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June  2017, 22(4): 1231-1252. doi: 10.3934/dcdsb.2017060

Stability analysis of an enteropathogen population growing within a heterogeneous group of animals

Carles Barril , and  Àngel Calsina

Faculty of Sciences, Universitat Autónoma de Barcelona, 08193 Bellaterra, 08193 Bellaterra, Barcelona, Spain

* Corresponding authorr: carlesbarril@mat.uab.cat

Received  October 2015 Revised  November 2016 Published  February 2017

Fund Project: The first author is supported by Spanish Ministry of Education grant FPU13/04333.

An autonomous semi-linear hyperbolic pde system for the proliferation of bacteria within a heterogeneous population of animals is presented and analysed. It is assumed that bacteria grow inside the intestines and that they can be either attached to the epithelial wall or as free particles in the lumen. A condition involving ecological parameters is given, which can be used to decide the existence of endemic equilibria as well as local stability properties of the non-endemic one. Some implications on phage therapy are addressed.

Keywords: Mathematical epidemiology, phage therapy, steady state stability, spatially structured population, semilinear formulation, characteristic equation.
Mathematics Subject Classification: Primary:92D25, 35B40;Secondary:35L50.
Citation: Carles Barril, Àngel Calsina. Stability analysis of an enteropathogen population growing within a heterogeneous group of animals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1231-1252. doi: 10.3934/dcdsb.2017060
References:
[1]

R. J. Atterbury, M. A. van Bergen, F. Ortiz, M. A. Lovell, J. A. Harris, A. De Boer, J. A. Wagenaar, V. M. Allen and P. A. Barrow, Bacteriophage therapy to reduce salmonella colonization of broiler chickens, Applied and Environmental Microbiology, 73 (2007), 4543-4549.  doi: 10.1128/AEM.00049-07.  Google Scholar

[2]

M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment, Microbial Ecology, 41 (2001), 210-221.  doi: 10.1007/s002480000005.  Google Scholar

[3]

B. Boldin, Persistence and spread of gastro-intestinal infections: The case of enterotoxigenic escherichia coli in piglets, Bulletin of Mathematical Biology, 70 (2008), 2077-2101.  doi: 10.1007/s11538-008-9348-8.  Google Scholar

[4]

F. Brauer, Mathematical epidemiology is not an oxymoron BMC Public Health, 9 (2009), S2. doi: 10.1186/1471-2458-9-S1-S2.  Google Scholar

[5]

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

[6]

À. 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

[7]

S. Chow and J. K. Hale, Methods of Bifurcation Theory Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[8]

P. Clément, O. Diekmann, M. Gyllenberg, H. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups, Mathematische Annalen, 277 (1987), 709-725.  doi: 10.1007/BF01457866.  Google Scholar

[9]

P. Clément, O. Diekmann, M. Gyllenberg, H. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups. Ⅲ. nonlinear lipschitz continuous perturbations in the sun-reflexive case, Pitman Research Notes in Mathematics, 190 (1989), 67-89.   Google Scholar

[10]

M. E. Coleman, D. W. Dreesen and R. G. Wiegert, A simulation of microbial competition in the human colonic ecosystem, Applied and Environmental Microbiology, 62 (1996), 3632-3639.   Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[12]

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

[13]

O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of volterra functional equations in the light of suns and stars, SIAM Journal on Mathematical Analysis, 39 (2007), 1023-1069.  doi: 10.1137/060659211.  Google Scholar

[14]

K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[15]

F. Gaggìa, P. Mattarelli and B. Biavati, Probiotics and prebiotics in animal feeding for safe food production, International Journal of Food Microbiology, 141 (2010), S15-S28.   Google Scholar

[16]

H. W. Hethcote and J. W. van Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Mathematical Biosciences, 84 (1987), 85-118.  doi: 10.1016/0025-5564(87)90044-7.  Google Scholar

[17]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[18]

M. Lichtner, Variation of constants formula for hyperbolic systems, Journal of Applied Analysis, 15 (2009), 79-100.  doi: 10.1515/JAA.2009.79.  Google Scholar

[19]

B. M. Marshall and S. B. Levy, Food animals and antimicrobials: Impacts on human health, Clinical Microbiology Reviews, 24 (2011), 718-733.  doi: 10.1128/CMR.00002-11.  Google Scholar

[20]

R. Nagel and J. Poland, The critical spectrum of strongly continuous semigroup, Advances in Mathematics, 152 (2000), 120-133.  doi: 10.1006/aima.1998.1893.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

H. L. Smith, Models of virulent phage growth with application to phage therapy, SIAM Journal on Applied Mathematics, 68 (2008), 1717-1737.  doi: 10.1137/070704514.  Google Scholar

[23]

H. L. Smith and H. R. Thieme, Chemostats and epidemics: Competition for nutrients/hosts, Math. Biosci. Eng., 10 (2013), 1635-1650.  doi: 10.3934/mbe.2013.10.1635.  Google Scholar

