• Previous Article
    The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system
  • NHM Home
  • This Issue
  • Next Article
    Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity
March  2013, 8(1): 327-342. doi: 10.3934/nhm.2013.8.327

Spread of viral infection of immobilized bacteria

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.
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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[7]

E. Ellis and M. Delbrück, The growth of bacteriophage,, J. of Physiology, 22 (1939), 365. 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). 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. doi: 10.1137/110848360.

[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.

[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.

[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.

[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.

[14]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford University Press, (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). 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. 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. 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. 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. 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. doi: 10.1007/BF00279720.

[21]

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

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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[7]

E. Ellis and M. Delbrück, The growth of bacteriophage,, J. of Physiology, 22 (1939), 365. 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). 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. doi: 10.1137/110848360.

[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.

[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.

[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.

[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.

[14]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford University Press, (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). 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. 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. 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. 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. 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. doi: 10.1007/BF00279720.

[21]

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

[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Tsanou Berge, Samuel Bowong, Jean Lubuma, Martin Luther Mann Manyombe. Modeling Ebola Virus Disease transmissions with reservoir in a complex virus life ecology. Mathematical Biosciences & Engineering, 2018, 15 (1) : 21-56. doi: 10.3934/mbe.2018002

[19]

Elamin H. Elbasha. Model for hepatitis C virus transmissions. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1045-1065. doi: 10.3934/mbe.2013.10.1045

[20]

Nicola Bellomo, Youshan Tao. Stabilization in a chemotaxis model for virus infection. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 105-117. doi: 10.3934/dcdss.2020006

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]