American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity
  • NHM Home
  • This Issue
  • Next Article
    The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system
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), 35-74. 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), 1113-1148. doi: 10.1137/110822967.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[7]

E. Ellis and M. Delbrück, The growth of bacteriophage, J. of Physiology, 22 (1939), 365-384. 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), 178101. 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), 670-688. doi: 10.1137/110848360.  Google Scholar

[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.  Google Scholar

[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.  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), 3-24. 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), 219-233. doi: 10.1007/s002850200144.  Google Scholar

[14]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, New York, 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), 031909. 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), 3-44. 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), 951-979. 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), 337-351. 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), 94-121. 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), 173-187. doi: 10.1007/BF00279720.  Google Scholar

[21]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 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), 430-470. 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), 183-218. 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), 1540-1549. 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), 365-373. 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), 35-74. 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), 1113-1148. doi: 10.1137/110822967.  Google Scholar

[3]

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

[4]

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

[5]

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

[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.  Google Scholar

[7]

E. Ellis and M. Delbrück, The growth of bacteriophage, J. of Physiology, 22 (1939), 365-384. 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), 178101. 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), 670-688. doi: 10.1137/110848360.  Google Scholar

[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.  Google Scholar

[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.  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), 3-24. 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), 219-233. doi: 10.1007/s002850200144.  Google Scholar

[14]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, New York, 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), 031909. 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), 3-44. 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), 951-979. 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), 337-351. 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), 94-121. 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), 173-187. doi: 10.1007/BF00279720.  Google Scholar

[21]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 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), 430-470. 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), 183-218. 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), 1540-1549. 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), 365-373. Google Scholar

[1]

Louis D. Bergsman, James M. Hyman, Carrie A. Manore. A mathematical model for the spread of west nile virus in migratory and resident birds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 401-424. doi: 10.3934/mbe.2015009
[2]

Rongsong Liu, Jiangping Shuai, Jianhong Wu, Huaiping Zhu. Modeling spatial spread of west nile virus and impact of directional dispersal of birds. Mathematical Biosciences & Engineering, 2006, 3 (1) : 145-160. doi: 10.3934/mbe.2006.3.145
[3]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[4]

Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 417-436. doi: 10.3934/dcdsb.2016.21.417
[5]

Yun Tian, Yu Bai, Pei Yu. Impact of delay on HIV-1 dynamics of fighting a virus with another virus. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1181-1198. doi: 10.3934/mbe.2014.11.1181
[6]

Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4959-4985. doi: 10.3934/dcds.2021064
[7]

Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283
[8]

Cuicui Jiang, Kaifa Wang, Lijuan Song. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1233-1246. doi: 10.3934/mbe.2017063
[9]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[10]

Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for virus-immune models with infinite delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3093-3114. doi: 10.3934/dcdsb.2015.20.3093
[11]

Wenjie Zuo, Junping Shi. Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1179-1200. doi: 10.3934/cpaa.2018057
[12]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[13]

Carlos Castillo-Chavez, Bingtuan Li, Haiyan Wang. Some recent developments on linear determinacy. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1419-1436. doi: 10.3934/mbe.2013.10.1419
[14]

Wendi Wang. Population dispersal and disease spread. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 797-804. doi: 10.3934/dcdsb.2004.4.797
[15]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483
[16]

Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443
[17]

Shigui Ruan, Wendi Wang, Simon A. Levin. The effect of global travel on the spread of SARS. Mathematical Biosciences & Engineering, 2006, 3 (1) : 205-218. doi: 10.3934/mbe.2006.3.205
[18]

Guangying Lv, Mingxin Wang. Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 415-433. doi: 10.3934/dcdsb.2010.13.415
[19]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[20]

Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences