2013, 10(1): 37-57. doi: 10.3934/mbe.2013.10.37

Distributed delays in a hybrid model of tumor-Immune system interplay

1. 

Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 Milan, Italy, Italy

2. 

Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, I-20141 Milan

Received  July 2012 Revised  September 2012 Published  December 2012

A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect.
    We here present a hybrid model with a generic delay kernel accounting that, due to many complex phenomena such as chemical transportation and cellular differentiation, the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model with two well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process.
    Via simulations and parametric sensitivity analysis techniques we $(i)$ relate tumor mass growth with the two kernels, we $(ii)$ measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and $(iii)$ we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.
Citation: Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences & Engineering, 2013, 10 (1) : 37-57. doi: 10.3934/mbe.2013.10.37
References:
[1]

S. A. Agarwala, "New Applications of Cancer Immunotherapy,", S. A. Agarwala (Guest Editor), (2003), 29.   Google Scholar

[2]

R. Barbuti, G. Caravagna, A. Maggiolo-Schettini and P. Milazzo, Delay stochastic simulation of biological systems: A purely delayed approach,, C.Priami et al.(Eds.): Trans. Comp. Sys. Bio. XIII, 6575 (2011), 61.   Google Scholar

[3]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation,, PLoS Comp. Bio., 2 (2006).   Google Scholar

[4]

N. Bellomo and G. Forni, Complex multicellular systems and Immune competition: New paradigms looking for a mathematical theory,, Curr. Top. Dev. Bio., 81 (2008), 485.  doi: 10.1016/S0070-2153(07)81017-9.  Google Scholar

[5]

E. Beretta, V. Capasso and F. Rinaldi, Global stability results for a generalized Lotka-Volterra system with distributed delays,, J. Math. Bio., 26 (1988), 661.   Google Scholar

[6]

I. Bleumer, E. Oosterwijk, P. de Mulder and P. F. Mulders, Immunotherapy for renal cell carcinoma,, Europ. Urol., 44 (2003), 65.  doi: 10.1016/S0302-2838(03)00191-X.  Google Scholar

[7]

N. Blumberg, C. Chuang-Stein and J. M. Heal, The relationship of blood transfusion, tumor staging and cancer recurrence,, Transf., 30 (1990), 291.  doi: 10.1046/j.1537-2995.1990.30490273432.x.  Google Scholar

[8]

K. B. Blyuss and Y. N. Kyrychko, Stability and bifurcations in an epidemic model with varying immunity period,, Bull. Math. Bio., 72 (2010), 490.  doi: 10.1007/s11538-009-9458-y.  Google Scholar

[9]

L. Bortolussi, Automata and (stochastic) programs. The hybrid automata lattice of a stochastic program,, J. Log. Comp., (2011).  doi: 10.1093/logcom/exr045.  Google Scholar

[10]

L. Bortolussi and A. Policriti, The importance of being (a little bit) discrete,, ENTCS, 229 (2009), 75.   Google Scholar

[11]

M. Bravetti and R. Gorrieri, The theory of interactive generalized semi-Markov processes,, Theoret. Comp. Sci., 282 (2002), 5.   Google Scholar

[12]

N. Burić and D. Todorović, Dynamics of delay-differential equations modelling immunology of tumor growth,, Cha. Sol. Fract., 13 (2002), 645.   Google Scholar

[13]

G. Caravagna, "Formal Modeling and Simulation of Biological Systems With Delays,", Ph.D. Thesis, (2011).   Google Scholar

[14]

G. Caravagna, A. d'Onofrio, P. Milazzo and R. Barbuti, Antitumour Immune surveillance through stochastic oscillations,, J. Th. Biology, 265 (2010), 336.   Google Scholar

[15]

G. Caravagna, A. Graudenzi, M.Antoniotti, G. Mauri and A. d'Onofrio, Effects of delayed Immune-response in tumor Immune-system interplay,, Proc. of the First Int. Work. on Hybrid Systems and Biology (HSB), 92 (2012), 106.   Google Scholar

