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December  2019, 24(12): 6607-6620. doi: 10.3934/dcdsb.2019158

Coinfection in a stochastic model for bacteriophage systems

Xavier Bardina 1, , Sílvia Cuadrado 1, and  Carles Rovira 2,,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Avinguda de l'Eix Central, 08193 Bellaterra, Spain
2. 

Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

* Corresponding author: Carles Rovira

Received  February 2018 Revised  March 2019 Published  December 2019 Early access  July 2019

Fund Project: X. Bardina is partially supported by the grant MTM2015-67802-P from MINECO, S. Cuadrado is partially supported by the grant MTM2017-84214-C2-2-P from MICINN and C. Rovira is partially supported by the grant MTM2015-65092-P from MINECO/FEDER, UE.

A system modeling bacteriophage treatments with coinfections in a noisy context is analysed. We prove that in a small noise regime, the system converges in the long term to a bacteria-free equilibrium. Moreover, we compare the treatment with coinfection with the treatment without coinfection, showing how coinfection affects the convergence to the bacteria-free equilibrium.

Keywords: Bacteriophage, coinfection, stochastic fluctuations.
Mathematics Subject Classification: Primary: 92D25; Secondary: 93E03, 34F05, 60H10.
Citation: Xavier Bardina, Sílvia Cuadrado, Carles Rovira. Coinfection in a stochastic model for bacteriophage systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6607-6620. doi: 10.3934/dcdsb.2019158
References:
[1]

S. T. Abedon, Lysis from without, Bacteriophage, 1 (2011), 46-49.  doi: 10.4161/bact.1.1.13980.

[2]

S. T. Abedon, Bacteriophage secondary infection, Virologica Sinica, 30 (2015), 3-10.  doi: 10.1007/s12250-014-3547-2.

[3]

S. T. Abedon, P. García, P. Mullany and R. Aminov, Phage therapy: Past, present and future, Frontiers in Microbiology, 8 (2017), 201700981. doi: 10.3389/978-2-88945-251-4.

[4]

S. Alizon, Co-infection and super-infection models in evolutionary epidemiology, Interface Focus, 3 (2013), 20130031. doi: 10.1098/rsfs.2013.0031.

[5]

M. van Baalen and M. W. Sabelis, The dynamics of multiple infection and the evolution of virulence, Am. Nat., 146 (1995), 881-910.  doi: 10.1086/285830.

[6]

X. Bardina, D. Bascompte, C. Rovira and S. Tindel, An analysis of a stochastic model for bacteriophage systems, Mathematical Biosciences, 241 (2013), 99-108.  doi: 10.1016/j.mbs.2012.09.009.

[7]

C. Bardina, D. Spricigo, M. Corts and M. Llagostera, Significance of the bacteriophage treatment schedule in reducing salmonella colonization of poultry, Applied and Environmental Microbiology, 78 (2012), 6600-6607.  doi: 10.1128/AEM.01257-12.

[8]

C. Barril, À. Calsina and J. Ripoll, On the reproduction number of a gut microbiota model, Bull. Math. Biol., 79 (2017), 2727-2746.  doi: 10.1007/s11538-017-0352-8.

[9]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Anal. Real World Appl., 2 (2001), 35-74.  doi: 10.1016/S0362-546X(99)00285-0.

[10]

B. J. M. Bohannan and R. E. Lenski, Linking genetic change to community evolution: Insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000), 362-377.  doi: 10.1046/j.1461-0248.2000.00161.x.

[11]

S. N. Busenberg and K. L. Cooke, The effect of integral conditions in certain equations modelling epidemics and population growth, J. Math. Biol., 10 (1980), 13-32.  doi: 10.1007/BF00276393.

[12]

B. J. Cairns, A. R. Timms, V. A. A. Jansen, I. F. Connerton and R. J. H. Payne, Quantitative models of in vitro bacteriophage host dynamics and their application to phage therapy, PLoS Pathog., 5 (2009), e1000253. doi: 10.1371/journal.ppat.1000253.

[13]

À. Calsina, J. M. Palmada and J. Ripoll, Optimal latent period in a bacteriophage population model structured by infection-age, Math. Models and Methods in Appl. Sci., 21 (2011), 693-718.  doi: 10.1142/S0218202511005180.

[14]

A. Campbell, Conditions for the existence of bacteriophages, Evolution, 15 (1961), 153-165.  doi: 10.1111/j.1558-5646.1961.tb03139.x.

[15]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. -O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, Springer, 1995. doi: 10.1007/978-1-4612-4206-2.

[16]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Mathematics, 2093. Springer, 2014. doi: 10.1007/978-3-319-02231-4.

[17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. 
[18]

B. R. Levin and J. J. Bull, Population and evolutionary dynamics of phage therapy., Nature Reviews Microbiology, 2 (2004), 166-173.  doi: 10.1038/nrmicro822.

[19]

B. R. Levin and J. J. Bull, Phage therapy revisited: The population biology of a bacterial infection and its treatment with bacteriophage and antibiotics, Am. Nat., 147 (1996), 881-898.  doi: 10.1086/285884.

[20]

B. R. Levin, F. M. Stewart and L. Chao, Resouce-limited growth, competition, and predation: A model an experimental studies with bacteria and bacteriophage, Am. Nat., 111 (1977), 3-24.  doi: 10.1086/283134.

[21]

J. Mosquera and F. R. Adler, Evolution of virulence: A unified framework for coinfection and superinfection, J. Theor. Biol., 195 (1998), 293-313.  doi: 10.1006/jtbi.1998.0793.

[22]

R. J. H. Payne and V. A. A. Jansen, Pharmacokinetic principles of bacteriophage therapy, Clin. Pharmacokinetics, 42 (2003), 315-325.  doi: 10.2165/00003088-200342040-00002.

