2014, 11(3): 403-425. doi: 10.3934/mbe.2014.11.403

Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042

Received  April 2012 Revised  May 2013 Published  January 2014

Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.
Citation: Edward J. Allen. Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology. Mathematical Biosciences & Engineering, 2014, 11 (3) : 403-425. doi: 10.3934/mbe.2014.11.403
References:
[1]

W. Aiello and H. Freedman, A time-delay model of single-species growth with stage structure, Mathematical Biosciences, 101 (1990), 139-153. doi: 10.1016/0025-5564(90)90019-U.

[2]

E. Allen, S. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics and Stochastics Reports, 64 (1998), 117-142. doi: 10.1080/17442509808834159.

[3]

E. Allen, Modeling With Itô Stochastic Differential Equations, Springer, Dordrecht, 2007.

[4]

E. Allen, L. Allen, A. Arciniega and P. Greenwood, Construction of equivalent stochastic differential equation models, Stoch. Anal. Appl., 26 (2008), 274-297. doi: 10.1080/07362990701857129.

[5]

L. Allen, An Introduction to Stochastic Processes with Applications to Biology, 2nd edition, CRC Press, Boca Raton, 2010.

[6]

L. Allen, An Introduction to Mathematical Biology, Prentice Hall, Upper Saddle River, 2007.

[7]

J. Al-Omari and A. Tallafha, Modelling and analysis of stage-structured population model with state-dependent maturation delay and harvesting, Int. Journal of Math. Analysis, 1 (2007), 391-407.

[8]

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.

[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]

L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Modell., 34 (2010), 1405-1417. doi: 10.1016/j.apm.2009.08.027.

[11]

G. Bocharov and F. Rihan, Numerical modeling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4.

[12]

E. Cabaña, The vibrating string forced by white noise, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 15 (1970), 111-130. doi: 10.1007/BF00531880.

[13]

S. Chisholm, S. Frankel, R. Goericke, R. Olson, B. Palenik, J. Waterbury, L. West-Johnsrud and E. Zettler, Prochlorococcus-marinus nov gen-nov sp - an oxyphototrophic marine prokaryote containing divinyl chlorophyll-a and chlorophyll-B, Archives of Microbiology, 157 (1992), 297-300. doi: 10.1007/BF00245165.

[14]

J. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics 20, Springer, Berlin, Heidelberg, New York, 1977.

[15]

J. Filée, F. Tétart, C. Suttle and H. Krisch, Marine T4-type bacteriophages, a ubiquitous component of the dark matter of the biosphere, Proc. Nat. Acad. Sci., 102 (2005), 12471-12476. doi: 10.1073/pnas.0503404102.

[16]

J. Fuhrman, Marine viruses and their biogeochemical and ecological effects, Nature, 399 (1999), 541-548.

[17]

T. Gard, Introduction to Stochastic Differential Equations, Marcel Decker, New York, 1987.

[18]

D. Gillespie, Exact stochastic simulation of coupled chemical reactions, The Journal of Physical Chemistry, 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[19]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluweer Academic Publishers, Dordrecht, The Netherlands, 1992.

[20]

S. Gourley and S. Ruan, Dynamics of the diffusive Nicholson's blowfies equation with distributed delay, Proceedings of the Royal Society of Edinburgh, 130 (2000), 1275-1291. doi: 10.1017/S0308210500000688.

[21]

S. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM J. Appl. Math., 65 (2005), 550-566. doi: 10.1137/S0036139903436613.

[22]

W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[23]

T. Hilleman, Environmental Biology, Science Publishers, Enfield, New Hampshire, 2009. doi: 10.1201/b10187.

[24]

V. Jansen, R. Nisbet and W. Gurney, Generation cycles in stage structured populations, Bulletin of Mathematical Biology, 52 (1990), 375-396. doi: 10.1016/S0092-8240(05)80217-4.

[25]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York, 1992.

[26]

P. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.

[27]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993.

[28]

T. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential equations, Journal of Applied Probability, 8 (1971), 344-356. doi: 10.2307/3211904.