[16]

G. Caravagna and J. Hillston, Bio-PEPAd: A non-Markovian extension of Bio-PEPA,, Th. Comp. Sc., 419 (2012), 26.   Google Scholar

[17]

G. Caravagna, G. Mauri and A. d'Onofrio, The interplay of intrinsic and extrinsic bounded noises in genetic networks,, Submitted. Preprint at , ().   Google Scholar

[18]

V. Costanza and J. H. Seinfeld, Stochastic sensitivity analysis in chemical kinetics,, J. Chem. Phys., 74 (1981), 3852.   Google Scholar

[19]

D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables,, Proc. Cambridge Phil. Soc., 51 (1955), 433.   Google Scholar

[20]

F. Crauste, Stability and hopf bifurcation for a first-order delay differential equation with distributed delay,, in, (2010), 263.   Google Scholar

[21]

P. R. D'Argenio, J.-P. Katoen and E. Brinksma, A stochastic automata model and its algebraic approach,, Proc. 5th Int. Workshop on Process Algebra and Performance Modeling, (1997), 97.   Google Scholar

[22]

C. Damiani and P. Lecca, A novel method for parameter sensitivity analysis of stochastic complex systems,, in, (2012).   Google Scholar

[23]

A. d'Onofrio, Tumor-Immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy,, Math. Mod. Meth. App. Sci., 16 (2006), 1375.   Google Scholar

[24]

A. d'Onofrio, Tumor evasion from Immune system control: Strategies of a MISS to become a MASS,, Ch. Sol. Fract., 31 (2007), 261.   Google Scholar

[25]

A. d'Onofrio and P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases,, J. Th. Bio., 256 (2009), 473.   Google Scholar

[26]

A. d'Onofrio, On the interaction between the Immune system and an exponentially replicating pathogen,, Math. Biosc. Eng., 7 (2010), 579.   Google Scholar

[27]

A. d'Onofrio, G. Caravagna and R. Barbuti, Fine-tuning anti-tumor immunotherapies via stochastic simulations,, BMC Bioinformatics, 13 (2012).   Google Scholar

[28]

A. d'Onofrio, Tumour evasion from Immune system control as bounded-noise induced transition,, Phys. Rev. E, 81 (2010).   Google Scholar

[29]

A. d'Onofrio and A. Ciancio, A simple biophysical model of tumor evasion form Immune control,, Phys. Rev. E, 84 (2011).   Google Scholar

[30]

M. Al Tameemi, M. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the Immune system: consequences of brief encounters,, Biology Direct, (2012).   Google Scholar

[31]

A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of Tumor-Immune system interaction,, Math. Comp. Mod., 51 (2010), 572.   Google Scholar

[32]

H. H. A. Davis, Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models,, J. Roy. Stat. So. Series B, 46 (1984), 353.   Google Scholar

[33]

R. J. DeBoer, P. Hogeweg, F. Hub, J. Dullens, R. A. DeWeger and W. DenOtter, Macrophage T Lymphocyte interactions in the anti-tumor Immune response: A mathematical model,, J. Immunol., 134 (1985), 2748.   Google Scholar

[34]

L. G. De Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated Immune response to tumor growth,, Cancer Res., 65 (2005), 7950.   Google Scholar

[35]

V. T. De Vito Jr., J. Hellman and S. A. Rosenberg, "Cancer: Principles and Practice of Oncology,", J. P. Lippincott. 2005., (2005).   Google Scholar

[36]

G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of Cancer Immunoediting,, Ann. Rev. of Immun., 22 (2004), 322.   Google Scholar

[37]

P. Ehrlich, Ueber den jetzigen Stand der Karzinomforschung,, Ned. Tijdschr. Geneeskd., 5 (1909), 273.   Google Scholar

[38]

