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October  2021, 26(10): 5227-5249. doi: 10.3934/dcdsb.2020341

Qualitative analysis of a simple tumor-immune system with time delay of tumor action

1. 

Department of Mathematics, Shaanxi University of Science and Technology, Xi'an, 710021, China

2. 

Department of Immunology, Xi'an Medical University, Xi'an 710021, China

3. 

Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China, Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA

* Corresponding author: Jianquan Li; ** Equal contributor: Dian Zhang

Received  May 2019 Revised  October 2020 Published  October 2021 Early access  November 2020

Fund Project: Jianquan Li is supported by the National Natural Science Fund of China (Nos. 11971281, 12071268), Dian Zhang is supported by the Project of Xi'an Medical University (No. 2018GJFY05), Jia Li is supported by the National Natural Science Fund of China (No. 12071095), Xiaolin Lin is supported by the National Natural Science Fund of China (No. 11771259)

In this paper, we propose a simple tumor-immune system model with time delay of tumor action, where two kinds of effects of the tumor cells (i.e. stimulation and neutralization) on the effector cells are considered. The local stability of the model is obtained by analyzing the characteristic equations of the model at the corresponding equilibria, the sufficient conditions on the global stability are found by applying the Fluctuation Lemma and constructing the different convergent sequences. The obtained results show that, compared to the results for the model without time delay, the time delay of tumor action can affect the stability of tumor equilibrium of the model as the stimulation effect of the tumor cells is strong enough, while the delay is harmless for the stability of tumor equilibrium under the neutralization of tumor cells. For the appropriate neutralization of tumor cells on effector cells, the bistability of the tumor free equilibrium and the stronger tumor equilibrium can appear. In the case of stimulation of tumor cells, the sufficiently large time delay can lead to the appearance of a stable periodic solution by Hopf bifurcation, and the numerical simulation illustrates that the amplitude of the periodic orbit increases with time delay. We also discuss the dependence of the tumor equilibrium and the time delay threshold, determining the stability of the tumor equilibrium, on tumor action. The related conditions determining dynamics of the model are expressed by certain formulae with biological meanings.

Citation: Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5227-5249. doi: 10.3934/dcdsb.2020341
References:
[1]

P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dyn. Syst., 12 (2013), 1847-1888.  doi: 10.1137/120887898.

[2]

P. Bi and H. Xiao, Hopf bifurcation for tumor-immune competition systems with delay, Electron. J. Diff. Equ., 2014 (2014), 1-13. 

[3]

P. Bi and H. Xiao, Bifurcations of tumor-immune competition systems with delay, Abstract and Applied Analysis, 2014 (2014), 1-12.  doi: 10.1155/2014/723159.

[4]

N. Buric and T. Dragana, Dynamics of delay differential equations modeling immunology of tumor growth, Chaos Solitons Fractals, 13 (2002), 645-655.  doi: 10.1016/S0960-0779(00)00275-7.

[5]

A. d'OnofrioF. GattiP. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour immune system interaction, Math. Comput. Model., 51 (2010), 572-591.  doi: 10.1016/j.mcm.2009.11.005.

[6]

Y. DongG. HuangR. Miyazaki and Y. Takeuchi, Dynamics in a tumor immune system with time delays, Appl. Math. Comput., 252 (2015), 99-113.  doi: 10.1016/j.amc.2014.11.096.

[7]

M. Galach, Dynamics of the tumor-immune system competition–the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406. 

[8]

J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.

[9]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

W. M. HirschH. Hanisch and J.-P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.  doi: 10.1002/cpa.3160380607.

[11]

P. JohansenT. StorniL. Rettig and et al., Antigen kinetics determines immune reactivity, Proc. Natl. Acad. Sci., 105 (2008), 5189-5194.  doi: 10.1073/pnas.0706296105.

[12]

S. Khajanchi and S. Banerjee, Stability and bifurcation ananlysis of delay induced tumor immune interactional model, Appl. Math. Comput., 248 (2014), 652-671.  doi: 10.1016/j.amc.2014.10.009.

[13]

S. Khajanchi and J. J. Nieto, Mathematical modeling of tumor-immune competitive system, considering the role of time delay, Appl. Math. Comput., 340 (2019), 180-205.  doi: 10.1016/j.amc.2018.08.018.

[14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. 
[15]

V. A. KuznetsovI. A. MakalkinM. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors:Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.  doi: 10.1007/BF02460644.

[16]

N. Nishida and M. Kudo, Immunological microenvironment of hepatocellular carcinoma and its clinical implication, Oncology, 92 (2017), 40-49.  doi: 10.1159/000451015.

