# 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 & 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, (2011). Google Scholar [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. doi: 10.1016/j.biosystems.2009.01.007. Google Scholar [3] C. A. Braumann, Itô versus Stratonovich calculus in random population growth,, Math. Biosci., 206 (2007), 81. doi: 10.1016/j.mbs.2004.09.002. Google Scholar [4] C. A. Braumann, Growth and extinction of populations in randomly varying environments,, Comput. Math. Appl., 56 (2008), 631. doi: 10.1016/j.camwa.2008.01.006. Google Scholar [5] J. Cresson and S. Darses, Stochastic embedding of dynamical systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2736519. Google Scholar [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. Google Scholar [7] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics,, J. Math. Anal. Appl., 341 (2008), 1084. doi: 10.1016/j.jmaa.2007.11.005. Google Scholar [8] L. C. Evans, An Introduction to Stochastic Differential Equations,, Lecture Notes, (2013). doi: 10.1090/mbk/082. Google Scholar [9] U. Forys, Global analysis of Marchuk's model in a case of weak immune system,, Math. Comput. Modelling Vol., 25 (1997), 97. doi: 10.1016/S0895-7177(97)00042-3. Google Scholar [10] R. F. Fox, Stochastic versions of the Hodgkin-Huxley equations,, Biophys. J., 72 (1997), 2068. doi: 10.1016/S0006-3495(97)78850-7. Google Scholar [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. Google Scholar [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, (2009), 247. Google Scholar [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. doi: 10.1016/j.apm.2011.05.027. Google Scholar [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). doi: 10.1186/1471-2105-13-S5-S8. Google Scholar [15] A. Kamina, R. W. Makuch and H. Zhao, A stochastic modeling of early HIV-1 population dynamics,, Math. Biosci., 170 (2001), 187. doi: 10.1016/S0025-5564(00)00069-9. Google Scholar [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. Google Scholar [17] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology,, Bull. Math. Biol., 41 (1979), 469. Google Scholar [18] M.-L. Linne and T. O. Jalonen, Simulations of the cultured granule neuron excitability,, Neurocomputing, 52-54 (2003), 52. doi: 10.1016/S0925-2312(02)00804-4. Google Scholar [19] H. Lisei and D. Julitz, A stochastic model for the growth of Cancer tumors,, Studia Univ., LIII (2008), 39. Google Scholar [20] R. M. May, Stability in randomly fluctuating versus deterministic environments,, Am. Nat., 107 (1973), 621. doi: 10.1086/282863. Google Scholar [21] A. Milian, Stochastic viability and comparison theorem,, Coll. Math., 68 (1995), 297. Google Scholar [22] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, $6^{th}$ edition, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar [23] N. H. Pavel, Differential Equations, Flow Invariance and Applications,, Pitman, (1984). Google Scholar [24] A. S. Perelson, D. E. Kirschner and R. J. DeBoer, Dynamics of HIV infection of CD4$^+$ T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A. Google Scholar [25] A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response,, Science, 271 (1996), 497. doi: 10.1126/science.271.5248.497. Google Scholar [26] G. Rosenkranz, Growth models with stochastic differential equations. An example from tumor immunology,, Math. Biosci., 75 (1985), 175. doi: 10.1016/0025-5564(85)90036-7. Google Scholar [27] A. Saarinen, M.-L. Linne and O. Yli-Harja, Stochastic differential equation model for cerebellar granule cell excitability,, Plos Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000004. Google Scholar [28] A. Saarinen, M.