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Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells
Distributed delays in a hybrid model of tumorImmune system interplay
1.  Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I20126 Milan, Italy, Italy 
2.  Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, I20141 Milan 
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 tumorinduced 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 twodimensional stochastic process is interlinked to a multidimensional meanfield system. We instantiate the model with two wellknown 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 delayinduced tumor eradication.
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S. A. Agarwala, "New Applications of Cancer Immunotherapy,", S. A. Agarwala (Guest Editor), (2003), 29. 
[2] 
R. Barbuti, G. Caravagna, A. MaggioloSchettini 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. 
[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). 
[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/S00702153(07)810179. 
[5] 
E. Beretta, V. Capasso and F. Rinaldi, Global stability results for a generalized LotkaVolterra system with distributed delays,, J. Math. Bio., 26 (1988), 661. 
[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/S03022838(03)00191X. 
[7] 
N. Blumberg, C. ChuangStein and J. M. Heal, The relationship of blood transfusion, tumor staging and cancer recurrence,, Transf., 30 (1990), 291. doi: 10.1046/j.15372995.1990.30490273432.x. 
[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/s115380099458y. 
[9] 
L. Bortolussi, Automata and (stochastic) programs. The hybrid automata lattice of a stochastic program,, J. Log. Comp., (2011). doi: 10.1093/logcom/exr045. 
[10] 
L. Bortolussi and A. Policriti, The importance of being (a little bit) discrete,, ENTCS, 229 (2009), 75. 
[11] 
M. Bravetti and R. Gorrieri, The theory of interactive generalized semiMarkov processes,, Theoret. Comp. Sci., 282 (2002), 5. 
[12] 
N. Burić and D. Todorović, Dynamics of delaydifferential equations modelling immunology of tumor growth,, Cha. Sol. Fract., 13 (2002), 645. 
[13] 
G. Caravagna, "Formal Modeling and Simulation of Biological Systems With Delays,", Ph.D. Thesis, (2011). 
[14] 
G. Caravagna, A. d'Onofrio, P. Milazzo and R. Barbuti, Antitumour Immune surveillance through stochastic oscillations,, J. Th. Biology, 265 (2010), 336. 
[15] 
G. Caravagna, A. Graudenzi, M.Antoniotti, G. Mauri and A. d'Onofrio, Effects of delayed Immuneresponse in tumor Immunesystem interplay,, Proc. of the First Int. Work. on Hybrid Systems and Biology (HSB), 92 (2012), 106. 
[16] 
G. Caravagna and J. Hillston, BioPEPAd: A nonMarkovian extension of BioPEPA,, Th. Comp. Sc., 419 (2012), 26. 
[17] 
G. Caravagna, G. Mauri and A. d'Onofrio, The interplay of intrinsic and extrinsic bounded noises in genetic networks,, Submitted. Preprint at , (). 
[18] 
V. Costanza and J. H. Seinfeld, Stochastic sensitivity analysis in chemical kinetics,, J. Chem. Phys., 74 (1981), 3852. 
[19] 
D. R. Cox, The analysis of nonMarkovian stochastic processes by the inclusion of supplementary variables,, Proc. Cambridge Phil. Soc., 51 (1955), 433. 
[20] 
F. Crauste, Stability and hopf bifurcation for a firstorder delay differential equation with distributed delay,, in, (2010), 263. 
[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. 
[22] 
C. Damiani and P. Lecca, A novel method for parameter sensitivity analysis of stochastic complex systems,, in, (2012). 
[23] 
A. d'Onofrio, TumorImmune system interaction: modeling the tumorstimulated proliferation of effectors and immunotherapy,, Math. Mod. Meth. App. Sci., 16 (2006), 1375. 
[24] 
A. d'Onofrio, Tumor evasion from Immune system control: Strategies of a MISS to become a MASS,, Ch. Sol. Fract., 31 (2007), 261. 
[25] 
A. d'Onofrio and P. Manfredi, Informationrelated changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases,, J. Th. Bio., 256 (2009), 473. 
[26] 
A. d'Onofrio, On the interaction between the Immune system and an exponentially replicating pathogen,, Math. Biosc. Eng., 7 (2010), 579. 