[17]

J. PrietoI. Melero and B. Sangro, Immunological landscape and immunotherapy of hepatocellular carcinoma, Nat. Rev. Gastroenterol. Hepatol., 12 (2015), 681-700.  doi: 10.1038/nrgastro.2015.173.

[18]

F. A. RihanM. SafanM. A. Abdeen and D. A. Rahman, Qualitative and computational analysis of a mathematical model for tumor-immune interactions, Journal of Applied Mathematics, 2012 (2012), 1-19.  doi: 10.1155/2012/475720.

[19]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[20]

M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol., 47 (2003), 270–294. doi: 10.1007/s00285-003-0211-0.

[21]

R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate, Nonlinear Dyn., 61 (2010), 229-239.  doi: 10.1007/s11071-009-9644-3.

[22]

R. Yafia, Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response, Nonl. Anal.: Real World Appl., 8 (2007), 1359-1369.  doi: 10.1016/j.nonrwa.2006.08.003.

[23]

R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math., 67 (2007), 1693-1703.  doi: 10.1137/060657947.

[24]

R. Yafia, Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response, Discrete Dynamics in Nature and Society, 2006 (2006), Art. ID 95296, 9 pp. doi: 10.1155/DDNS/2006/95296.

[25]

R. Yafia, Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response, Discrete Dynamics in Nature and Society, 2006 (2006), Art. ID 58463, 13 pp. doi: 10.1155/DDNS/2006/58463.

show all references

References:
[1]

P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dyn. Syst., 12 (2013), 1847-1888.  doi: 10.1137/120887898.

[2]

P. Bi and H. Xiao, Hopf bifurcation for tumor-immune competition systems with delay, Electron. J. Diff. Equ., 2014 (2014), 1-13. 

[3]

P. Bi and H. Xiao, Bifurcations of tumor-immune competition systems with delay, Abstract and Applied Analysis, 2014 (2014), 1-12.  doi: 10.1155/2014/723159.

[4]

N. Buric and T. Dragana, Dynamics of delay differential equations modeling immunology of tumor growth, Chaos Solitons Fractals, 13 (2002), 645-655.  doi: 10.1016/S0960-0779(00)00275-7.

[5]

A. d'OnofrioF. GattiP. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour immune system interaction, Math. Comput. Model., 51 (2010), 572-591.  doi: 10.1016/j.mcm.2009.11.005.

[6]

Y. DongG. HuangR. Miyazaki and Y. Takeuchi, Dynamics in a tumor immune system with time delays, Appl. Math. Comput., 252 (2015), 99-113.  doi: 10.1016/j.amc.2014.11.096.

[7]

M. Galach, Dynamics of the tumor-immune system competition–the effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406. 

[8]

J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.

[9]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[10]

W. M. HirschH. Hanisch and J.-P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.  doi: 10.1002/cpa.3160380607.

[11]

P. JohansenT. StorniL. Rettig and et al., Antigen kinetics determines immune reactivity, Proc. Natl. Acad. Sci., 105 (2008), 5189-5194.  doi: 10.1073/pnas.0706296105.

[12]

S. Khajanchi and S. Banerjee, Stability and bifurcation ananlysis of delay induced tumor immune interactional model, Appl. Math. Comput., 248 (2014), 652-671.  doi: 10.1016/j.amc.2014.10.009.

[13]

S. Khajanchi and J. J. Nieto, Mathematical modeling of tumor-immune competitive system, considering the role of time delay, Appl. Math. Comput., 340 (2019), 180-205.  doi: 10.1016/j.amc.2018.08.018.

[14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. 
[15]

V. A. KuznetsovI. A. MakalkinM. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors:Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.  doi: 10.1007/BF02460644.

[16]

N. Nishida and M. Kudo, Immunological microenvironment of hepatocellular carcinoma and its clinical implication, Oncology, 92 (2017), 40-49.  doi: 10.1159/000451015.

[17]

J. PrietoI. Melero and B. Sangro, Immunological landscape and immunotherapy of hepatocellular carcinoma, Nat. Rev. Gastroenterol. Hepatol., 12 (2015), 681-700.  doi: 10.1038/nrgastro.2015.173.

[18]

F. A. RihanM. SafanM. A. Abdeen and D. A. Rahman, Qualitative and computational analysis of a mathematical model for tumor-immune interactions, Journal of Applied Mathematics, 2012 (2012), 1-19.  doi: 10.1155/2012/475720.

