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October  2017, 14(5&6): 1361-1377. doi: 10.3934/mbe.2017070

A bacteriophage model based on CRISPR/Cas immune system in a chemostat

Mengshi Shu , Rui Fu and  Wendi Wang ,

Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Wendi Wang

Received  May 31, 2016 Accepted  November 21, 2016 Published  May 2017

Fund Project: Supported in part by the NSF of China (11571284)

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.

Keywords: Bacteriophage, bifurcation, bistability, backward, evolution.
Mathematics Subject Classification: Primary: 92D30; Secondary: 34D20.
Citation: Mengshi Shu, Rui Fu, Wendi Wang. A bacteriophage model based on CRISPR/Cas immune system in a chemostat. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1361-1377. doi: 10.3934/mbe.2017070
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

D. H. Duckworth, Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976), 793-802. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

P. Gómez and A. Buckling, Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011), 106-109. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763. doi: 10.1137/0134064. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

T. Li, Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011), 1297-1303. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993. Google Scholar

[38]

L. M. Proctor and J. A. Fuhrman, Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990), 60-62. doi: 10.1038/343060a0. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[45]

R. A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

X. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

D. H. Duckworth, Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976), 793-802. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

P. Gómez and A. Buckling, Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011), 106-109. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763. doi: 10.1137/0134064. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

T. Li, Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011), 1297-1303. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993. Google Scholar

[38]

L. M. Proctor and J. A. Fuhrman, Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990), 60-62. doi: 10.1038/343060a0. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[45]

R. A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

X. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

Figure 1.  Bifurcation graphs for $\kappa\geq\mu$ in case $(C1)$. Panel (a) shows the forward bifurcation with $\varepsilon=0.01$. Panel (b) indicates the backward bifurcation with $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents the branch point $E_1$
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Figure 2.  Bifurcation graphs for case $(C2)$. Panel (a) shows the forward bifurcation with $\varepsilon=0.3$ and panel (b) demonstrates the backward bifurcation with $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents a branch point. The solid lines represent stable branches and dashed lines mean unstable branches
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Figure 3.  Bifurcation graphs for case $(C3)$. Panel (a) shows the forward bifurcation at $E_1$ and a transcritical bifurcation at $E_3$ when $\varepsilon=0.1$ and panel (b) demonstrates the bistable phenomena between $E_1, ~E_3$ and a transcritical bifurcation at $E_4$ when $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents a branch point. The solid lines represent stable branches and dashed lines mean unstable branches
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Figure 4.  Graphs of bistable behaviors in case $(C1)$ with $\kappa\geq\mu$. Panel (a) shows the bistability of the infection-free equilibrium and an coexist equilibrium where $\varepsilon=0.01, ~b=130$. Panel (b) indicates the bistable coexistence of the infection-free equilibria with a stable periodic solution where $\varepsilon=0.9, ~b=260$
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