Mean-square stability of a stochastic model for bacteriophage infection with time delays. (English) Zbl 1134.92036

Carletti, Margherita

Math. Biosci. 210, No. 2, 395-414 (2007).

Summary: We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delays. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for a very long time.

MSC:

92D40	Ecology
34K50	Stochastic functional-differential equations
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)
93E15	Stochastic stability in control theory
34K20	Stability theory of functional-differential equations
65C20	Probabilistic models, generic numerical methods in probability and statistics
92D30	Epidemiology

Keywords:

stochastic stability; numerical methods for stochastic delay differential equations; bacteriophage infection

Software:

dde23

References:

[1]	Abedon, S., Phage ecology, (Calendar, R., The Bacteriophages (2004), Oxford University Press: Oxford University Press Oxford)
[2]	Allen, L. J.S., An Introduction to Stochastic Process with Applications to Biology (2003), Prentice Hall: Prentice Hall New Jersey · Zbl 1205.60001
[3]	Baker, C. T.H.; Buckwar, E., Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math., 3, 315 (2000) · Zbl 0974.65008
[4]	Baker, C. T.H.; Buckwar, E., Exponential stability in \(p\)-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comp. Appl. Math., 184, 2, 404 (2005) · Zbl 1081.65011
[5]	Bandyopadhyay, M.; Chattopadhyay, J., Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18, 913 (2005) · Zbl 1078.34035
[6]	Beltrami, E.; Carroll, T. O., Modeling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32, 57 (1994) · Zbl 0825.92122
[7]	Beretta, E.; Carletti, M.; Solimano, F., On the effects of environmental fluctuations in a simple model of bacteria-bacteriophage interaction, Can. Appl. Math. Quart., 8, 4, 321 (2000) · Zbl 1049.34521
[8]	Beretta, E.; Kolmanowskii, V.; Shaikhet, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comp. Simul., 45, 269 (1998) · Zbl 1017.92504
[9]	Beretta, E.; Kuang, Y., Modeling and analysis of a marine bacteriophage infection, Math. Biosci., 149, 57 (1998) · Zbl 0946.92012
[10]	Beretta, E.; Kuang, Y., Modeling and analysis of a marine bacteriophage infection with latency period, Nonlin. Anal. RWA, 2, 35 (2001) · Zbl 1015.92049
[11]	Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 1144 (2002) · Zbl 1013.92034
[12]	Bergh, O.; Borsheim, K. Y.; Bratback, G.; Heldal, M., High abundance of viruses found in aquatic environments, Nature, 340, 467 (1989)
[13]	Bohannan, B. J.M.; Lenski, R. E., The effect of resource enrichment on a chemostat community of bacteria and phage, Ecology, 780, 2303 (1997)
[14]	Bohannan, B. J.M.; Lenski, R. E., Linking genetic change to community evolution: insights from studies of bacteria and bacteriophages, Ecol. Lett., 3, 362 (2000)
[15]	Brsheim, K. Y., Native marine bacteriophages, FEMS Microbiol. Ecol., 102, 141 (1992)
[17]	Campbell, A., Conditions for the existence of bacteriophage, Evolution, 15, 153 (1961)
[18]	Carletti, M., On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 175, 2, 117 (2002) · Zbl 0987.92027
[19]	Carletti, M., Numerical solution of stochastic differential problems in the biosciences, J. Comp. Appl. Math., 185, 2, 422 (2006) · Zbl 1077.65007
[20]	Carletti, M., Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection, Math. Med. Biol., 23, 297 (2006) · Zbl 1117.92032
[21]	Carletti, M.; Beretta, E., Numerical detection of instability regions for delay models with delay dependent parameters, J. Comp. Appl. Math., 205, 835 (2007) · Zbl 1124.34053
[22]	Carletti, M.; Burrage, K.; Burrage, P. M., Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Math. Comp. Simul., 64, 271 (2004) · Zbl 1039.65005
[23]	Chao, L.; Levin, B. R.; Stewart, F. M., A complex community in a simple habitat: an experimental study with bacteria and phage, Ecology, 58, 369 (1977)