H. Enderling, L. Hlatky and P. Hahnfeldt, Immunoediting: Evidence of the multifaceted role of the immune system in self-metastatic tumor growth,, Theoretical Biology and Medical Modelling, 9 (2012).   Google Scholar

[39]

M. Farkas, "Periodic Motions,", Springer-Verlag, (1994).   Google Scholar

[40]

P. Feng, Dynamics of a segmentation clock model with discrete and distributed delays,, Int. J. Biomath., 3 (2010), 1.   Google Scholar

[41]

M. Galach, Dynamics of the tumour-Immune system competition: The effect of time delay,, Int. J. App. Math. and Comp. Sci., 13 (2003), 395.   Google Scholar

[42]

C. W. Gardiner, "Handbook of Stochastic Methods,", (2nd edition). Springer. 1985., (1985).   Google Scholar

[43]

R. Gatti, et al., Cyclic Leukocytosis in Chronic Myelogenous Leukemia: New Perspectives on Pathogenesis and Therapy,, Blood, 41 (1973), 771.   Google Scholar

[44]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,, J. of Comp. Phys., 22 (1976), 403.   Google Scholar

[45]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions,, J. of Phys. Chem., 81 (1977), 2340.   Google Scholar

[46]

P. W. Glynn, On the role of generalized semi-markov processes in simulation output analysis,, Proc. of the 15th conference on Winter simulation, 1 (1983), 39.   Google Scholar

[47]

R. Gunawan, Y. Cao, L. Petzold and F. J. Doyle III, Sensitivity analysis of discrete stochastic systems,, Biophys. J., 88 (2005), 2530.   Google Scholar

[48]

S. A. Gourley and S.Ruan, Dynamics of the diffusive Nicholson blowflies equation with distributed delay,, Proc. Roy. Soc. Edinburgh A, 130 (2000), 1275.   Google Scholar

[49]

Y. Han and Y. Song, Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays,, Nonlin. Dyn., 69 (2011), 357.   Google Scholar

[50]

R. Jessop, "Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays,", University of Waterloo, (2011).   Google Scholar

[51]

C. H. June, Adoptive T cell therapy for cancer in the clinic,, J. Clin. Invest., 117 (2007), 1466.   Google Scholar

[52]

J. M. Kaminski, J. B. Summers, M. B. Ward, M. R. Huber and B. Minev, Immunotherapy and prostate cancer,, Canc. Treat. Rev., 29 (2004), 199.   Google Scholar

[53]

B. J. Kennedy, Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy,, Blood, 35 (1970), 751.   Google Scholar

[54]

D. Kirschner, J. C. Arciero and T. L. Jackson, A mathematical model of tumor-Immune evasion and siRNA treatment,, Discr. Cont. Dyn. Systems, 4 (2004), 39.   Google Scholar

[55]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-Immune interaction,, J. Math. Biol., 37 (1998), 235.   Google Scholar

[56]

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

[57]

Y. Kuang, Delay differential equations,, Sourcebook in Theoretical Ecology, (2011).   Google Scholar

[58]

K. A. Kuznetsov and G. D. Knott, Modeling tumor regrowth and immunotherapy,, Math. Comp. Mod., 33 (2001).   Google Scholar

[59]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol., 56 (1994), 295.   Google Scholar

[60]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Sc., 197 (1977), 287.   Google Scholar

[61]

R. M. C. May and A. R. McLean, "Theoretical Ecology: Principles and Applications,", Oxford University Press, (2007).   Google Scholar

[62]

B. C. Mehta and M. B. Agarwal, Cyclic oscillations in leukocyte count in chronic myeloid leukemia,, A. Hem. 63 (1980), 63 (1980), 68.   Google Scholar

[63]

J. D. Murray, "Mathematical Biology,", third edition, (2003).   Google Scholar

[64]

D. Pardoll, Does the Immune system see tumours as foreign or self?,, Ann. Rev. Immun., 21 (2003), 807.   Google Scholar

[65]

