Spread of viral infection of immobilized bacteria

Don A. Jones; Hal L. Smith; Horst R. Thieme

doi:10.3934/nhm.2013.8.327

Article Contents

2013, Volume 8, Issue 1: 327-342. Doi: 10.3934/nhm.2013.8.327

This issue Previous Article The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system Next Article Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity

Spread of viral infection of immobilized bacteria

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA 85287
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

Received: March 31, 2012

Revised: September 30, 2012

Published: February 2013

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

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.