[29]

E. Kutter and A. Sulakvelidze, Bacteriophages: Biology and Applications, CRC Press, Boca Raton, Florida, 2004. doi: 10.1201/9780203491751.

[30]

P. Langevin, Sur la théorie du mouvement brownien, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530-533.

[31]

J. Li, Y. Kuang and C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, J. Theor. Biol., 242 (2006), 722-735. doi: 10.1016/j.jtbi.2006.04.002.

[32]

S. Liu, Z. Liu and J. Tang, A delayed marine bacteriophage infection model, Applied Mathematics Letters, 20 (2007), 702-706. doi: 10.1016/j.aml.2006.06.017.

[33]

A. Makroglou, I. Karaoustas, J. Li and Y. Kuang, Delay differential equation models in diabetes modeling: A review, in EOLSS encyclopedia, (eds. A. de Gaetano, P. Palumbo), 2011.

[34]

C. Munn, Marine Microbiology: Ecology and Applications, 2nd edition, Garland Science, Taylor and Francis Group, New York, 2011.

[35]

D. Oreopoulos, R. Lindeman, D. VanderJagt, A. Tzamaloukas, H. Bhagavan and P. Garry, Renal excretion of ascorbic acid: Effect of age and sex, Journal of the American Collge of Nutrition, 12 (1993), 537-542. doi: 10.1080/07315724.1993.10718349.

[36]

J. Paul, M. Sullivan, A. Segall and F. Rohwer, Marine phage genomics, Comparative Biochemistry and Physiology Part B, 133 (2002), 463-476. doi: 10.1016/S1096-4959(02)00168-9.

[37]

S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, ( eds. O. Arino, M. Hbid and E. Ait Dads), Springer, Berlin, 2006, 477-517. doi: 10.1007/1-4020-3647-7_11.

[38]

E. Shapiro, H. Tillil, K. Polonsky, V. Fang, A. Rubenstein and E. Van Cauter, Oscillations in insulin secretion during constant glucose infusion in normal man: Relationship to changes in plasma glucose, The Journal of Clinical Endocrinology & Metabolism, 67 (1988), 307-314. doi: 10.1210/jcem-67-2-307.

[39]

C. Simon, G. Brandenberger and M. Follenius, Ultradian oscillations of plasma glucose, insulin, and C-peptide in man during continuous enteral nutrition, The Journal of Clinical Endocrinology & Metabolism, 64 (1987), 669-674. doi: 10.1210/jcem-64-4-669.

[40]

J. Sturis, K. Polonsky, E. Mosekilde and E. Van Cauter, Computer-model for mechanisms underlying ultradian oscillations of insulin and glucose, Am. J. of Physiol. Endocrinol. Metab., 260 (1991), E801-E809.

[41]

Y. Su, J. Wei and J. Shi, Bifurcation analysis in a delayed diffusive Nicholson's blowflies equation, Nonlinear Analysis: Real World Applications, 11 (2010), 1692-1703. doi: 10.1016/j.nonrwa.2009.03.024.

[42]

M. Sullivan, J. Waterbury and S. Chisholm, Cyanophages infecting the oceanic cyanobacterium Prochlorococcus, Nature, 424 (2003), 1047-1051. doi: 10.1038/nature01929.

[43]

M. Sullivan, M. Coleman, P. Weigele, F. Rohwer and S. Chisholm, Three Prochlorococcus Cyanophage genomes: Signature features and ecological interpretations, PLoS Biology, 3 (2005), 790-806. doi: 10.1371/journal.pbio.0030144.

[44]

C. Suttle, Marine viruses major players in the global ecosystem, Nature Reviews Microbiology, 5 (2007), 801-812. doi: 10.1038/nrmicro1750.

[45]

I. Tolic, E. Mosekilde and J. Sturis, Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion, J. Theor. Biol., 207 (2000), 361-375.