D. Rodriguez-Perez, O. Sotolongo-Grau, R. Espinosa, R. O. Sotolongo-Costa, J. A. Santos Miranda and J. C. Antoranz, Assessment of cancer immunotherapy outcome in terms of the Immune response time features,, Math. Med. and Bio., 24 (2007), 287.   Google Scholar

[66]

P. Martin, S. Martin, P. Burton and I. Roitt, "Roitt's Essential Immunology,", Wiley-Blackwell, (2011).   Google Scholar

[67]

S. Ruan, Delay differential Eequation in single species dynamics,, in, 1 (): 477.   Google Scholar

[68]

A. Sohrabi, J. Sandoz, J. S. Spratt and H. C. Polk, Recurrence of breast cancer: Obesity, tumor size, and axillary lymph node metastases,, JAMA, 244 (1980), 264.   Google Scholar

[69]

H. Tsao, A. B. Cosimi and A. J. Sober, Ultra-late recurrence (15 years or longer) of cutaneous melanoma,, Cancer, 79 (1997), 2361.   Google Scholar

[70]

A. P. Vicari, G. Caux and G. Trinchieri, Tumor escape from Immune surveillance through dendritic cell inactivation,, Sem. Canc. Biol., 2 (2002), 33.   Google Scholar

[71]

M. Villasana and A. Radunskaya, A delay differential equation model for tumour growth,, J. of Math. Bio., 47 (2003), 270.   Google Scholar

[72]

H. Vodopick, E. M. Rupp, C. L. Edwards, F. A. Goswitz and J. J. Beauchamp, Spontaneous cyclic leukocytosis and thrombocytosis in chronic granulocytic leukemia,, New Engl. J. of Med., 286 (1972), 284.   Google Scholar

[73]

T. L. Whiteside, Tumor-induced death of Immune cells: Its mechanisms and consequences,, Sem. Canc. Biol., 12 (2002), 43.   Google Scholar

[74]

E. C. Zeeman, Stability of dynamical systems,, Nonlin., 1 (1988), 115.   Google Scholar

[75]

C. H. Zhang and Y. Xiang-Ping, Stability and Hopf bifurcations in a delayed predator-prey system with a distributed delay,, Int. J. Bifur. Chaos Appl. Sci. Eng., 19 (2009), 2283.   Google Scholar

show all references

References:
[1]

S. A. Agarwala, "New Applications of Cancer Immunotherapy,", S. A. Agarwala (Guest Editor), (2003), 29.   Google Scholar

[2]

R. Barbuti, G. Caravagna, A. Maggiolo-Schettini and P. Milazzo, Delay stochastic simulation of biological systems: A purely delayed approach,, C.Priami et al.(Eds.): Trans. Comp. Sys. Bio. XIII, 6575 (2011), 61.   Google Scholar

[3]

M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation,, PLoS Comp. Bio., 2 (2006).   Google Scholar

[4]

N. Bellomo and G. Forni, Complex multicellular systems and Immune competition: New paradigms looking for a mathematical theory,, Curr. Top. Dev. Bio., 81 (2008), 485.  doi: 10.1016/S0070-2153(07)81017-9.  Google Scholar

[5]

E. Beretta, V. Capasso and F. Rinaldi, Global stability results for a generalized Lotka-Volterra system with distributed delays,, J. Math. Bio., 26 (1988), 661.   Google Scholar

[6]

I. Bleumer, E. Oosterwijk, P. de Mulder and P. F. Mulders, Immunotherapy for renal cell carcinoma,, Europ. Urol., 44 (2003), 65.  doi: 10.1016/S0302-2838(03)00191-X.  Google Scholar

[7]

N. Blumberg, C. Chuang-Stein and J. M. Heal, The relationship of blood transfusion, tumor staging and cancer recurrence,, Transf., 30 (1990), 291.  doi: 10.1046/j.1537-2995.1990.30490273432.x.  Google Scholar

