# American Institute of Mathematical Sciences

September  2016, 21(7): 2145-2168. doi: 10.3934/dcdsb.2016041

## Stochastic models in biology and the invariance problem

 1 Laboratoire de Mathématiques Appliquées de Pau, Université de Pau et des Pays de l'Adour, Avenue de l'Université, BP 1155, 64013 Pau Cedex, France 2 Institut Pluridisciplinaire de Recherches Appliquées, Université de Pau et des Pays de l'Adour, Avenue de l'Université, BP 1155, 64013 Pau Cedex, France 3 Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010 Graz, Austria

Received  December 2015 Revised  February 2016 Published  August 2016

Invariance is a crucial property for many mathematical models of biological or biomedical systems, meaning that the solutions necessarily take values in a given range. This property reflects physical or biological constraints of the system and is independent of the model under consideration. While most classical deterministic models respect invariance, many recent stochastic extensions violate this fundamental property. Based on an invariance criterion for systems of stochastic differential equations we discuss several stochastic models exhibiting this behavior and propose classes of modified, admissible models as possible resolutions. Numerical simulations are presented to illustrate the model behavior.
Citation: Jacky Cresson, Bénédicte Puig, Stefanie Sonner. Stochastic models in biology and the invariance problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2145-2168. doi: 10.3934/dcdsb.2016041
##### References:
 [1] L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, $2^{nd}$ edition, Chapman & Hall, Boca Raton, 2011. [2] A. J. Arenas, G. González-Parra and J.-A. Moraño, Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia, Spain, BioSystems, 96 (2009), 206-212. doi: 10.1016/j.biosystems.2009.01.007. [3] C. A. Braumann, Itô versus Stratonovich calculus in random population growth, Math. Biosci., 206 (2007), 81-107. doi: 10.1016/j.mbs.2004.09.002. [4] C. A. Braumann, Growth and extinction of populations in randomly varying environments, Comput. Math. Appl., 56 (2008), 631-644. doi: 10.1016/j.camwa.2008.01.006. [5] J. Cresson and S. Darses, Stochastic embedding of dynamical systems, J. Math. Phys., 48 (2007), 072703, 54pp. doi: 10.1063/1.2736519. [6] J. Cresson, B. Puig and S. Sonner, Validating stochastic models: Invariance criteria for systems of stochastic differential equations and the selection of a stochastic Hodgkin-Huxley type model, Int. J. Biomath. Biostat., 2 (2013), 111-122. [7] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101. doi: 10.1016/j.jmaa.2007.11.005. [8] L. C. Evans, An Introduction to Stochastic Differential Equations, Lecture Notes, UC Berkley, 2013. doi: 10.1090/mbk/082. [9] U. Forys, Global analysis of Marchuk's model in a case of weak immune system, Math. Comput. Modelling Vol., 25 (1997), 97-106. doi: 10.1016/S0895-7177(97)00042-3. [10] R. F. Fox, Stochastic versions of the Hodgkin-Huxley equations, Biophys. J., 72 (1997), 2068-2074. doi: 10.1016/S0006-3495(97)78850-7. [11] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol., 117 (1952), 500-544. [12] R. Horhat, R. Horhat and D. Opris, The simulation of a stochastic model for tumor-immune system, Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS BEBI'09, World Scientific and Engineering Academy and Society (WSEAS), (2009), 247-252. [13] Z. Huang, Q. Yang and J. Cao, Stochastic stability and bifurcation for the chronic state in Marchuk's model with noise, Appl. Math. Model., 35 (2011), 5842-5855. doi: 