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Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse
A bacteriophage model based on CRISPR/Cas immune system in a chemostat
Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
Clustered regularly interspaced short palindromic repeats (CRISPRs) along with Cas proteins are a widespread immune system across bacteria and archaea. In this paper, a mathematical model in a chemostat is proposed to investigate the effect of CRISPR/Cas on the bacteriophage dynamics. It is shown that the introduction of CRISPR/Cas can induce a backward bifurcation and transcritical bifurcation. Numerical simulations reveal the coexistence of a stable infection-free equilibrium with an infection equilibrium, or a stable infection-free equilibrium with a stable periodic solution.
References:
[1] |
L. J. Allen and S. W. Vidurupola,
Impact of variability in stochastic models of bacteria-phage dynamics applicable to phage therapy, Stochastic Analysis and Applications, 32 (2014), 427-449.
doi: 10.1080/07362994.2014.889922. |
[2] |
I. Aviram and A. Rabinovitch,
Bactria and lytic phage coexistence in a chemostat with periodic nutrient supply, Bulletin of Mathematical Biology, 76 (2014), 225-244.
doi: 10.1007/s11538-013-9917-3. |
[3] |
E. Beretta and Y. Kuang,
Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76.
doi: 10.1016/S0025-5564(97)10015-3. |
[4] |
E. Beretta and Y. Kuang,
Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001), 35-74.
doi: 10.1016/S0362-546X(99)00285-0. |
[5] |
B. J. 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. |
[6] |
S. J. Brouns, M. M. Jore and M. Lundgren,
Small CRISPR RNAs guide antiviral defense in prokaryotes, Science, 321 (2008), 960-964.
doi: 10.1126/science.1159689. |
[7] |
A. Buckling and M. Brockhurst,
Bacteria-virus coevolution, Evolutionary Systems Biology, 751 (2012), 347-370.
doi: 10.1007/978-1-4614-3567-9_16. |
[8] |
J. J. Bull, C. S. Vegge and M. Schmerer, Phenotypic resistance and the dynamics of bacterial escape from phage control PloS One, 9 (2014), e94690.
doi: 10.1371/journal.pone.0094690. |
[9] |
B. J. Cairns, A. R. Timms and V. Jansen, Quantitative models of vitro bacteriophage-host dynamics and their application to phage therapy PLoS Pathog, 5 (2009), e1000253.
doi: 10.1371/journal.ppat.1000253. |
[10] |
A. Calsina and J. J. Rivaud,
A size structured model for bacteria-phages interaction, Nonlinear Analysis: Real World Applications, 15 (2014), 100-117.
doi: 10.1016/j.nonrwa.2013.06.004. |
[11] |
A. Campbell,
Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165.
doi: 10.2307/2406076. |
[12] |
C. L. Carrillo, R. J. Atterbury and A. El-Shibiny,
Bacteriophage therapy to reduce Campylobacter jejuni colonization of broiler chickens, Applied and Environmental Microbiology, 71 (2005), 6554-6563.
doi: 10.1128/AEM.71.11.6554-6563.2005. |
[13] |
J. J. Dennehy, What can phages tell us about host-pathogen coevolution?
International Journal of Evolutionary Biology, 2012 (2012), Article ID 396165, 12 pages.
doi: 10.1155/2012/396165. |
[14] |
H. Deveau, J. E. Garneau and S. Moineau,
CRISPR/Cas system and its role in phage-bacteria interactions, Annual Review of Microbiology, 64 (2010), 475-493.
doi: 10.1146/annurev.micro.112408.134123. |
[15] |
A. Dhooge, W. Govaerts and Y. A. Kuznetsov,
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans Math Software, 29 (2003), 141-164.
doi: 10.1145/779359.779362. |
[16] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[17] |
D. H. Duckworth,
Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976), 793-802.
|
[18] |
P. C. Fineran and E. Charpentier,
Memory of viral infections by CRISPR-Cas adaptive immune systems: Acquisition of new information, Virology, 434 (2012), 202-209.
doi: 10.1016/j.virol.2012.10.003. |
[19] |
J. E. Garneau and M. Dupuis,
The CRISPR/Cas bacterial immune system cleaves bacteriophage and plasmid DNA, Nature, 468 (2010), 67-71.
doi: 10.1038/nature09523. |
[20] |
P. Gómez and A. Buckling,
Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011), 106-109.
