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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 & 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. doi: 10.1016/S0362-546X(99)00285-0. Google Scholar

[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. doi: 10.1137/110822967. Google Scholar

[3]

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

[4]

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

[5]

O. Diekmann, Limiting behaviour in an epidemic model,, Nonlinear Analysis, 1 (1977), 459. Google Scholar

[6]

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

[7]

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

[8]

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

[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. doi: 10.1137/110848360. Google Scholar

[10]

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

[11]

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

[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. doi: 10.1086/283134. Google Scholar

[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. doi: 10.1007/s002850200144. Google Scholar

[14]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford University Press, (2000). Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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. doi: 10.1515/crll.1979.306.94. Google Scholar

[20]

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

[21]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003). Google Scholar

[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. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar

[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. doi: 10.1007/s002850200145. Google Scholar

[24]

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

[25]

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

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. doi: 10.1016/S0362-546X(99)00285-0. Google Scholar

[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. doi: 10.1137/110822967. Google Scholar

[3]

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

[4]

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

[5]

O. Diekmann, Limiting behaviour in an epidemic model,, Nonlinear Analysis, 1 (1977), 459. Google Scholar

[6]

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

[7]

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

[8]

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

[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. doi: 10.1137/110848360. Google Scholar

[10]

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

[11]

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

[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. doi: 10.1086/283134. Google Scholar

[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. doi: 10.1007/s002850200144. Google Scholar

[14]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford University Press, (2000). Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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. doi: 10.1515/crll.1979.306.94. Google Scholar

[20]

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

[21]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003). Google Scholar

[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. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar

[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. doi: 10.1007/s002850200145. Google Scholar

[24]

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

[25]

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

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