10.1016/j.apm.2011.05.027. [14] S. K. Jha and C. J. Langmead, Exploring behaviors of stochastic differential equation models of biological systems using change of measures, BMC Bioinformatics, 13 (2012), S8, 10 pp. doi: 10.1186/1471-2105-13-S5-S8. [15] A. Kamina, R. W. Makuch and H. Zhao, A stochastic modeling of early HIV-1 population dynamics, Math. Biosci., 170 (2001), 187-198. doi: 10.1016/S0025-5564(00)00069-9. [16] V. Kuznetsov, I. Makalkin, M. Taylor and A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. [17] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bull. Math. Biol., 41 (1979), 469-490. [18] M.-L. Linne and T. O. Jalonen, Simulations of the cultured granule neuron excitability, Neurocomputing, 52-54 (2003), 583-590. doi: 10.1016/S0925-2312(02)00804-4. [19] H. Lisei and D. Julitz, A stochastic model for the growth of Cancer tumors, Studia Univ. "Babes-Bolay", Math., LIII (2008), 39-56. [20] R. M. May, Stability in randomly fluctuating versus deterministic environments, Am. Nat., 107 (1973), 621-650. doi: 10.1086/282863. [21] A. Milian, Stochastic viability and comparison theorem, Coll. Math., 68 (1995), 297-316. [22] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, $6^{th}$ edition, Springer-Verlag, Berlin-Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6. [23] N. H. Pavel, Differential Equations, Flow Invariance and Applications, Pitman, Boston, 1984. [24] A. S. Perelson, D. E. Kirschner and R. J. DeBoer, Dynamics of HIV infection of CD4$^+$ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. [25] A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499. doi: 10.1126/science.271.5248.497. [26] G. Rosenkranz, Growth models with stochastic differential equations. An example from tumor immunology, Math. Biosci., 75 (1985), 175-186. doi: 10.1016/0025-5564(85)90036-7. [27] A. Saarinen, M.-L. Linne and O. Yli-Harja, Stochastic differential equation model for cerebellar granule cell excitability, Plos Comput. Biol., 4 (2008), e1000004, 11 pp. doi: 10.1371/journal.pcbi.1000004. [28] A. Saarinen, M.-L. Linne and O. Yli-Harja, Modeling single neuron behavior using stochastic differential equations, Neurocomputing, 69 (2006), 1091-1096. doi: 10.1016/j.neucom.2005.12.052. [29] C. Surulescu and N. Surulescu, On some stochastic differential models with applications to biological problems, ECMTB, 14 (2011), 106-117. [30] C. Surulescu and N. Surulescu, Some classes of stochastic differential equations as an alternative modeling approach to biomedical problems, in Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Mathematics, Springer, 2102 (2013), 269-307. doi: 10.1007/978-3-319-03080-7_9. [31] I. Swameye, T. G. Müller, J. Timmer, O. Sandra and U. Klingmuller, Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling, PNAS, 100 (2003), 1028-1033. doi: 10.1073/pnas.0237333100. [32] G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci.,101 (1990), 237-284. doi: 10.1016/0025-5564(90)90021-P. [33] H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics, J. Theor. Biol., 195 (1998), 451-463. doi: 10.1006/jtbi.1998.0806. [34] M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140-178. doi: 10.1016/0040-5809(77)90040-5. [35] N. G. van Kampen, Itô versus Stratonovich, J. Stat. Phys., 24 (1981), 175-187. doi: 10.1007/BF01007642. [36] W. Walter, Gewöhnliche Differential gleichungen, $7^{th}$ edition, Springer-Verlag, Berlin-Heidelberg, 2000. [37] Y. Yuan and L. J. Allen, Stochastic models for virus and immune system dynamics, Math. Biosci., 234 (2011), 84-94. doi: 10.1016/j.mbs.2011.08.007.