[24]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[25]

W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

show all references

References:
[1]

R. J. Atterbury, M. A. van Bergen, F. Ortiz, M. A. Lovell, J. A. Harris, A. De Boer, J. A. Wagenaar, V. M. Allen and P. A. Barrow, Bacteriophage therapy to reduce salmonella colonization of broiler chickens, Applied and Environmental Microbiology, 73 (2007), 4543-4549.  doi: 10.1128/AEM.00049-07.  Google Scholar

[2]

M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment, Microbial Ecology, 41 (2001), 210-221.  doi: 10.1007/s002480000005.  Google Scholar

[3]

B. Boldin, Persistence and spread of gastro-intestinal infections: The case of enterotoxigenic escherichia coli in piglets, Bulletin of Mathematical Biology, 70 (2008), 2077-2101.  doi: 10.1007/s11538-008-9348-8.  Google Scholar

[4]

F. Brauer, Mathematical epidemiology is not an oxymoron BMC Public Health, 9 (2009), S2. doi: 10.1186/1471-2458-9-S1-S2.  Google Scholar

[5]

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

[6]

À. 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

[7]

S. Chow and J. K. Hale, Methods of Bifurcation Theory Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[8]

P. Clément, O. Diekmann, M. Gyllenberg, H. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups, Mathematische Annalen, 277 (1987), 709-725.  doi: 10.1007/BF01457866.  Google Scholar

[9]

P. Clément, O. Diekmann, M. Gyllenberg, H. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups. Ⅲ. nonlinear lipschitz continuous perturbations in the sun-reflexive case, Pitman Research Notes in Mathematics, 190 (1989), 67-89.   Google Scholar

[10]

M. E. Coleman, D. W. Dreesen and R. G. Wiegert, A simulation of microbial competition in the human colonic ecosystem, Applied and Environmental Microbiology, 62 (1996), 3632-3639.   Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[12]

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

[13]

O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of volterra functional equations in the light of suns and stars, SIAM Journal on Mathematical Analysis, 39 (2007), 1023-1069.  doi: 10.1137/060659211.  Google Scholar

[14]

K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[15]

F. Gaggìa, P. Mattarelli and B. Biavati, Probiotics and prebiotics in animal feeding for safe food production, International Journal of Food Microbiology, 141 (2010), S15-S28.   Google Scholar

[16]

H. W. Hethcote and J. W. van Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Mathematical Biosciences, 84 (1987), 85-118.  doi: 10.1016/0025-5564(87)90044-7.  Google Scholar

[17]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[18]

M. Lichtner, Variation of constants formula for hyperbolic systems, Journal of Applied Analysis, 15 (2009), 79-100.  doi: 10.1515/JAA.2009.79.  Google Scholar

[19]

B. M. Marshall and S. B. Levy, Food animals and antimicrobials: Impacts on human health, Clinical Microbiology Reviews, 24 (2011), 718-733.  doi: 10.1128/CMR.00002-11.  Google Scholar

[20]

R. Nagel and J. Poland, The critical spectrum of strongly continuous semigroup, Advances in Mathematics, 152 (2000), 120-133.  doi: 10.1006/aima.1998.1893.  Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

H. L. Smith, Models of virulent phage growth with application to phage therapy, SIAM Journal on Applied Mathematics, 68 (2008), 1717-1737.  doi: 10.1137/070704514.  Google Scholar

[23]

H. L. Smith and H. R. Thieme, Chemostats and epidemics: Competition for nutrients/hosts, Math. Biosci. Eng., 10 (2013), 1635-1650.  doi: 10.3934/mbe.2013.10.1635.  Google Scholar

[24]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[25]

W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

Figure Scheme 1.   
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Figure 1.  Bifurcation diagram showing epidemic progression (dark regions) or eradication (white regions) in a system with two hosts. The changing parameters are the fraction of bacteriophage given to the first host ($q_0^1/(q_0^1+q_0^2)$ ranging from 0 to 1) and its detachment rate ($\delta_1$ ranging from 0 to 1.5). The bacterial and bacteriophage distributions along the intestine once the system has converged to the equilibria are shown for two different set of parameters A and B. The dashed line refers to attached bacteria while gray color is used for host one and black for host two. The other fix parameters used to do the numeric simulations are the total bacteriophage dose per time unit $q_0^1+q_0^2=11$ and: $c_h=l_h=1$, $\gamma_1^h(u)=1-u$, $\gamma_2^h(v)=1-v$, $\alpha_h =4$, $b=4$, $\kappa_1^h=0.06$, $\kappa_2^h=0.1$, $\lambda_1^h=\lambda_2^h=0.1$, $\mu_1=0.4$ and $\mu_2=0.1$ for all $h\in{1,2}$ and $\delta_2=0.5$. Notice that the two hosts only differ in their detachment rate ($\delta_1$ and $\delta_2$) and the treatment received ($q_0^1$ and $q_0^2$).
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