[8]

K. B. Blyuss and Y. N. Kyrychko, Stability and bifurcations in an epidemic model with varying immunity period,, Bull. Math. Bio., 72 (2010), 490.  doi: 10.1007/s11538-009-9458-y.  Google Scholar

[9]

L. Bortolussi, Automata and (stochastic) programs. The hybrid automata lattice of a stochastic program,, J. Log. Comp., (2011).  doi: 10.1093/logcom/exr045.  Google Scholar

[10]

L. Bortolussi and A. Policriti, The importance of being (a little bit) discrete,, ENTCS, 229 (2009), 75.   Google Scholar

[11]

M. Bravetti and R. Gorrieri, The theory of interactive generalized semi-Markov processes,, Theoret. Comp. Sci., 282 (2002), 5.   Google Scholar

[12]

N. Burić and D. Todorović, Dynamics of delay-differential equations modelling immunology of tumor growth,, Cha. Sol. Fract., 13 (2002), 645.   Google Scholar

[13]

G. Caravagna, "Formal Modeling and Simulation of Biological Systems With Delays,", Ph.D. Thesis, (2011).   Google Scholar

[14]

G. Caravagna, A. d'Onofrio, P. Milazzo and R. Barbuti, Antitumour Immune surveillance through stochastic oscillations,, J. Th. Biology, 265 (2010), 336.   Google Scholar

[15]

G. Caravagna, A. Graudenzi, M.Antoniotti, G. Mauri and A. d'Onofrio, Effects of delayed Immune-response in tumor Immune-system interplay,, Proc. of the First Int. Work. on Hybrid Systems and Biology (HSB), 92 (2012), 106.   Google Scholar

[16]

G. Caravagna and J. Hillston, Bio-PEPAd: A non-Markovian extension of Bio-PEPA,, Th. Comp. Sc., 419 (2012), 26.   Google Scholar

[17]

G. Caravagna, G. Mauri and A. d'Onofrio, The interplay of intrinsic and extrinsic bounded noises in genetic networks,, Submitted. Preprint at , ().   Google Scholar

[18]

V. Costanza and J. H. Seinfeld, Stochastic sensitivity analysis in chemical kinetics,, J. Chem. Phys., 74 (1981), 3852.   Google Scholar

[19]

D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables,, Proc. Cambridge Phil. Soc., 51 (1955), 433.   Google Scholar

[20]

F. Crauste, Stability and hopf bifurcation for a first-order delay differential equation with distributed delay,, in, (2010), 263.   Google Scholar

[21]

P. R. D'Argenio, J.-P. Katoen and E. Brinksma, A stochastic automata model and its algebraic approach,, Proc. 5th Int. Workshop on Process Algebra and Performance Modeling, (1997), 97.   Google Scholar

[22]

C. Damiani and P. Lecca, A novel method for parameter sensitivity analysis of stochastic complex systems,, in, (2012).   Google Scholar

[23]

A. d'Onofrio, Tumor-Immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy,, Math. Mod. Meth. App. Sci., 16 (2006), 1375.   Google Scholar

[24]

A. d'Onofrio, Tumor evasion from Immune system control: Strategies of a MISS to become a MASS,, Ch. Sol. Fract., 31 (2007), 261.   Google Scholar

[25]

A. d'Onofrio and P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases,, J. Th. Bio., 256 (2009), 473.   Google Scholar

[26]

A. d'Onofrio, On the interaction between the Immune system and an exponentially replicating pathogen,, Math. Biosc. Eng., 7 (2010), 579.   Google Scholar

[27]

A. d'Onofrio, G. Caravagna and R. Barbuti, Fine-tuning anti-tumor immunotherapies via stochastic simulations,, BMC Bioinformatics, 13 (2012).   Google Scholar

[28]

A. d'Onofrio, Tumour evasion from Immune system control as bounded-noise induced transition,, Phys. Rev. E, 81 (2010).   Google Scholar