-L. Linne and O. Yli-Harja, Modeling single neuron behavior using stochastic differential equations,, Neurocomputing, 69 (2006), 1091. doi: 10.1016/j.neucom.2005.12.052. Google Scholar [29] C. Surulescu and N. Surulescu, On some stochastic differential models with applications to biological problems,, ECMTB, 14 (2011), 106. Google Scholar [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, 2102 (2013), 269. doi: 10.1007/978-3-319-03080-7_9. Google Scholar [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. doi: 10.1073/pnas.0237333100. Google Scholar [32] G. W. Swan, Role of optimal control theory in cancer chemotherapy,, Math. Biosci., 101 (1990), 237. doi: 10.1016/0025-5564(90)90021-P. Google Scholar [33] H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics,, J. Theor. Biol., 195 (1998), 451. doi: 10.1006/jtbi.1998.0806. Google Scholar [34] M. Turelli, Random environments and stochastic calculus,, Theor. Popul. Biol., 12 (1977), 140. doi: 10.1016/0040-5809(77)90040-5. Google Scholar [35] N. G. van Kampen, Itô versus Stratonovich,, J. Stat. Phys., 24 (1981), 175. doi: 10.1007/BF01007642. Google Scholar [36] W. Walter, Gewöhnliche Differential gleichungen,, $7^{th}$ edition, (2000). Google Scholar [37] Y. Yuan and L. J. Allen, Stochastic models for virus and immune system dynamics,, Math. Biosci., 234 (2011), 84. doi: 10.1016/j.mbs.2011.08.007. Google Scholar

show all references

##### References:
 [1] L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology,, $2^{nd}$ edition, (2011). Google Scholar [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. doi: 10.1016/j.biosystems.2009.01.007. Google Scholar [3] C. A. Braumann, Itô versus Stratonovich calculus in random population growth,, Math. Biosci., 206 (2007), 81. doi: 10.1016/j.mbs.2004.09.002. Google Scholar [4] C. A. Braumann, Growth and extinction of populations in randomly varying environments,, Comput. Math. Appl., 56 (2008), 631. doi: 10.1016/j.camwa.2008.01.006. Google Scholar [5] J. Cresson and S. Darses, Stochastic embedding of dynamical systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2736519. Google Scholar [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. Google Scholar [7] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics,, J. Math. Anal. Appl., 341 (2008), 1084. doi: 10.1016/j.jmaa.2007.11.005. Google Scholar [8] L. C. Evans, An Introduction to Stochastic Differential Equations,, Lecture Notes, (2013). doi: 10.1090/mbk/082. Google Scholar [9] U. Forys, Global analysis of Marchuk's model in a case of weak immune system,, Math. Comput. Modelling Vol., 25 (1997), 97. doi: 10.1016/S0895-7177(97)00042-3. Google Scholar [10] R. F. Fox, Stochastic versions of the Hodgkin-Huxley equations,, Biophys. J., 72 (1997), 2068. doi: 10.1016/S0006-3495(97)78850-7. Google Scholar [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. Google Scholar [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, (2009), 247. Google Scholar [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. doi: 10.1016/j.apm.2011.05.027. Google Scholar [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). doi: 10.1186/1471-2105-13-S5-S8. Google Scholar [15] A. Kamina, R. W. Makuch and H. Zhao, A stochastic modeling of early HIV-1 population dynamics,, Math. Biosci., 170 (2001), 187. doi: 10.1016/S0025-5564(00)00069-9. Google Scholar [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. Google Scholar [17] R. Lefever and W. Horsthemke, Bistability in fluctuating environments. Implications in tumor immunology,, Bull. Math. Biol., 41 (1979), 469. Google Scholar [18] M.-L. Linne and T. O. Jalonen, Simulations of the cultured granule neuron excitability,, Neurocomputing, 52-54 (2003), 52. doi: 10.1016/S0925-2312(02)00804-4. Google Scholar [19] H. Lisei and D. Julitz, A stochastic model for the growth of Cancer tumors,, Studia Univ., LIII (2008), 39. Google Scholar [20] R. M. May, Stability in randomly fluctuating versus deterministic environments,, Am. Nat., 107 (1973), 621. doi: 10.1086/282863. Google Scholar [21] A. Milian, Stochastic viability and comparison theorem,, Coll. Math., 68 (1995), 297. Google Scholar [22] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, $6^{th}$ edition, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar [23] N. H. Pavel, Differential Equations, Flow Invariance and Applications,, Pitman, (1984). Google Scholar [24] A. S. Perelson, D. E. Kirschner and R. J. DeBoer, Dynamics of HIV infection of CD4$^+$ T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A. Google Scholar [25] A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response,, Science, 271 (1996), 497. doi: 10.1126/science.271.5248.497. Google Scholar [26] G. Rosenkranz, Growth models with stochastic differential equations. An example from tumor immunology,, Math. Biosci., 75 (1985), 175. doi: 10.1016/0025-5564(85)90036-7. Google Scholar [27] A. Saarinen, M.