# American Institute of Mathematical Sciences

March  2016, 21(2): 471-496. doi: 10.3934/dcdsb.2016.21.471

## Spread of phage infection of bacteria in a petri dish

 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  January 2015 Revised  May 2015 Published  November 2015

We extend our previous work on the spatial spread of phage infection of immobile bacteria on an agar coated plate by explicitly including loss of viruses by both adsorption to bacteria and by decay of free viruses and by including a distributed virus latent period and distributed burst size rather than fixed values of these key parameters. We extend earlier results on the spread of virus and on the existence of traveling wave solutions when the basic reproductive number for virus, $\mathcal{R}_0$, exceeds one and we compare the results with those obtained in earlier work. Finally, we formulate and analyze a model of multiple virus strains competing to infect a common bacterial host in a petri dish.
Citation: Don A. Jones, Hal L. Smith, Horst R. Thieme. Spread of phage infection of bacteria in a petri dish. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 471-496. doi: 10.3934/dcdsb.2016.21.471
##### 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, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783.  Google Scholar [7] O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal.: Theory, Methods, Appl., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9.  Google Scholar [8] 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 [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 Journal on Applied Mathematics, 72 (2012), 670-688. doi: 10.1137/110848360.  Google Scholar [10] D. A. Jones, H. L. Smith and H. R. Thieme, Spread of viral infection of immobilized bacteria, Networks and Heterogeneous Media, 8 (2013), 327-342. doi: 10.3934/nhm.2013.8.327.  Google Scholar [11] 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 [12] 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 [13] 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 [14] 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 [15] M. T. Madigan and J. M. Martinko, Brock Biology of Microorganisms, 11 ed, Pearson Prentice Hall, Upper Saddle River, NJ, 2006. Google Scholar [16] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000.  Google Scholar [17] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar [18] 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 [19] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity invariance, comparison and convergence, J. reine und angewandte Mathematik, 413 (1991), 1-35.  Google Scholar [20] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Math., 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar [21] H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, J. of Math. Biol., 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4.  Google Scholar [22] H. L. Smith and H. R. Thieme, A reaction-diffusion system with time-delay modeling virus plaque formation, Canadian Applied Math. Quarterly, 19 (2011), 385-399.  Google Scholar [23] G. Stent, Molecular Biology Of Bacterial Viruses, W.H. Freeman and Co., London, 1963. Google Scholar [24] H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biology, 4 (1977), 337-351. doi: 10.1007/BF00275082.  Google Scholar [25] 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 [26] H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biology, 8 (1979), 173-187. doi: 10.1007/BF00279720.  Google Scholar [27] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003.  Google Scholar [28] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Eqn., 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar [29] 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 [30] 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 [31] 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, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783.  Google Scholar [7] O. Diekmann and H. G. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal.: Theory, Methods, Appl., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9.  Google Scholar [8] 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 [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 Journal on Applied Mathematics, 72 (2012), 670-688. doi: 10.1137/110848360.  Google Scholar [10] D. A. Jones, H. L. Smith and H. R. Thieme, Spread of viral infection of immobilized bacteria, Networks and Heterogeneous Media, 8 (2013), 327-342. doi: 10.3934/nhm.2013.8.327.  Google Scholar [11] 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 [12] 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 [13] 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 [14] 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 [15] M. T. Madigan and J. M. Martinko, Brock Biology of Microorganisms, 11 ed, Pearson Prentice Hall, Upper Saddle River, NJ, 2006. Google Scholar [16] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000.  Google Scholar [17] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar [18] 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 [19] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity invariance, comparison and convergence, J. reine und angewandte Mathematik, 413 (1991), 1-35.  Google Scholar [20] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Math., 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar [21] H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, J. of Math. Biol., 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4.  Google Scholar [22] H. L. Smith and H. R. Thieme, A reaction-diffusion system with time-delay modeling virus plaque formation, Canadian Applied Math. Quarterly, 19 (2011), 385-399.  Google Scholar [23] G. Stent, Molecular Biology Of Bacterial Viruses, W.H. Freeman and Co., London, 1963. Google Scholar [24] H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biology, 4 (1977), 337-351. doi: 10.1007/BF00275082.  Google Scholar [25] 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 [26] H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biology, 8 (1979), 173-187. doi: 10.1007/BF00279720.  Google Scholar [27] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003.  Google Scholar [28] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Eqn., 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar [29] 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 [30] 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 [31] J. Yin and L. You, Amplification and spread of viruses in a growing plaque, J. Theor. Biol., 200 (1999), 365-373. Google Scholar
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