: 10.1137/0521064.

[13]

R. Nagel (ed.), One Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer, Berlin, 1986.

[14]

M. T. Osterholm, K. A. Moore, N. S. Kelley, L. M. Brosseau, G. Wong, F. A. Murphy, C. J. Peters, J. W. LeDuc, P. K. Russell, M. V. Herp, J. Kapetshi, J. J. T. Muyembe, B. K. Ilunga, J. E. Strong, A. Grolla, A. Wolz, B. Kargbo, D. K. Kargbo, P. Formenty, D. A. Sanders and G. P. LondKobinger, Transmission of the Ebola Viruses: What we know and what we do not know, mBio, American Society for Microbiology, 8 (2017), http://mbio.asm.org/content/6/2/e00137-15.full doi: 10.1128/mBio.00137-15.

[15]

A. Pazy, Semigroups of Operators and Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

J. Richardson, Deadly Ebola Virus Linked to Bush Meat, Food Safety News, 2012, http://www.foodsafetynews.com/2012/09/deadly-african-ebola-virus-linked-to-bushmeat/#.WUrBP_4o47Z

[17]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983.

[18]

R. Swanepool, P. Leman, F. Bart, N. Zachariades, L. Brack, P. Rollins, F, Ksiazek and C. Peters, Experimental inoculation of plants with ebola, Emerging Infectious Diseases, 1996,321–325.

[19]

R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[20]

P. D. WalshK. A. AbernethyM. BermejoR. BeyersP. DeWachterM. E. AkouB. HuijbregtsD. I. MamboungaA. K. TohamA. M. KilbournS. A. LahmS. LatourF. MaiselsC. MbinaY. MihindouS. N. ObiangE. N. EffaM. P. StarkeyP. TelferM. ThibaultC. E. G. TutinL. J. T. White and D. S. Wilkie, Catastrophic ape decline in western equatorial Africa, Nature, 422 (2003), 611-614.  doi: 10.1038/nature01566.

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