[46]

J. Walsh, An introduction to stochastic partial differential equations, in Notes in Mathematics,, Volume 1180, (eds. A. Dold and B. Eckmann), Springer-Verlag, Berlin, 1986, 265-439. doi: 10.1007/BFb0074920.

show all references

References:
[1]

W. Aiello and H. Freedman, A time-delay model of single-species growth with stage structure, Mathematical Biosciences, 101 (1990), 139-153. doi: 10.1016/0025-5564(90)90019-U.

[2]

E. Allen, S. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stochastics and Stochastics Reports, 64 (1998), 117-142. doi: 10.1080/17442509808834159.

[3]

E. Allen, Modeling With Itô Stochastic Differential Equations, Springer, Dordrecht, 2007.

[4]

E. Allen, L. Allen, A. Arciniega and P. Greenwood, Construction of equivalent stochastic differential equation models, Stoch. Anal. Appl., 26 (2008), 274-297. doi: 10.1080/07362990701857129.

[5]

L. Allen, An Introduction to Stochastic Processes with Applications to Biology, 2nd edition, CRC Press, Boca Raton, 2010.

[6]

L. Allen, An Introduction to Mathematical Biology, Prentice Hall, Upper Saddle River, 2007.

[7]

J. Al-Omari and A. Tallafha, Modelling and analysis of stage-structured population model with state-dependent maturation delay and harvesting, Int. Journal of Math. Analysis, 1 (2007), 391-407.

[8]

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.

[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]

L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Modell., 34 (2010), 1405-1417. doi: 10.1016/j.apm.2009.08.027.

[11]

G. Bocharov and F. Rihan, Numerical modeling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4.

[12]

E. Cabaña, The vibrating string forced by white noise, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 15 (1970), 111-130. doi: 10.1007/BF00531880.

[13]

S. Chisholm, S. Frankel, R. Goericke, R. Olson, B. Palenik, J. Waterbury, L. West-Johnsrud and E. Zettler, Prochlorococcus-marinus nov gen-nov sp - an oxyphototrophic marine prokaryote containing divinyl chlorophyll-a and chlorophyll-B, Archives of Microbiology, 157 (1992), 297-300. doi: 10.1007/BF00245165.

[14]

J. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics 20, Springer, Berlin, Heidelberg, New York, 1977.

[15]

J. Filée, F. Tétart, C. Suttle and H. Krisch, Marine T4-type bacteriophages, a ubiquitous component of the dark matter of the biosphere, Proc. Nat. Acad. Sci., 102 (2005), 12471-12476. doi: 10.1073/pnas.0503404102.

[16]

J. Fuhrman, Marine viruses and their biogeochemical and ecological effects, Nature, 399 (1999), 541-548.

[17]

T. Gard, Introduction to Stochastic Differential Equations, Marcel Decker, New York, 1987.

[18]

D. Gillespie, Exact stochastic simulation of coupled chemical reactions, The Journal of Physical Chemistry, 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[19]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluweer Academic Publishers, Dordrecht, The Netherlands, 1992.

[20]

S. Gourley and S. Ruan, Dynamics of the diffusive Nicholson's blowfies equation with distributed delay, Proceedings of the Royal Society of Edinburgh, 130 (2000), 1275-1291. doi: 10.1017/S0308210500000688.

[21]

S. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM J. Appl. Math., 65 (2005), 550-566. doi: 10.1137/S0036139903436613.

[22]

W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[23]

T. Hilleman, Environmental Biology, Science Publishers, Enfield, New Hampshire, 2009. doi: 10.1201/b10187.

[24]

V. Jansen, R. Nisbet and W. Gurney, Generation cycles in stage structured populations, Bulletin of Mathematical Biology, 52 (1990), 375-396. doi: 10.1016/S0092-8240(05)80217-4.

[25]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York, 1992.

[26]

P. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.

[27]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993.

[28]

T. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential equations, Journal of Applied Probability, 8 (1971), 344-356. doi: 10.2307/3211904.

[29]

E. Kutter and A. Sulakvelidze, Bacteriophages: Biology and Applications, CRC Press, Boca Raton, Florida, 2004. doi: 10.1201/9780203491751.