[27] 
A. d'Onofrio, G. Caravagna and R. Barbuti, Finetuning antitumor immunotherapies via stochastic simulations,, BMC Bioinformatics, 13 (2012). 
[28] 
A. d'Onofrio, Tumour evasion from Immune system control as boundednoise induced transition,, Phys. Rev. E, 81 (2010). 
[29] 
A. d'Onofrio and A. Ciancio, A simple biophysical model of tumor evasion form Immune control,, Phys. Rev. E, 84 (2011). 
[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). 
[31] 
A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delayinduced oscillatory dynamics of TumorImmune system interaction,, Math. Comp. Mod., 51 (2010), 572. 
[32] 
H. H. A. Davis, Piecewise deterministic Markov processes: a general class of nondiffusion stochastic models,, J. Roy. Stat. So. Series B, 46 (1984), 353. 
[33] 
R. J. DeBoer, P. Hogeweg, F. Hub, J. Dullens, R. A. DeWeger and W. DenOtter, Macrophage T Lymphocyte interactions in the antitumor Immune response: A mathematical model,, J. Immunol., 134 (1985), 2748. 
[34] 
L. G. De Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cellmediated Immune response to tumor growth,, Cancer Res., 65 (2005), 7950. 
[35] 
V. T. De Vito Jr., J. Hellman and S. A. Rosenberg, "Cancer: Principles and Practice of Oncology,", J. P. Lippincott. 2005., (2005). 
[36] 
G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of Cancer Immunoediting,, Ann. Rev. of Immun., 22 (2004), 322. 
[37] 
P. Ehrlich, Ueber den jetzigen Stand der Karzinomforschung,, Ned. Tijdschr. Geneeskd., 5 (1909), 273. 
[38] 
H. Enderling, L. Hlatky and P. Hahnfeldt, Immunoediting: Evidence of the multifaceted role of the immune system in selfmetastatic tumor growth,, Theoretical Biology and Medical Modelling, 9 (2012). 
[39]  
[40] 
P. Feng, Dynamics of a segmentation clock model with discrete and distributed delays,, Int. J. Biomath., 3 (2010), 1. 
[41] 
M. Galach, Dynamics of the tumourImmune system competition: The effect of time delay,, Int. J. App. Math. and Comp. Sci., 13 (2003), 395. 
[42] 
C. W. Gardiner, "Handbook of Stochastic Methods,", (2nd edition). Springer. 1985., (1985). 
[43] 
R. Gatti, et al., Cyclic Leukocytosis in Chronic Myelogenous Leukemia: New Perspectives on Pathogenesis and Therapy,, Blood, 41 (1973), 771. 
[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. 
[45] 
D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions,, J. of Phys. Chem., 81 (1977), 2340. 
[46] 
P. W. Glynn, On the role of generalized semimarkov processes in simulation output analysis,, Proc. of the 15th conference on Winter simulation, 1 (1983), 39. 
[47] 
R. Gunawan, Y. Cao, L. Petzold and F. J. Doyle III, Sensitivity analysis of discrete stochastic systems,, Biophys. J., 88 (2005), 2530. 
[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. 
[49] 
Y. Han and Y. Song, Stability and Hopf bifurcation in a threeneuron unidirectional ring with distributed delays,, Nonlin. Dyn., 69 (2011), 357. 
[50] 
R. Jessop, "Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays,", University of Waterloo, (2011). 
[51] 
C. H. June, Adoptive T cell therapy for cancer in the clinic,, J. Clin. Invest., 117 (2007), 1466. 
[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. 
[53] 
B. J. Kennedy, Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy,, Blood, 35 (1970), 751. 
[54] 
D. Kirschner, J. C. Arciero and T. L. Jackson, A mathematical model of tumorImmune evasion and siRNA treatment,, Discr. Cont. Dyn. Systems, 4 (2004), 39. 
[55] 
D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumorImmune interaction,, J. Math. Biol., 37 (1998), 235. 
[56] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). 
[57] 
Y. Kuang, Delay differential equations,, Sourcebook in Theoretical Ecology, (2011). 
[58] 
K. A. Kuznetsov and G. D. Knott, Modeling tumor regrowth and immunotherapy,, Math. Comp. Mod., 33 (2001). 
[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. 