|
[21] |
S. A. Gourley and Y. Kuang,
A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal of Applied Mathematics, 65 (2004), 550-566.
doi: 10.1137/S0036139903436613. |
[22] |
J. K. Hale and S. M. V. Lunel,
Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[23] |
Z. Han and H. L. Smith,
Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat, Mathematical Biosciences and Engineering, 9 (2012), 737-765.
doi: 10.3934/mbe.2012.9.737. |
[24] |
S. B. Hsu, S. Hubbell and P. Waltman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[25] |
S. B. Hsu,
Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.
doi: 10.1137/0134064. |
[26] |
P. Horvath and R. Barrangou,
CRISPR/Cas the immune system of bacteria and archaea, Science, 327 (2010), 167-170.
doi: 10.1126/science.1179555. |
[27] |
J. Iranzo, A. E. Lobkovsky and Y. I. Wolf,
Evolutionary dynamics of the prokaryotic adaptive immunity system CRISPR-Cas in an explicit ecological context, Journal of Bacteriology, 195 (2013), 3834-3844.
doi: 10.1128/JB.00412-13. |
[28] |
B. R. Levin, F. M. Stewart and L. Chao,
Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977), 3-24.
doi: 10.1086/283134. |
[29] |
B. R. Levin, Nasty viruses, costly plasmids, population dynamics, and the conditions for establishing and maintaining CRISPR-mediated adaptive immunity in bacteria PLoS Genet, 6 (2010), e1001171.
doi: 10.1371/journal.pgen.1001171. |
[30] |
B. R. Levin, S. Moineau and M. Bushman, The population and evolutionary dynamics of phage and bacteria with CRISPR-mediated immunity PLoS Genet, 9 (2013), e1003312.
doi: 10.1371/journal.pgen.1003312. |
[31] |
T. Li,
Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011), 1297-1303.
|
[32] |
M. Lin, H. F. Huo and Y. N. Li,
A competitive model in a chemostat with nutrient recycling and antibiotic treatment, Nonlinear Analysis: Real World Applications, 13 (2012), 2540-2555.
doi: 10.1016/j.nonrwa.2012.02.016. |
[33] |
Y. Ma and H. Chang,
A review on immune system of the bacteria and its self versus non-self discrimination, Chinese Veterinary Science, 42 (2012), 657-660.
|
[34] |
L. A. Marraffini and E. J. Sontheimer,
Self versus non-self discrimination during CRISPR RNA-directed immunity, Nature, 463 (2010), 568-571.
doi: 10.1038/nature08703. |
[35] |
S. Matsuzaki, M. Rashel and J. Uchiyama,
Bacteriophage therapy: a revitalized therapy against bacterial infectious diseases, Journal of Infection and Chemotherapy, 11 (2005), 211-219.
doi: 10.1007/s10156-005-0408-9. |
[36] |
K. Northcott and M. Imran,
Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat, Journal of Mathematical Biology, 64 (2012), 1043-1086.
doi: 10.1007/s00285-011-0439-z. |
[37] |
L. Perko,
Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993. |
[38] |
L. M. Proctor and J. A. Fuhrman,
Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990), 60-62.
doi: 10.1038/343060a0. |
[39] |
J. Reeks, J. H. Naismith and M. F. White,
CRISPR interference: A structural perspective, Biochemical Journal, 453 (2013), 155-166.
doi: 10.1042/BJ20130316. |
[40] |
G. Robledo, F. Grognard and J. L. Gouzé,
Global stability for a model of competition in the chemostat with microbial inputs, Nonlinear Analysis: Real World Applications, 13 (2012), 582-598.
doi: 10.1016/j.nonrwa.2011.07.049. |
[41] |
K. D. Seed and D. W. Lazinski,
A bacteriophage encodes its own CRISPR/Cas adaptive response to evade host innate immunity, Nature, 494 (2013), 489-491.
doi: 10.1038/nature11927. |
[42] |
H. L. Smith and H. R. Thieme,
Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012), 951-979.
doi: 10.1007/s00285-011-0434-4. |
[43] |
H. R. Thieme,
Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[44] |
H. R. Thieme,
Persistence under relaxed point-dissipativity with application to an endemic model, SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[45] |
R. A. Usmani,
Applied Linear Algebra, Marcel Dekker, New York, 1987. |
[46] |
W. Wang and X. Zhao,
An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112.