[30]

P. Langevin, Sur la théorie du mouvement brownien, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530-533.

[31]

J. Li, Y. Kuang and C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, J. Theor. Biol., 242 (2006), 722-735. doi: 10.1016/j.jtbi.2006.04.002.

[32]

S. Liu, Z. Liu and J. Tang, A delayed marine bacteriophage infection model, Applied Mathematics Letters, 20 (2007), 702-706. doi: 10.1016/j.aml.2006.06.017.

[33]

A. Makroglou, I. Karaoustas, J. Li and Y. Kuang, Delay differential equation models in diabetes modeling: A review, in EOLSS encyclopedia, (eds. A. de Gaetano, P. Palumbo), 2011.

[34]

C. Munn, Marine Microbiology: Ecology and Applications, 2nd edition, Garland Science, Taylor and Francis Group, New York, 2011.

[35]

D. Oreopoulos, R. Lindeman, D. VanderJagt, A. Tzamaloukas, H. Bhagavan and P. Garry, Renal excretion of ascorbic acid: Effect of age and sex, Journal of the American Collge of Nutrition, 12 (1993), 537-542. doi: 10.1080/07315724.1993.10718349.

[36]

J. Paul, M. Sullivan, A. Segall and F. Rohwer, Marine phage genomics, Comparative Biochemistry and Physiology Part B, 133 (2002), 463-476. doi: 10.1016/S1096-4959(02)00168-9.

[37]

S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, ( eds. O. Arino, M. Hbid and E. Ait Dads), Springer, Berlin, 2006, 477-517. doi: 10.1007/1-4020-3647-7_11.

[38]

E. Shapiro, H. Tillil, K. Polonsky, V. Fang, A. Rubenstein and E. Van Cauter, Oscillations in insulin secretion during constant glucose infusion in normal man: Relationship to changes in plasma glucose, The Journal of Clinical Endocrinology & Metabolism, 67 (1988), 307-314. doi: 10.1210/jcem-67-2-307.

[39]

C. Simon, G. Brandenberger and M. Follenius, Ultradian oscillations of plasma glucose, insulin, and C-peptide in man during continuous enteral nutrition, The Journal of Clinical Endocrinology & Metabolism, 64 (1987), 669-674. doi: 10.1210/jcem-64-4-669.

[40]

J. Sturis, K. Polonsky, E. Mosekilde and E. Van Cauter, Computer-model for mechanisms underlying ultradian oscillations of insulin and glucose, Am. J. of Physiol. Endocrinol. Metab., 260 (1991), E801-E809.

[41]

Y. Su, J. Wei and J. Shi, Bifurcation analysis in a delayed diffusive Nicholson's blowflies equation, Nonlinear Analysis: Real World Applications, 11 (2010), 1692-1703. doi: 10.1016/j.nonrwa.2009.03.024.

[42]

M. Sullivan, J. Waterbury and S. Chisholm, Cyanophages infecting the oceanic cyanobacterium Prochlorococcus, Nature, 424 (2003), 1047-1051. doi: 10.1038/nature01929.

[43]

M. Sullivan, M. Coleman, P. Weigele, F. Rohwer and S. Chisholm, Three Prochlorococcus Cyanophage genomes: Signature features and ecological interpretations, PLoS Biology, 3 (2005), 790-806. doi: 10.1371/journal.pbio.0030144.

[44]

C. Suttle, Marine viruses major players in the global ecosystem, Nature Reviews Microbiology, 5 (2007), 801-812. doi: 10.1038/nrmicro1750.

[45]

I. Tolic, E. Mosekilde and J. Sturis, Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion, J. Theor. Biol., 207 (2000), 361-375.

[46]

J. Walsh, An introduction to stochastic partial differential equations, in Notes in Mathematics,, Volume 1180, (eds. A. Dold and B. Eckmann), Springer-Verlag, Berlin, 1986, 265-439. doi: 10.1007/BFb0074920.

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