[60] 
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Sc., 197 (1977), 287. 
[61] 
R. M. C. May and A. R. McLean, "Theoretical Ecology: Principles and Applications,", Oxford University Press, (2007). 
[62] 
B. C. Mehta and M. B. Agarwal, Cyclic oscillations in leukocyte count in chronic myeloid leukemia,, A. Hem. 63 (1980), 63 (1980), 68. 
[63] 
J. D. Murray, "Mathematical Biology,", third edition, (2003). 
[64] 
D. Pardoll, Does the Immune system see tumours as foreign or self?,, Ann. Rev. Immun., 21 (2003), 807. 
[65] 
D. RodriguezPerez, O. SotolongoGrau, R. Espinosa, R. O. SotolongoCosta, 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. 
[66] 
P. Martin, S. Martin, P. Burton and I. Roitt, "Roitt's Essential Immunology,", WileyBlackwell, (2011). 
[67] 
S. Ruan, Delay differential Eequation in single species dynamics,, in, 1 (): 477. 
[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. 
[69] 
H. Tsao, A. B. Cosimi and A. J. Sober, Ultralate recurrence (15 years or longer) of cutaneous melanoma,, Cancer, 79 (1997), 2361. 
[70] 
A. P. Vicari, G. Caux and G. Trinchieri, Tumor escape from Immune surveillance through dendritic cell inactivation,, Sem. Canc. Biol., 2 (2002), 33. 
[71] 
M. Villasana and A. Radunskaya, A delay differential equation model for tumour growth,, J. of Math. Bio., 47 (2003), 270. 
[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. 
[73] 
T. L. Whiteside, Tumorinduced death of Immune cells: Its mechanisms and consequences,, Sem. Canc. Biol., 12 (2002), 43. 
[74] 
E. C. Zeeman, Stability of dynamical systems,, Nonlin., 1 (1988), 115. 
[75] 
C. H. Zhang and Y. XiangPing, Stability and Hopf bifurcations in a delayed predatorprey system with a distributed delay,, Int. J. Bifur. Chaos Appl. Sci. Eng., 19 (2009), 2283. 
show all references
References:
[1] 
S. A. Agarwala, "New Applications of Cancer Immunotherapy,", S. A. Agarwala (Guest Editor), (2003), 29. 
[2] 
R. Barbuti, G. Caravagna, A. MaggioloSchettini 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. 
[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). 
[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/S00702153(07)810179. 
[5] 
E. Beretta, V. Capasso and F. Rinaldi, Global stability results for a generalized LotkaVolterra system with distributed delays,, J. Math. Bio., 26 (1988), 661. 
[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/S03022838(03)00191X. 
[7] 
N. Blumberg, C. ChuangStein and J. M. Heal, The relationship of blood transfusion, tumor staging and cancer recurrence,, Transf., 30 (1990), 291. doi: 10.1046/j.15372995.1990.30490273432.x. 
[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/s115380099458y. 
[9] 
L. Bortolussi, Automata and (stochastic) programs. The hybrid automata lattice of a stochastic program,, J. Log. Comp., (2011). doi: 10.1093/logcom/exr045. 
[10] 
L. Bortolussi and A. Policriti, The importance of being (a little bit) discrete,, ENTCS, 229 (2009), 75. 
[11] 
M. Bravetti and R. Gorrieri, The theory of interactive generalized semiMarkov processes,, Theoret. Comp. Sci., 282 (2002), 5. 
[12] 
N. Burić and D. Todorović, Dynamics of delaydifferential equations modelling immunology of tumor growth,, Cha. Sol. Fract., 13 (2002), 645. 
[13] 
G. Caravagna, "Formal Modeling and Simulation of Biological Systems With Delays,", Ph.D. Thesis, (2011). 
[14] 
G. Caravagna, A. d'Onofrio, P. Milazzo and R. Barbuti, Antitumour Immune surveillance through stochastic oscillations,, J. Th. Biology, 265 (2010), 336. 