-L. Linne and O. Yli-Harja, Stochastic differential equation model for cerebellar granule cell excitability,, Plos Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000004. Google Scholar [28] A. Saarinen, M.-L. Linne and O. Yli-Harja, Modeling single neuron behavior using stochastic differential equations,, Neurocomputing, 69 (2006), 1091. doi: 10.1016/j.neucom.2005.12.052. Google Scholar [29] C. Surulescu and N. Surulescu, On some stochastic differential models with applications to biological problems,, ECMTB, 14 (2011), 106. Google Scholar [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, 2102 (2013), 269. doi: 10.1007/978-3-319-03080-7_9. Google Scholar [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. doi: 10.1073/pnas.0237333100. Google Scholar [32] G. W. Swan, Role of optimal control theory in cancer chemotherapy,, Math. Biosci., 101 (1990), 237. doi: 10.1016/0025-5564(90)90021-P. Google Scholar [33] H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics,, J. Theor. Biol., 195 (1998), 451. doi: 10.1006/jtbi.1998.0806. Google Scholar [34] M. Turelli, Random environments and stochastic calculus,, Theor. Popul. Biol., 12 (1977), 140. doi: 10.1016/0040-5809(77)90040-5. Google Scholar [35] N. G. van Kampen, Itô versus Stratonovich,, J. Stat. Phys., 24 (1981), 175. doi: 10.1007/BF01007642. Google Scholar [36] W. Walter, Gewöhnliche Differential gleichungen,, $7^{th}$ edition, (2000). Google Scholar [37] Y. Yuan and L. J. Allen, Stochastic models for virus and immune system dynamics,, Math. Biosci., 234 (2011), 84. doi: 10.1016/j.mbs.2011.08.007. Google Scholar
 [1] Katarzyna PichÓr, Ryszard Rudnicki. Stability of stochastic semigroups and applications to Stein's neuronal model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 377-385. doi: 10.3934/dcdsb.2018026 [2] 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 [3] Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375 [4] Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737 [5] Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257 [6] Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43 [7] Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017 [8] Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282 [9] Jun Zhou, Jun Shen, Weinian Zhang. A powered Gronwall-type inequality and applications to stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7207-7234. doi: 10.3934/dcds.2016114 [10] 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 [11] Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Florian K. Richter. A generalization of Kátai's orthogonality criterion with applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2581-2612. doi: 10.3934/dcds.2019108 [12] Alison M. Melo, Leandro B. Morgado, Paulo R. Ruffino. Decomposition of stochastic flows generated by Stratonovich SDEs with jumps. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3209-3218. doi: 10.3934/dcdsb.2016094 [13] Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351 [14] Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 [15] Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191 [16] Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407 [17] Christine Burggraf, Wilfried Grecksch, Thomas Glauben. Stochastic control of individual's health investments. Conference Publications, 2015, 2015 (special) : 159-168. doi: 10.3934/proc.2015.0159 [18] Hitoshi Ishii, Paola Loreti, Maria Elisabetta Tessitore. A PDE approach to stochastic invariance. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 651-664. doi: 10.3934/dcds.2000.6.651 [19] Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733 [20] Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244

2018 Impact Factor: 1.008