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. doi: 10.1016/0025-5564(90)90019-U. Google Scholar

[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. doi: 10.1080/17442509808834159. Google Scholar

[3]

E. Allen, Modeling With Itô Stochastic Differential Equations,, Springer, (2007). Google Scholar

[4]

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

[5]

L. Allen, An Introduction to Stochastic Processes with Applications to Biology,, 2nd edition, (2010). Google Scholar

[6]

L. Allen, An Introduction to Mathematical Biology,, Prentice Hall, (2007). Google Scholar

[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. Google Scholar

[8]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection,, Mathematical Biosciences, 149 (1998), 57. doi: 10.1016/S0025-5564(97)10015-3. Google Scholar

[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. doi: 10.1016/S0362-546X(99)00285-0. Google Scholar

[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. doi: 10.1016/j.apm.2009.08.027. Google Scholar

[11]

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

[12]

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

[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. doi: 10.1007/BF00245165. Google Scholar

[14]

J. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Lecture Notes in Biomathematics 20, (1977). Google Scholar

[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. doi: 10.1073/pnas.0503404102. Google Scholar

[16]

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

[17]

T. Gard, Introduction to Stochastic Differential Equations,, Marcel Decker, (1987). Google Scholar

[18]

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

[19]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluweer Academic Publishers, (1992). Google Scholar

[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. doi: 10.1017/S0308210500000688. Google Scholar

[21]

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

[22]

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

[23]

T. Hilleman, Environmental Biology,, Science Publishers, (2009). doi: 10.1201/b10187. Google Scholar

[24]

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

[25]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer-Verlag, (1992). Google Scholar

[26]

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

[27]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Press, (1993). Google Scholar

[28]

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

[29]

E. Kutter and A. Sulakvelidze, Bacteriophages: Biology and Applications,, CRC Press, (2004). doi: 10.1201/9780203491751. Google Scholar

[30]

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

[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. doi: 10.1016/j.jtbi.2006.04.002. Google Scholar

[32]

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

[33]

A. Makroglou, I. Karaoustas, J. Li and Y. Kuang, Delay differential equation models in diabetes modeling: A review,, in EOLSS encyclopedia, (2011). Google Scholar

[34]

C. Munn, Marine Microbiology: Ecology and Applications,, 2nd edition, (2011). Google Scholar

[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. doi: 10.1080/07315724.1993.10718349. Google Scholar

[36]

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

[37]

S. Ruan, Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications, (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[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. doi: 10.1210/jcem-67-2-307. Google Scholar

[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. doi: 10.1210/jcem-64-4-669. Google Scholar

[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). Google Scholar

[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. doi: 10.1016/j.nonrwa.2009.03.024. Google Scholar

[42]

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

[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. doi: 10.1371/journal.pbio.0030144. Google Scholar

[44]

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

[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. Google Scholar

[46]

J. Walsh, An introduction to stochastic partial differential equations,, in Notes in Mathematics, (1180), 265. doi: 10.1007/BFb0074920. Google Scholar

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. doi: 10.1016/0025-5564(90)90019-U. Google Scholar

[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. doi: 10.1080/17442509808834159. Google Scholar

[3]

E. Allen, Modeling With Itô Stochastic Differential Equations,, Springer, (2007). Google Scholar

[4]

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

[5]

L. Allen, An Introduction to Stochastic Processes with Applications to Biology,, 2nd edition, (2010). Google Scholar

[6]

L. Allen, An Introduction to Mathematical Biology,, Prentice Hall, (2007). Google Scholar

[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. Google Scholar

[8]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection,, Mathematical Biosciences, 149 (1998), 57. doi: 10.1016/S0025-5564(97)10015-3. Google Scholar

[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. doi: 10.1016/S0362-546X(99)00285-0. Google Scholar

[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. doi: 10.1016/j.apm.2009.08.027. Google Scholar

[11]

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

[12]

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

[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. doi: 10.1007/BF00245165. Google Scholar

[14]

J. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Lecture Notes in Biomathematics 20, (1977). Google Scholar

[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. doi: 10.1073/pnas.0503404102. Google Scholar

[16]

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

[17]

T. Gard, Introduction to Stochastic Differential Equations,, Marcel Decker, (1987). Google Scholar

[18]

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

[19]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluweer Academic Publishers, (1992). Google Scholar

[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. doi: 10.1017/S0308210500000688. Google Scholar

[21]

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

[22]

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

[23]

T. Hilleman, Environmental Biology,, Science Publishers, (2009). doi: 10.1201/b10187. Google Scholar

[24]

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

[25]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer-Verlag, (1992). Google Scholar

[26]

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

[27]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Press, (1993). Google Scholar

[28]

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

[29]

E. Kutter and A. Sulakvelidze, Bacteriophages: Biology and Applications,, CRC Press, (2004). doi: 10.1201/9780203491751. Google Scholar

[30]

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

[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. doi: 10.1016/j.jtbi.2006.04.002. Google Scholar

[32]

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

[33]

A. Makroglou, I. Karaoustas, J. Li and Y. Kuang, Delay differential equation models in diabetes modeling: A review,, in EOLSS encyclopedia, (2011). Google Scholar

[34]

C. Munn, Marine Microbiology: Ecology and Applications,, 2nd edition, (2011). Google Scholar

[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. doi: 10.1080/07315724.1993.10718349. Google Scholar

[36]

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

[37]

S. Ruan, Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications, (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[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. doi: 10.1210/jcem-67-2-307. Google Scholar

[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. doi: 10.1210/jcem-64-4-669. Google Scholar

[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). Google Scholar

[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. doi: 10.1016/j.nonrwa.2009.03.024. Google Scholar

[42]

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

[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. doi: 10.1371/journal.pbio.0030144. Google Scholar

[44]

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

[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. Google Scholar

[46]

J. Walsh, An introduction to stochastic partial differential equations,, in Notes in Mathematics, (1180), 265. doi: 10.1007/BFb0074920. Google Scholar

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