[15] 
G. Caravagna, A. Graudenzi, M.Antoniotti, G. Mauri and A. d'Onofrio, Effects of delayed Immuneresponse in tumor Immunesystem interplay,, Proc. of the First Int. Work. on Hybrid Systems and Biology (HSB), 92 (2012), 106. 
[16] 
G. Caravagna and J. Hillston, BioPEPAd: A nonMarkovian extension of BioPEPA,, Th. Comp. Sc., 419 (2012), 26. 
[17] 
G. Caravagna, G. Mauri and A. d'Onofrio, The interplay of intrinsic and extrinsic bounded noises in genetic networks,, Submitted. Preprint at , (). 
[18] 
V. Costanza and J. H. Seinfeld, Stochastic sensitivity analysis in chemical kinetics,, J. Chem. Phys., 74 (1981), 3852. 
[19] 
D. R. Cox, The analysis of nonMarkovian stochastic processes by the inclusion of supplementary variables,, Proc. Cambridge Phil. Soc., 51 (1955), 433. 
[20] 
F. Crauste, Stability and hopf bifurcation for a firstorder delay differential equation with distributed delay,, in, (2010), 263. 
[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. 
[22] 
C. Damiani and P. Lecca, A novel method for parameter sensitivity analysis of stochastic complex systems,, in, (2012). 
[23] 
A. d'Onofrio, TumorImmune system interaction: modeling the tumorstimulated proliferation of effectors and immunotherapy,, Math. Mod. Meth. App. Sci., 16 (2006), 1375. 
[24] 
A. d'Onofrio, Tumor evasion from Immune system control: Strategies of a MISS to become a MASS,, Ch. Sol. Fract., 31 (2007), 261. 
[25] 
A. d'Onofrio and P. Manfredi, Informationrelated changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases,, J. Th. Bio., 256 (2009), 473. 
[26] 
A. d'Onofrio, On the interaction between the Immune system and an exponentially replicating pathogen,, Math. Biosc. Eng., 7 (2010), 579. 
[27] 
A. d'Onofrio, G. Caravagna and R. Barbuti, Finetuning antitumor immunotherapies via stochastic simulations,, BMC Bioinformatics, 13 (2012). 
[28] 
A. d'Onofrio, Tumour evasion from Immune system control as boundednoise induced transition,, Phys. Rev. E, 81 (2010). 
[29] 
A. d'Onofrio and A. Ciancio, A simple biophysical model of tumor evasion form Immune control,, Phys. Rev. E, 84 (2011). 
[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). 
[31] 
A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delayinduced oscillatory dynamics of TumorImmune system interaction,, Math. Comp. Mod., 51 (2010), 572. 
[32] 
H. H. A. Davis, Piecewise deterministic Markov processes: a general class of nondiffusion stochastic models,, J. Roy. Stat. So. Series B, 46 (1984), 353. 
[33] 
R. J. DeBoer, P. Hogeweg, F. Hub, J. Dullens, R. A. DeWeger and W. DenOtter, Macrophage T Lymphocyte interactions in the antitumor Immune response: A mathematical model,, J. Immunol., 134 (1985), 2748. 
[34] 
L. G. De Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cellmediated Immune response to tumor growth,, Cancer Res., 65 (2005), 7950. 
[35] 
V. T. De Vito Jr., J. Hellman and S. A. Rosenberg, "Cancer: Principles and Practice of Oncology,", J. P. Lippincott. 2005., (2005). 
[36] 
G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of Cancer Immunoediting,, Ann. Rev. of Immun., 22 (2004), 322. 
[37] 
P. Ehrlich, Ueber den jetzigen Stand der Karzinomforschung,, Ned. Tijdschr. Geneeskd., 5 (1909), 273. 
[38] 
H. Enderling, L. Hlatky and P. Hahnfeldt, Immunoediting: Evidence of the multifaceted role of the immune system in selfmetastatic tumor growth,, Theoretical Biology and Medical Modelling, 9 (2012). 
[39]  
[40] 
P. Feng, Dynamics of a segmentation clock model with discrete and distributed delays,, Int. J. Biomath., 3 (2010), 1. 
[41] 
M. Galach, Dynamics of the tumourImmune system competition: The effect of time delay,, Int. J. App. Math. and Comp. Sci., 13 (2003), 395. 