doi: 10.1016/j.mbs.2002.11.001. |
[47] |
R. J. Weld, C. Butts and J. A. Heinemann,
Models of phage growth and their applicability to phage therapy, Journal of Theoretical Biology, 227 (2004), 1-11.
doi: 10.1016/S0022-5193(03)00262-5. |
[48] |
X. Zhao,
Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003.
doi: 10.1007/978-0-387-21761-1. |
show all references
References:
[1] |
L. J. Allen and S. W. Vidurupola,
Impact of variability in stochastic models of bacteria-phage dynamics applicable to phage therapy, Stochastic Analysis and Applications, 32 (2014), 427-449.
doi: 10.1080/07362994.2014.889922. |
[2] |
I. Aviram and A. Rabinovitch,
Bactria and lytic phage coexistence in a chemostat with periodic nutrient supply, Bulletin of Mathematical Biology, 76 (2014), 225-244.
doi: 10.1007/s11538-013-9917-3. |
[3] |
E. Beretta and Y. Kuang,
Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76.
doi: 10.1016/S0025-5564(97)10015-3. |
[4] |
E. Beretta and Y. Kuang,
Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001), 35-74.
doi: 10.1016/S0362-546X(99)00285-0. |
[5] |
B. J. 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. |
[6] |
S. J. Brouns, M. M. Jore and M. Lundgren,
Small CRISPR RNAs guide antiviral defense in prokaryotes, Science, 321 (2008), 960-964.
doi: 10.1126/science.1159689. |
[7] |
A. Buckling and M. Brockhurst,
Bacteria-virus coevolution, Evolutionary Systems Biology, 751 (2012), 347-370.
doi: 10.1007/978-1-4614-3567-9_16. |
[8] |
J. J. Bull, C. S. Vegge and M. Schmerer, Phenotypic resistance and the dynamics of bacterial escape from phage control PloS One, 9 (2014), e94690.
doi: 10.1371/journal.pone.0094690. |
[9] |
B. J. Cairns, A. R. Timms and V. Jansen, Quantitative models of vitro bacteriophage-host dynamics and their application to phage therapy PLoS Pathog, 5 (2009), e1000253.
doi: 10.1371/journal.ppat.1000253. |
[10] |
A. Calsina and J. J. Rivaud,
A size structured model for bacteria-phages interaction, Nonlinear Analysis: Real World Applications, 15 (2014), 100-117.
doi: 10.1016/j.nonrwa.2013.06.004. |
[11] |
A. Campbell,
Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165.
doi: 10.2307/2406076. |
[12] |
C. L. Carrillo, R. J. Atterbury and A. El-Shibiny,
Bacteriophage therapy to reduce Campylobacter jejuni colonization of broiler chickens, Applied and Environmental Microbiology, 71 (2005), 6554-6563.
doi: 10.1128/AEM.71.11.6554-6563.2005. |
[13] |
J. J. Dennehy, What can phages tell us about host-pathogen coevolution?
International Journal of Evolutionary Biology, 2012 (2012), Article ID 396165, 12 pages.
doi: 10.1155/2012/396165. |
[14] |
H. Deveau, J. E. Garneau and S. Moineau,
CRISPR/Cas system and its role in phage-bacteria interactions, Annual Review of Microbiology, 64 (2010), 475-493.
doi: 10.1146/annurev.micro.112408.134123. |
[15] |
A. Dhooge, W. Govaerts and Y. A. Kuznetsov,
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans Math Software, 29 (2003), 141-164.
doi: 10.1145/779359.779362. |
[16] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[17] |
D. H. Duckworth,
Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976), 793-802.
|
[18] |
P. C. Fineran and E. Charpentier,
Memory of viral infections by CRISPR-Cas adaptive immune systems: Acquisition of new information, Virology, 434 (2012), 202-209.
doi: 10.1016/j.virol.2012.10.003. |
[19] |
J. E. Garneau and M. Dupuis,
The CRISPR/Cas bacterial immune system cleaves bacteriophage and plasmid DNA, Nature, 468 (2010), 67-71.
doi: 10.1038/nature09523. |
[20] |
P. Gómez and A. Buckling,
Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011), 106-109.
|
[21] |
S. A. Gourley and Y. Kuang,
A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal of Applied Mathematics, 65 (2004), 550-566.
doi: 10.1137/S0036139903436613. |
[22] |
J. K. Hale and S. M. V. Lunel,
Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[23] |
Z. Han and H. L. Smith,
Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat, Mathematical Biosciences and Engineering, 9 (2012), 737-765.