show all references

##### References:
 [1] L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, $2^{nd}$ edition, Chapman & Hall, Boca Raton, 2011. [2] A. J. Arenas, G. González-Parra and J.-A. Moraño, Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia, Spain, BioSystems, 96 (2009), 206-212. doi: 10.1016/j.biosystems.2009.01.007. [3] C. A. Braumann, Itô versus Stratonovich calculus in random population growth, Math. Biosci., 206 (2007), 81-107. doi: 10.1016/j.mbs.2004.09.002. [4] C. A. Braumann, Growth and extinction of populations in randomly varying environments, Comput. Math. Appl., 56 (2008), 631-644. doi: 10.1016/j.camwa.2008.01.006. [5] J. Cresson and S. Darses, Stochastic embedding of dynamical systems, J. Math. Phys., 48 (2007), 072703, 54pp. doi: 10.1063/1.2736519. [6] J. Cresson, B. Puig and S. Sonner, Validating stochastic models: Invariance criteria for systems of stochastic differential equations and the selection of a stochastic Hodgkin-Huxley type model, Int. J. Biomath. Biostat., 2 (2013), 111-122. [7] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101. doi: 10.1016/j.jmaa.2007.11.005. [8] L. C. Evans, An Introduction to Stochastic Differential Equations, Lecture Notes, UC Berkley, 2013. doi: 10.1090/mbk/082. [9] U. Forys, Global analysis of Marchuk's model in a case of weak immune system, Math. Comput. Modelling Vol., 25 (1997), 97-106. doi: 10.1016/S0895-7177(97)00042-3. [10] R. F. Fox, Stochastic versions of the Hodgkin-Huxley equations, Biophys. J., 72 (1997), 2068-2074. doi: 10.1016/S0006-3495(97)78850-7. [11] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol., 117 (1952), 500-544. [12] R. Horhat, R. Horhat and D. Opris, The simulation of a stochastic model for tumor-immune system, Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS BEBI'09, World Scientific and Engineering Academy and Society (WSEAS), (2009), 247-252. [13] Z. Huang, Q. Yang and J. Cao, Stochastic stability and bifurcation for the chronic state in Marchuk's model with noise, Appl. Math. Model., 35 (2011), 5842-5855. doi: 10.1016/j.apm.2011.05.027. [14] S. K. Jha and C. J. Langmead, Exploring behaviors of stochastic differential equation models of biological systems using change of measures, BMC Bioinformatics, 13 (2012), S8, 10 pp. doi: 10.1186/1471-2105-13-S5-S8. [15] A. Kamina, R. W. Makuch and H. Zhao, A stochastic modeling of early HIV-1 population dynamics, Math. Biosci., 170 (2001), 187-198. doi: 10.1016/S0025-5564(00)00069-9. [16] V. Kuznetsov, I. Makalkin, M. Taylor and A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. [17] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology, Bull. Math. Biol., 41 (1979), 469-490. [18] M.-L. Linne and T. O. Jalonen, Simulations of the cultured granule neuron excitability, Neurocomputing, 52-54 (2003), 583-590. doi: 10.1016/S0925-2312(02)00804-4. [19] H. Lisei and D. Julitz, A stochastic model for the growth of Cancer tumors, Studia Univ. "Babes-Bolay", Math., LIII (2008), 39-56. [20] R. M. May, Stability in randomly fluctuating versus deterministic environments, Am. Nat., 107 (1973), 621-650. doi: 10.1086/282863. [21] A. Milian, Stochastic viability and comparison theorem, Coll. Math., 68 (1995), 297-316. [22] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, $6^{th}$ edition, Springer-Verlag, Berlin-Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6. [23] N. H. Pavel, Differential Equations, Flow Invariance and Applications, Pitman, Boston, 1984. [24] A. S. Perelson, D. E. Kirschner and R. J. DeBoer, Dynamics of HIV infection of CD4$^+$ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. [25] A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497-499. doi: 10.1126/science.271.5248.497. [26] G. Rosenkranz, Growth models with stochastic differential equations. An example from tumor immunology, Math. Biosci., 75 (1985), 175-186. doi: 10.1016/0025-5564(85)90036-7. [27] A. Saarinen, M.-L. Linne and O. Yli-Harja, Stochastic differential equation model for cerebellar granule cell excitability, Plos Comput. Biol., 4 (2008), e1000004, 11 pp. doi: 10.1371/journal.pcbi.1000004. [28] A. Saarinen, M.-L. Linne and O. Yli-Harja, Modeling single neuron behavior using stochastic differential equations, Neurocomputing, 69 (2006), 1091-1096. doi: 10.1016/j.neucom.2005.12.052. [29] C. Surulescu and N. Surulescu, On some stochastic differential models with applications to biological problems, ECMTB, 14 (2011), 106-117. [30] C. Surulescu and N. Surulescu, Some classes of stochastic differential equations as an alternative modeling approach to biomedical problems, in Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Mathematics, Springer, 2102 (2013), 269-307. doi: 10.1007/978-3-319-03080-7_9. [31] I. Swameye, T. G. Müller, J. Timmer, O. Sandra and U. Klingmuller, Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling, PNAS, 100 (2003), 1028-1033. doi: 10.1073/pnas.0237333100. [32] G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci.,101 (1990), 237-284. doi: 10.1016/0025-5564(90)90021-P. [33] H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics, J. Theor. Biol., 195 (1998), 451-463. doi: 10.1006/jtbi.1998.0806. [34] M. Turelli, Random environments and stochastic calculus, Theor. Popul. Biol., 12 (1977), 140-178. doi: 10.1016/0040-5809(77)90040-5. [35] N. G. van Kampen, Itô versus Stratonovich, J. Stat. Phys., 24 (1981), 175-187. doi: 10.1007/BF01007642. [36] W. Walter, Gewöhnliche Differential gleichungen, $7^{th}$ edition, Springer-Verlag, Berlin-Heidelberg, 2000. [37] Y. Yuan and L. J. Allen, Stochastic models for virus and immune system dynamics, Math. Biosci., 234 (2011), 84-94. doi: 10.1016/j.mbs.2011.08.007.
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