[24]	Chattopadhyay, J.; Sarkar, R. R.; Chaki, S.; Bhattacharya, S., Effects of environmental fluctuations on the occurrence of malignant malaria—a model based study, Ecol. Model., 177, 179 (2004)
[25]	Fuhrman, A. J., Marine viruses and their biogeochemical and ecological effects, Nature, 399, 541 (1999)
[26]	Gourley, S. A.; Kuang, Y., A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM J. Appl. Math., 65, 2, 550 (2005) · Zbl 1068.92042
[27]	Hadas, H.; Einav, M.; Fishov, I.; Zaritsky, A., Bacteriophage T4 development depends on the physiology of its host Escherichia coli, Microbiology, 143, 179 (1997)
[28]	Hale, J. K.; Verduyn Lunel, S., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0787.34002
[29]	Jiang, S. C.; Paul, J. H., Seasonal and diel abundance of viruses and occurrence of lysogeny/bacteriocinogeny in the marine environment, Mar. Ecol. Prog. Ser., 104, 163 (1994)
[30]	Kolmanovskii, V.; Myshkis, A., Introduction to the Theory and Applications of Functional Differential Equations (1999), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0917.34001
[31]	Kuang, Y.; Fagan, W. F.; Loladze, I., Biodiversity, habitat area, resource growth rate and interference competition, Bull. Math. Biol., 65, 497 (2003) · Zbl 1334.92349
[32]	Kundu, K.; Bandopadhyay, M.; Chattopadhyay, J., Logistic growth functions may overcome extinction in a ratio-dependent predator-prey model: deterministic and stochastic approach, Nonlin. Studies, 12, 101 (2005) · Zbl 1070.92050
[33]	Kutter, E.; Guttman, B.; Carlson, K., The transition from host to phage metabolism after T4 infection, (Karam, J. D., Molecular Biology of Bacteriophage T4 (1994), American Society of Microbiology: American Society of Microbiology Washington DC), 343
[34]	Lenski, R. E.; Levin, B. R., Constrains on the coevolution of bacteria and virulent phage: a model, some experiments, and predictions for natural communities, Am. Natural., 125, 585 (1985)
[35]	Levin, B. R.; Stewart, F. M.; Chao, L., Resource-limited growth, competition, and predation: a model and experimental studies with bacteria and bacteriophage, Am. Nat., 111, 324 (1977)
[36]	Lythgoe, K. A.; Chao, L., mechanism of coexistence of a bacteria and bacteriophage in a spatially homogeneous environment, Ecol. Lett., 6, 326 (2003)
[37]	Mao, X., Stochastic Differential Equations and their Applications (1997), Horwood Publishing Ltd: Horwood Publishing Ltd Chichester
[38]	Moebus, K., Lytic and inhibition responses to bacteriophage among marine bacteria, with special reference to the origin of phage-host systems, Helgolander Meeresuntersuchungen, 36, 375 (1983)
[39]	Mohammed, S. E.A., Stochastic Functional Differential Equations (1984), Pitman: Pitman Boston Mass · Zbl 0497.60060
[40]	Nisbet, R. N.; Gurney, W. S.C., Modelling Fluctuating Populations (1982), John Wiley and Sons: John Wiley and Sons New York · Zbl 0593.92013
[41]	Proctor, L. M.; Fuhrman, J. A., Viral mortality of marine bacteria and cyanobacteria, Nature, 343, 60 (1990)
[42]	Proctor, L. M.; Okubo, A.; Fuhrman, J. A., Calibrating of phage-induced mortality in marine bacteria. Ultrastructural studies of marine bacteriophage development from one-step growth experiments, Microb. Ecol., 25, 161 (1993)
[43]	Rabinovitch, A.; Aviramb, I.; Zaritsky, A., Bacterial debrisan ecological mechanism for coexistence of bacteria and their viruses, J. Theor. Biol., 224, 377 (2003) · Zbl 1464.92150
[44]	Sarkar, R. R.; Chattopadhyay, J., The role of environmental stochasticity in a toxic phytoplankton—non-toxic phytoplankton—zooplankton system, Environmetrics, 14, 775 (2003)
[45]	Schrag, S. J.; Mittler, J. E., Host-parasite coexistence: the role of spacial refuges in stabilizing phage-bacteria interactions, Am. Natural., 148, 348 (1996)
[47]	Weld, R. J.; Butts, C.; Heinemann, J. A., Models of phage growth and their applicability to phage therapy, J. Theor. Biol., 227, 1 (2004) · Zbl 1439.92112