[42] 
C. W. Gardiner, "Handbook of Stochastic Methods,", (2nd edition). Springer. 1985., (1985). 
[43] 
R. Gatti, et al., Cyclic Leukocytosis in Chronic Myelogenous Leukemia: New Perspectives on Pathogenesis and Therapy,, Blood, 41 (1973), 771. 
[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. 
[45] 
D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions,, J. of Phys. Chem., 81 (1977), 2340. 
[46] 
P. W. Glynn, On the role of generalized semimarkov processes in simulation output analysis,, Proc. of the 15th conference on Winter simulation, 1 (1983), 39. 
[47] 
R. Gunawan, Y. Cao, L. Petzold and F. J. Doyle III, Sensitivity analysis of discrete stochastic systems,, Biophys. J., 88 (2005), 2530. 
[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. 
[49] 
Y. Han and Y. Song, Stability and Hopf bifurcation in a threeneuron unidirectional ring with distributed delays,, Nonlin. Dyn., 69 (2011), 357. 
[50] 
R. Jessop, "Stability and Hopf Bifurcation Analysis of Hopfield Neural Networks with a General Distribution of Delays,", University of Waterloo, (2011). 
[51] 
C. H. June, Adoptive T cell therapy for cancer in the clinic,, J. Clin. Invest., 117 (2007), 1466. 
[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. 
[53] 
B. J. Kennedy, Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy,, Blood, 35 (1970), 751. 
[54] 
D. Kirschner, J. C. Arciero and T. L. Jackson, A mathematical model of tumorImmune evasion and siRNA treatment,, Discr. Cont. Dyn. Systems, 4 (2004), 39. 
[55] 
D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumorImmune interaction,, J. Math. Biol., 37 (1998), 235. 
[56] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). 
[57] 
Y. Kuang, Delay differential equations,, Sourcebook in Theoretical Ecology, (2011). 
[58] 
K. A. Kuznetsov and G. D. Knott, Modeling tumor regrowth and immunotherapy,, Math. Comp. Mod., 33 (2001). 
[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. 
[60] 
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Sc., 197 (1977), 287. 
[61] 
R. M. C. May and A. R. McLean, "Theoretical Ecology: Principles and Applications,", Oxford University Press, (2007). 
[62] 
B. C. Mehta and M. B. Agarwal, Cyclic oscillations in leukocyte count in chronic myeloid leukemia,, A. Hem. 63 (1980), 63 (1980), 68. 
[63] 
J. D. Murray, "Mathematical Biology,", third edition, (2003). 
[64] 
D. Pardoll, Does the Immune system see tumours as foreign or self?,, Ann. Rev. Immun., 21 (2003), 807. 
[65] 
D. RodriguezPerez, O. SotolongoGrau, R. Espinosa, R. O. SotolongoCosta, 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. 
[66] 
P. Martin, S. Martin, P. Burton and I. Roitt, "Roitt's Essential Immunology,", WileyBlackwell, (2011). 
[67] 
S. Ruan, Delay differential Eequation in single species dynamics,, in, 1 (): 477. 
[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. 
[69] 
H. Tsao, A. B. Cosimi and A. J. Sober, Ultralate recurrence (15 years or longer) of cutaneous melanoma,, Cancer, 79 (1997), 2361. 
[70] 
A. P. Vicari, G. Caux and G. Trinchieri, Tumor escape from Immune surveillance through dendritic cell inactivation,, Sem. Canc. Biol., 2 (2002), 33. 
[71] 
M. Villasana and A. Radunskaya, A delay differential equation model for tumour growth,, J. of Math. Bio., 47 (2003), 270. 
[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. 
[73] 
T. L. Whiteside, Tumorinduced death of Immune cells: Its mechanisms and consequences,, Sem. Canc. Biol., 12 (2002), 43. 
[74] 
E. C. Zeeman, Stability of dynamical systems,, Nonlin., 1 (1988), 115. 
[75] 
C. H. Zhang and Y. XiangPing, Stability and Hopf bifurcations in a delayed predatorprey system with a distributed delay,, Int. J. Bifur. Chaos Appl. Sci. Eng., 19 (2009), 2283. 
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