[29]

A. d'Onofrio and A. Ciancio, A simple biophysical model of tumor evasion form Immune control,, Phys. Rev. E, 84 (2011).   Google Scholar

[30]

M. Al Tameemi, M. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the Immune system: consequences of brief encounters,, Biology Direct, (2012).   Google Scholar

[31]

A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of Tumor-Immune system interaction,, Math. Comp. Mod., 51 (2010), 572.   Google Scholar

[32]

H. H. A. Davis, Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models,, J. Roy. Stat. So. Series B, 46 (1984), 353.   Google Scholar

[33]

R. J. DeBoer, P. Hogeweg, F. Hub, J. Dullens, R. A. DeWeger and W. DenOtter, Macrophage T Lymphocyte interactions in the anti-tumor Immune response: A mathematical model,, J. Immunol., 134 (1985), 2748.   Google Scholar

[34]

L. G. De Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated Immune response to tumor growth,, Cancer Res., 65 (2005), 7950.   Google Scholar

[35]

V. T. De Vito Jr., J. Hellman and S. A. Rosenberg, "Cancer: Principles and Practice of Oncology,", J. P. Lippincott. 2005., (2005).   Google Scholar

[36]

G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of Cancer Immunoediting,, Ann. Rev. of Immun., 22 (2004), 322.   Google Scholar

[37]

P. Ehrlich, Ueber den jetzigen Stand der Karzinomforschung,, Ned. Tijdschr. Geneeskd., 5 (1909), 273.   Google Scholar

[38]

H. Enderling, L. Hlatky and P. Hahnfeldt, Immunoediting: Evidence of the multifaceted role of the immune system in self-metastatic tumor growth,, Theoretical Biology and Medical Modelling, 9 (2012).   Google Scholar

[39]

M. Farkas, "Periodic Motions,", Springer-Verlag, (1994).   Google Scholar

[40]

P. Feng, Dynamics of a segmentation clock model with discrete and distributed delays,, Int. J. Biomath., 3 (2010), 1.   Google Scholar

[41]

M. Galach, Dynamics of the tumour-Immune system competition: The effect of time delay,, Int. J. App. Math. and Comp. Sci., 13 (2003), 395.   Google Scholar

[42]

C. W. Gardiner, "Handbook of Stochastic Methods,", (2nd edition). Springer. 1985., (1985).   Google Scholar

[43]

R. Gatti, et al., Cyclic Leukocytosis in Chronic Myelogenous Leukemia: New Perspectives on Pathogenesis and Therapy,, Blood, 41 (1973), 771.   Google Scholar

[44]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,, J. of Comp. Phys., 22 (1976), 403.   Google Scholar

[45]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions,, J. of Phys. Chem., 81 (1977), 2340.   Google Scholar

[46]

P. W. Glynn, On the role of generalized semi-markov processes in simulation output analysis,, Proc. of the 15th conference on Winter simulation, 1 (1983), 39.   Google Scholar

[47]

R. Gunawan, Y. Cao, L. Petzold and F. J. Doyle III, Sensitivity analysis of discrete stochastic systems,, Biophys. J., 88 (2005), 2530.   Google Scholar

[48]

S. A. Gourley and S.Ruan, Dynamics of the diffusive Nicholson blowflies equation with distributed delay,, Proc. Roy. Soc. Edinburgh A, 130 (2000), 1275.   Google Scholar

[49]

Y. Han and Y. Song, Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays,, Nonlin. Dyn., 69 (2011), 357.   Google Scholar

[50]

R. Jessop, "Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays,", University of Waterloo, (2011).   Google Scholar

[51]

C. H. June, Adoptive T cell therapy for cancer in the clinic,, J. Clin. Invest., 117 (2007), 1466.   Google Scholar

[52]

J. M. Kaminski, J. B. Summers, M. B. Ward, M. R. Huber and B. Minev, Immunotherapy and prostate cancer,, Canc. Treat. Rev., 29 (2004), 199.   Google Scholar

