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March  2013, 8(1): 327-342. doi: 10.3934/nhm.2013.8.327

Spread of viral infection of immobilized bacteria

Don A. Jones 1, , Hal L. Smith 2, and  Horst R. Thieme 3,

1. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA 85287, United States
2. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804
3. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States

Received  April 2012 Revised  October 2012 Published  April 2013

A reaction diffusion system with a distributed time delay is proposed for virus spread on bacteria immobilized on an agar-coated plate. A distributed delay explicitly accounts for a virus latent period of variable duration. The model allows the number of virus progeny released when an infected cell lyses to depend on the duration of the latent period. A unique spreading speed for virus infection is established and traveling wave solutions are shown to exist.
Keywords: linear determinacy, traveling wave solution, distributed delay., virus spread, Virus plaque, speed of spread.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Don A. Jones, Hal L. Smith, Horst R. Thieme. Spread of viral infection of immobilized bacteria. Networks and Heterogeneous Media, 2013, 8 (1) : 327-342. doi: 10.3934/nhm.2013.8.327
References:
[1]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency, Nonlinear Analysis RWA, 2 (2001), 35-74. doi: 10.1016/S0362-546X(99)00285-0.

[2]

C. Beaumont, J.-B. Burie, A. Ducrot and P. Zongo, Propogation of Salmonella within an industrial hens house, SIAM J. Appl. Math., 72 (2012), 1113-1148. doi: 10.1137/110822967.

[3]

A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165. doi: 10.2307/2406076.

[4]

P. DeLeenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[5]

O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Analysis, TMA, 1 (1977), 459-470.

[6]

O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Analysis, TMA, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9.

[7]

E. Ellis and M. Delbrück, The growth of bacteriophage, J. of Physiology, 22 (1939), 365-384. doi: 10.1085/jgp.22.3.365.

[8]

J. Fort and V. Mendez, Time-delayed spread of viruses in growing plaques, Physical Review Letters, 89 (2002), 178101. doi: 10.1103/PhysRevLett.89.178101.

[9]

D. A. Jones, G. Röst, H. L. Smith and H. R.Thieme, On spread of phage infection of bacteria in a petri dish, SIAM J. Appl. Math., 72 (2012), 670-688. doi: 10.1137/110848360.

[10]

A. L. Koch, The growth of viral plaques during enlargement phase, J. Theor. Biol., 6 (1964), 413-431. doi: 10.1016/0022-5193(64)90056-6.

[11]

Y. Lee and J. Yin, Imaging the propagation of viruses, Communication to the Editor, Biotechnology and Bioengineering, 52 (1996), 438-442. doi: 10.1002/(SICI)1097-0290(19961105)52:3<438::AID-BIT11>3.0.CO;2-F.

[12]

B. Levin, F. Stewart and L. Chao, Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, Amer. Naturalist, 111 (1977), 3-24. doi: 10.1086/283134.

[13]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.

[14]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, New York, 2000.

[15]

V. Ortega-Cejas, J. Fort, V. Mendez and D. Campos, Approximate solution to the speed of spreading viruses, Physical Review E, 69 (2004), 031909. doi: 10.1103/PhysRevE.69.031909.

[16]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[17]

H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, J. Math. Biol., 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4.

[18]

H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977), 337-351. doi: 10.1007/BF00275082.

[19]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121. doi: 10.1515/crll.1979.306.94.

[20]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.

[21]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.

[22]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, JDE, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[23]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperation models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[24]

J. Yin and J. S. McCaskill, Replication of viruses in a growing plaque: A reaction-diffusion model, Biophysics J., 61 (1992), 1540-1549. doi: 10.1016/S0006-3495(92)81958-6.

[25]

J. Yin and L. You, Amplification and spread of viruses in a growing plaque, J. Theor. Biol., 200 (1999), 365-373.

show all references

References:
[1]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency, Nonlinear Analysis RWA, 2 (2001), 35-74. doi: 10.1016/S0362-546X(99)00285-0.

[2]

C. Beaumont, J.-B. Burie, A. Ducrot and P. Zongo, Propogation of Salmonella within an industrial hens house, SIAM J. Appl. Math., 72 (2012), 1113-1148. doi: 10.1137/110822967.

[3]

A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165. doi: 10.2307/2406076.

[4]

P. DeLeenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[5]

O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Analysis, TMA, 1 (1977), 459-470.

[6]

O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Analysis, TMA, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9.

[7]

E. Ellis and M. Delbrück, The growth of bacteriophage, J. of Physiology, 22 (1939), 365-384. doi: 10.1085/jgp.22.3.365.

[8]

J. Fort and V. Mendez, Time-delayed spread of viruses in growing plaques, Physical Review Letters, 89 (2002), 178101. doi: 10.1103/PhysRevLett.89.178101.

[9]

D. A. Jones, G. Röst, H. L. Smith and H. R.Thieme, On spread of phage infection of bacteria in a petri dish, SIAM J. Appl. Math., 72 (2012), 670-688. doi: 10.1137/110848360.

[10]

A. L. Koch, The growth of viral plaques during enlargement phase, J. Theor. Biol., 6 (1964), 413-431. doi: 10.1016/0022-5193(64)90056-6.

[11]

Y. Lee and J. Yin, Imaging the propagation of viruses, Communication to the Editor, Biotechnology and Bioengineering, 52 (1996), 438-442. doi: 10.1002/(SICI)1097-0290(19961105)52:3<438::AID-BIT11>3.0.CO;2-F.

[12]

B. Levin, F. Stewart and L. Chao, Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, Amer. Naturalist, 111 (1977), 3-24. doi: 10.1086/283134.

[13]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.

[14]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, New York, 2000.

[15]

V. Ortega-Cejas, J. Fort, V. Mendez and D. Campos, Approximate solution to the speed of spreading viruses, Physical Review E, 69 (2004), 031909. doi: 10.1103/PhysRevE.69.031909.

[16]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[17]

H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, J. Math. Biol., 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4.

[18]

H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977), 337-351. doi: 10.1007/BF00275082.

[19]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121. doi: 10.1515/crll.1979.306.94.

[20]

H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.

[21]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.

[22]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, JDE, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[23]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperation models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[24]

J. Yin and J. S. McCaskill, Replication of viruses in a growing plaque: A reaction-diffusion model, Biophysics J., 61 (1992), 1540-1549. doi: 10.1016/S0006-3495(92)81958-6.

[25]

J. Yin and L. You, Amplification and spread of viruses in a growing plaque, J. Theor. Biol., 200 (1999), 365-373.

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