[23]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[24]

H. L. Smith and R. T. Trevino, Bacteriophage infection dynamics: Multiple host binding sites, Math. Model. Nat. Phenom., 4 (2009), 109-134.  doi: 10.1051/mmnp/20094604.

[25]

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.

[26]

G. Stent, Molecular Biology of Bacterial Viruses, W. H. Freeman and Co., London, 1963.

[27]

R. Weld, C. Butts and J. Heinemann, Models of phage growth and their applicability to phage therapy, J. Theor. Biol., 227 (2004), 1-11.  doi: 10.1016/S0022-5193(03)00262-5.

show all references

References:
[1]

S. T. Abedon, Lysis from without, Bacteriophage, 1 (2011), 46-49.  doi: 10.4161/bact.1.1.13980.

[2]

S. T. Abedon, Bacteriophage secondary infection, Virologica Sinica, 30 (2015), 3-10.  doi: 10.1007/s12250-014-3547-2.

[3]

S. T. Abedon, P. García, P. Mullany and R. Aminov, Phage therapy: Past, present and future, Frontiers in Microbiology, 8 (2017), 201700981. doi: 10.3389/978-2-88945-251-4.

[4]

S. Alizon, Co-infection and super-infection models in evolutionary epidemiology, Interface Focus, 3 (2013), 20130031. doi: 10.1098/rsfs.2013.0031.

[5]

M. van Baalen and M. W. Sabelis, The dynamics of multiple infection and the evolution of virulence, Am. Nat., 146 (1995), 881-910.  doi: 10.1086/285830.

[6]

X. Bardina, D. Bascompte, C. Rovira and S. Tindel, An analysis of a stochastic model for bacteriophage systems, Mathematical Biosciences, 241 (2013), 99-108.  doi: 10.1016/j.mbs.2012.09.009.

[7]

C. Bardina, D. Spricigo, M. Corts and M. Llagostera, Significance of the bacteriophage treatment schedule in reducing salmonella colonization of poultry, Applied and Environmental Microbiology, 78 (2012), 6600-6607.  doi: 10.1128/AEM.01257-12.

[8]

C. Barril, À. Calsina and J. Ripoll, On the reproduction number of a gut microbiota model, Bull. Math. Biol., 79 (2017), 2727-2746.  doi: 10.1007/s11538-017-0352-8.

[9]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Anal. Real World Appl., 2 (2001), 35-74.  doi: 10.1016/S0362-546X(99)00285-0.

[10]

B. J. M. Bohannan and R. E. Lenski, Linking genetic change to community evolution: Insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000), 362-377.  doi: 10.1046/j.1461-0248.2000.00161.x.

[11]

S. N. Busenberg and K. L. Cooke, The effect of integral conditions in certain equations modelling epidemics and population growth, J. Math. Biol., 10 (1980), 13-32.  doi: 10.1007/BF00276393.

[12]

B. J. Cairns, A. R. Timms, V. A. A. Jansen, I. F. Connerton and R. J. H. Payne, Quantitative models of in vitro bacteriophage host dynamics and their application to phage therapy, PLoS Pathog., 5 (2009), e1000253. doi: 10.1371/journal.ppat.1000253.

[13]

À. Calsina, J. M. Palmada and J. Ripoll, Optimal latent period in a bacteriophage population model structured by infection-age, Math. Models and Methods in Appl. Sci., 21 (2011), 693-718.  doi: 10.1142/S0218202511005180.

[14]

A. Campbell, Conditions for the existence of bacteriophages, Evolution, 15 (1961), 153-165.  doi: 10.1111/j.1558-5646.1961.tb03139.x.

[15]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. -O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, Springer, 1995. doi: 10.1007/978-1-4612-4206-2.

[16]

R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Mathematics, 2093. Springer, 2014. doi: 10.1007/978-3-319-02231-4.

[17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. 
[18]

B. R. Levin and J. J. Bull, Population and evolutionary dynamics of phage therapy., Nature Reviews Microbiology, 2 (2004), 166-173.  doi: 10.1038/nrmicro822.

[19]

B. R. Levin and J. J. Bull, Phage therapy revisited: The population biology of a bacterial infection and its treatment with bacteriophage and antibiotics, Am. Nat., 147 (1996), 881-898.  doi: 10.1086/285884.

[20]

B. R. Levin, F. M. Stewart and L. Chao, Resouce-limited growth, competition, and predation: A model an experimental studies with bacteria and bacteriophage, Am. Nat., 111 (1977), 3-24.  doi: 10.1086/283134.

[21]

J. Mosquera and F. R. Adler, Evolution of virulence: A unified framework for coinfection and superinfection, J. Theor. Biol., 195 (1998), 293-313.  doi: 10.1006/jtbi.1998.0793.

[22]

R. J. H. Payne and V. A. A. Jansen, Pharmacokinetic principles of bacteriophage therapy, Clin. Pharmacokinetics, 42 (2003), 315-325.  doi: 10.2165/00003088-200342040-00002.

[23]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[24]

H. L. Smith and R. T. Trevino, Bacteriophage infection dynamics: Multiple host binding sites, Math. Model. Nat. Phenom., 4 (2009), 109-134.  doi: 10.1051/mmnp/20094604.

[25]

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.

[26]

G. Stent, Molecular Biology of Bacterial Viruses, W. H. Freeman and Co., London, 1963.

[27]

R. Weld, C. Butts and J. Heinemann, Models of phage growth and their applicability to phage therapy, J. Theor. Biol., 227 (2004), 1-11.  doi: 10.1016/S0022-5193(03)00262-5.

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