[53]

B. J. Kennedy, Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy,, Blood, 35 (1970), 751.   Google Scholar

[54]

D. Kirschner, J. C. Arciero and T. L. Jackson, A mathematical model of tumor-Immune evasion and siRNA treatment,, Discr. Cont. Dyn. Systems, 4 (2004), 39.   Google Scholar

[55]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-Immune interaction,, J. Math. Biol., 37 (1998), 235.   Google Scholar

[56]

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

[57]

Y. Kuang, Delay differential equations,, Sourcebook in Theoretical Ecology, (2011).   Google Scholar

[58]

K. A. Kuznetsov and G. D. Knott, Modeling tumor regrowth and immunotherapy,, Math. Comp. Mod., 33 (2001).   Google Scholar

[59]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol., 56 (1994), 295.   Google Scholar

[60]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Sc., 197 (1977), 287.   Google Scholar

[61]

R. M. C. May and A. R. McLean, "Theoretical Ecology: Principles and Applications,", Oxford University Press, (2007).   Google Scholar

[62]

B. C. Mehta and M. B. Agarwal, Cyclic oscillations in leukocyte count in chronic myeloid leukemia,, A. Hem. 63 (1980), 63 (1980), 68.   Google Scholar

[63]

J. D. Murray, "Mathematical Biology,", third edition, (2003).   Google Scholar

[64]

D. Pardoll, Does the Immune system see tumours as foreign or self?,, Ann. Rev. Immun., 21 (2003), 807.   Google Scholar

[65]

D. Rodriguez-Perez, O. Sotolongo-Grau, R. Espinosa, R. O. Sotolongo-Costa, J. A. Santos Miranda and J. C. Antoranz, Assessment of cancer immunotherapy outcome in terms of the Immune response time features,, Math. Med. and Bio., 24 (2007), 287.   Google Scholar

[66]

P. Martin, S. Martin, P. Burton and I. Roitt, "Roitt's Essential Immunology,", Wiley-Blackwell, (2011).   Google Scholar

[67]

S. Ruan, Delay differential Eequation in single species dynamics,, in, 1 (): 477.   Google Scholar

[68]

A. Sohrabi, J. Sandoz, J. S. Spratt and H. C. Polk, Recurrence of breast cancer: Obesity, tumor size, and axillary lymph node metastases,, JAMA, 244 (1980), 264.   Google Scholar

[69]

H. Tsao, A. B. Cosimi and A. J. Sober, Ultra-late recurrence (15 years or longer) of cutaneous melanoma,, Cancer, 79 (1997), 2361.   Google Scholar

[70]

A. P. Vicari, G. Caux and G. Trinchieri, Tumor escape from Immune surveillance through dendritic cell inactivation,, Sem. Canc. Biol., 2 (2002), 33.   Google Scholar

[71]

M. Villasana and A. Radunskaya, A delay differential equation model for tumour growth,, J. of Math. Bio., 47 (2003), 270.   Google Scholar

[72]

H. Vodopick, E. M. Rupp, C. L. Edwards, F. A. Goswitz and J. J. Beauchamp, Spontaneous cyclic leukocytosis and thrombocytosis in chronic granulocytic leukemia,, New Engl. J. of Med., 286 (1972), 284.   Google Scholar

[73]

T. L. Whiteside, Tumor-induced death of Immune cells: Its mechanisms and consequences,, Sem. Canc. Biol., 12 (2002), 43.   Google Scholar

[74]

E. C. Zeeman, Stability of dynamical systems,, Nonlin., 1 (1988), 115.   Google Scholar

[75]

C. H. Zhang and Y. Xiang-Ping, Stability and Hopf bifurcations in a delayed predator-prey system with a distributed delay,, Int. J. Bifur. Chaos Appl. Sci. Eng., 19 (2009), 2283.   Google Scholar

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