doi: 10.3934/mbe.2012.9.737. |
[24] |
S. B. Hsu, S. Hubbell and P. Waltman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[25] |
S. B. Hsu,
Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.
doi: 10.1137/0134064. |
[26] |
P. Horvath and R. Barrangou,
CRISPR/Cas the immune system of bacteria and archaea, Science, 327 (2010), 167-170.
doi: 10.1126/science.1179555. |
[27] |
J. Iranzo, A. E. Lobkovsky and Y. I. Wolf,
Evolutionary dynamics of the prokaryotic adaptive immunity system CRISPR-Cas in an explicit ecological context, Journal of Bacteriology, 195 (2013), 3834-3844.
doi: 10.1128/JB.00412-13. |
[28] |
B. R. Levin, F. M. Stewart and L. Chao,
Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977), 3-24.
doi: 10.1086/283134. |
[29] |
B. R. Levin, Nasty viruses, costly plasmids, population dynamics, and the conditions for establishing and maintaining CRISPR-mediated adaptive immunity in bacteria PLoS Genet, 6 (2010), e1001171.
doi: 10.1371/journal.pgen.1001171. |
[30] |
B. R. Levin, S. Moineau and M. Bushman, The population and evolutionary dynamics of phage and bacteria with CRISPR-mediated immunity PLoS Genet, 9 (2013), e1003312.
doi: 10.1371/journal.pgen.1003312. |
[31] |
T. Li,
Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011), 1297-1303.
|
[32] |
M. Lin, H. F. Huo and Y. N. Li,
A competitive model in a chemostat with nutrient recycling and antibiotic treatment, Nonlinear Analysis: Real World Applications, 13 (2012), 2540-2555.
doi: 10.1016/j.nonrwa.2012.02.016. |
[33] |
Y. Ma and H. Chang,
A review on immune system of the bacteria and its self versus non-self discrimination, Chinese Veterinary Science, 42 (2012), 657-660.
|
[34] |
L. A. Marraffini and E. J. Sontheimer,
Self versus non-self discrimination during CRISPR RNA-directed immunity, Nature, 463 (2010), 568-571.
doi: 10.1038/nature08703. |
[35] |
S. Matsuzaki, M. Rashel and J. Uchiyama,
Bacteriophage therapy: a revitalized therapy against bacterial infectious diseases, Journal of Infection and Chemotherapy, 11 (2005), 211-219.
doi: 10.1007/s10156-005-0408-9. |
[36] |
K. Northcott and M. Imran,
Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat, Journal of Mathematical Biology, 64 (2012), 1043-1086.
doi: 10.1007/s00285-011-0439-z. |
[37] |
L. Perko,
Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993. |
[38] |
L. M. Proctor and J. A. Fuhrman,
Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990), 60-62.
doi: 10.1038/343060a0. |
[39] |
J. Reeks, J. H. Naismith and M. F. White,
CRISPR interference: A structural perspective, Biochemical Journal, 453 (2013), 155-166.
doi: 10.1042/BJ20130316. |
[40] |
G. Robledo, F. Grognard and J. L. Gouzé,
Global stability for a model of competition in the chemostat with microbial inputs, Nonlinear Analysis: Real World Applications, 13 (2012), 582-598.
doi: 10.1016/j.nonrwa.2011.07.049. |
[41] |
K. D. Seed and D. W. Lazinski,
A bacteriophage encodes its own CRISPR/Cas adaptive response to evade host innate immunity, Nature, 494 (2013), 489-491.
doi: 10.1038/nature11927. |
[42] |
H. L. Smith and H. R. Thieme,
Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012), 951-979.
doi: 10.1007/s00285-011-0434-4. |
[43] |
H. R. Thieme,
Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[44] |
H. R. Thieme,
Persistence under relaxed point-dissipativity with application to an endemic model, SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[45] |
R. A. Usmani,
Applied Linear Algebra, Marcel Dekker, New York, 1987. |
[46] |
W. Wang and X. Zhao,
An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112.
doi: 10.1016/j.mbs.2002.11.001. |
[47] |
R. J. Weld, C. Butts and J. A. Heinemann,
Models of phage growth and their applicability to phage therapy, Journal of Theoretical Biology, 227 (2004), 1-11.
doi: 10.1016/S0022-5193(03)00262-5. |
[48] |
X. Zhao,
Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003.
doi: 10.1007/978-0-387-21761-1. |




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