# American Institute of Mathematical Sciences

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. [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. [3] E. Allen, Modeling With Itô Stochastic Differential Equations,, Springer, (2007). [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. [5] L. Allen, An Introduction to Stochastic Processes with Applications to Biology,, 2nd edition, (2010). [6] L. Allen, An Introduction to Mathematical Biology,, Prentice Hall, (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. [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. [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. [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. [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. [12] E. Cabaña, The vibrating string forced by white noise,, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 15 (1970), 111. 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. doi: 10.1007/BF00245165. [14] J. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Lecture Notes in Biomathematics 20, (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. doi: 10.1073/pnas.0503404102. [16] J. Fuhrman, Marine viruses and their biogeochemical and ecological effects,, Nature, 399 (1999), 541. [17] T. Gard, Introduction to Stochastic Differential Equations,, Marcel Decker, (1987). [18] D. Gillespie, Exact stochastic simulation of coupled chemical reactions,, The Journal of Physical Chemistry, 81 (1977), 2340. doi: 10.1021/j100540a008. [19] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluweer Academic Publishers, (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. 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. doi: 10.1137/S0036139903436613. [22] W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0. [23] T. Hilleman, Environmental Biology,, Science Publishers, (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. doi: 10.1016/S0092-8240(05)80217-4. [25] P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer-Verlag, (1992). [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. [27] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Press, (1993). [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. [29] E. Kutter and A. Sulakvelidze, Bacteriophages: Biology and Applications,, CRC Press, (2004). doi: 10.1201/9780203491751. [30] P. Langevin, Sur la théorie du mouvement brownien,, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530. [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. [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. [33] A. Makroglou, I. Karaoustas, J. Li and Y. Kuang, Delay differential equation models in diabetes modeling: A review,, in EOLSS encyclopedia, (2011). [34] C. Munn, Marine Microbiology: Ecology and Applications,, 2nd edition, (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. 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. doi: 10.1016/S1096-4959(02)00168-9. [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. [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. [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. [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). [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. [42] M. Sullivan, J. Waterbury and S. Chisholm, Cyanophages infecting the oceanic cyanobacterium Prochlorococcus,, Nature, 424 (2003), 1047. 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. doi: 10.1371/journal.pbio.0030144. [44] C. Suttle, Marine viruses major players in the global ecosystem,, Nature Reviews Microbiology, 5 (2007), 801. 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. [46] J. Walsh, An introduction to stochastic partial differential equations,, in Notes in Mathematics, (1180), 265. 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. 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. doi: 10.1080/17442509808834159. [3] E. Allen, Modeling With Itô Stochastic Differential Equations,, Springer, (2007). [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. [5] L. Allen, An Introduction to Stochastic Processes with Applications to Biology,, 2nd edition, (2010). [6] L. Allen, An Introduction to Mathematical Biology,, Prentice Hall, (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. [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. [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. [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. [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. [12] E. Cabaña, The vibrating string forced by white noise,, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 15 (1970), 111. 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. doi: 10.1007/BF00245165. [14] J. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Lecture Notes in Biomathematics 20, (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. doi: 10.1073/pnas.0503404102. [16] J. Fuhrman, Marine viruses and their biogeochemical and ecological effects,, Nature, 399 (1999), 541. [17] T. Gard, Introduction to Stochastic Differential Equations,, Marcel Decker, (1987). [18] D. Gillespie, Exact stochastic simulation of coupled chemical reactions,, The Journal of Physical Chemistry, 81 (1977), 2340. doi: 10.1021/j100540a008. [19] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluweer Academic Publishers, (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. 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. doi: 10.1137/S0036139903436613. [22] W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0. [23] T. Hilleman, Environmental Biology,, Science Publishers, (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. doi: 10.1016/S0092-8240(05)80217-4. [25] P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations,, Springer-Verlag, (1992). [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. [27] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Press, (1993). [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. [29] E. Kutter and A. Sulakvelidze, Bacteriophages: Biology and Applications,, CRC Press, (2004). doi: 10.1201/9780203491751. [30] P. Langevin, Sur la théorie du mouvement brownien,, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530. [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. [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. [33] A. Makroglou, I. Karaoustas, J. Li and Y. Kuang, Delay differential equation models in diabetes modeling: A review,, in EOLSS encyclopedia, (2011). [34] C. Munn, Marine Microbiology: Ecology and Applications,, 2nd edition, (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. 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. doi: 10.1016/S1096-4959(02)00168-9. [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. [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. [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. [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). [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. [42] M. Sullivan, J. Waterbury and S. Chisholm, Cyanophages infecting the oceanic cyanobacterium Prochlorococcus,, Nature, 424 (2003), 1047. 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. doi: 10.1371/journal.pbio.0030144. [44] C. Suttle, Marine viruses major players in the global ecosystem,, Nature Reviews Microbiology, 5 (2007), 801. 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. [46] J. Walsh, An introduction to stochastic partial differential equations,, in Notes in Mathematics, (1180), 265. doi: 10.1007/BFb0074920.
 [1] Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683 [2] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [3] Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533 [4] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [5] András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43 [6] Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 [7] Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065 [8] Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321 [9] P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220 [10] Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 [11] Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23 [12] Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2963-2991. doi: 10.3934/dcdsb.2014.19.2963 [13] Neville J. Ford, Stewart J. Norton. Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 367-382. doi: 10.3934/cpaa.2006.5.367 [14] David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135 [15] Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019052 [16] Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272 [17] Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-21. doi: 10.3934/dcdsb.2019062 [18] Jacky Cresson, Bénédicte Puig, Stefanie Sonner. Stochastic models in biology and the invariance problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2145-2168. doi: 10.3934/dcdsb.2016041 [19] Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715 [20] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

2018 Impact Factor: 1.313