[19]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[20]

M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol., 47 (2003), 270–294. doi: 10.1007/s00285-003-0211-0.

[21]

R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate, Nonlinear Dyn., 61 (2010), 229-239.  doi: 10.1007/s11071-009-9644-3.

[22]

R. Yafia, Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response, Nonl. Anal.: Real World Appl., 8 (2007), 1359-1369.  doi: 10.1016/j.nonrwa.2006.08.003.

[23]

R. Yafia, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J. Appl. Math., 67 (2007), 1693-1703.  doi: 10.1137/060657947.

[24]

R. Yafia, Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response, Discrete Dynamics in Nature and Society, 2006 (2006), Art. ID 95296, 9 pp. doi: 10.1155/DDNS/2006/95296.

[25]

R. Yafia, Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response, Discrete Dynamics in Nature and Society, 2006 (2006), Art. ID 58463, 13 pp. doi: 10.1155/DDNS/2006/58463.

Figure 1.  The stable periodic orbit of model (6) as $ {\mu} K>\eta $ and $ n{\sigma}<r\eta $. Here, $ r = 2.5, K = 2, n = 0.8, {\sigma} = 0.5, {\mu} = 4 $, $ \eta = 1.5 $ and $ {\tau} = 0.3>0.1943 = {\tau}_0. $ Model (6) has a unique tumor equilibrium $ P^*(0.3272, 2.6138) $, and a stable periodic orbit
Figure 2.  The bifurcation diagram of model (6) with respect to the time delay $ {\tau} $ when $ {\mu} K> \eta $ and $ n{\sigma}<r\eta $. Here, except for the time delay $ {\tau} $, the values of other parameters are the same as those for Figure 1
Figure 3.  The trajectories of $ T = T(t) $ and $ E = E(t) $ of model (6) corresponding to three delays $ {\tau} = 1, 5, 10 $ in case of $ 0<{\mu}<\frac{\eta}{K} $ and $ n{\sigma}<r\eta $, respectively. Other parameter values are $ r = 2.5, K = 2, n = 0.8, {\sigma} = 0.5, {\mu} = 0.4 $ and $ \eta = 1.5 $
Figure 4.  The trajectories of model (6) with two delays $ {\tau} = 0.3, 2.0 $ in Case C4 of Table 1, where the left column is for $ {\tau} = 0.3 $, the right column is for $ {\tau} = 2.0 $, and the initial conditions of model (6) are the same in the two columns. The first row is the orbits of model (6) in the $ T $-$ E $ plane, the second row is the trajectories of $ T = T(t) $, and the third row is those of $ E = E(t) $
Figure 5.  The graph of function $ {\tau}_0 = {\tau}_0({\mu}) $ defined by (16) with $ {\mu} >\frac{\eta}{K} $. Here, except for $ {\mu} $ and $ {\tau} $, the values of other parameters are the same as those for Figure 1, and the regions $ D_1 $ and $ D_2 $ correspond to the stable and unstable cases of the tumor equilibrium $ P^* $, respectively
Table 1.  The existence and the stability of all feasible equilibria of model (2) without time delay. Here, US stands for unstable, GAS-1 globally asymptotically stable in the first quadrant of the $ T $-$ E $ plane, GAS-2 globally asymptotically stable on the set $ \{(T, E): T\ge 0, E>0\} $, LAS locally asymptotically stable, SN saddle-node
Case Conditions $ P_0 $ $ P^* $ $ P_* $ $ P^*_* $
C1 $ {\mu}>0, n{\sigma}<r\eta $ US GAS-1
C2 $ {\mu}<0, \eta+ {\mu} K \ge 0, n{\sigma}<r\eta $ US GAS-1
C3 $ {\mu}<0, \eta+ {\mu} K< 0, n{\sigma}\le r\eta $ US GAS-1
C4 $ \begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0, \\ r\eta<n{\sigma}<-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array} $ LAS LAS US
C5 $ \begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0,\\ n{\sigma}=-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array} $ LAS SN
C6 $ {\mu}>0, n{\sigma}\ge r\eta $ GAS-2
C7 $ \begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0, \\n{\sigma}>-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array} $ GAS-2
Case Conditions $ P_0 $ $ P^* $ $ P_* $ $ P^*_* $
C1 $ {\mu}>0, n{\sigma}<r\eta $ US GAS-1
C2 $ {\mu}<0, \eta+ {\mu} K \ge 0, n{\sigma}<r\eta $ US GAS-1
C3 $ {\mu}<0, \eta+ {\mu} K< 0, n{\sigma}\le r\eta $ US GAS-1
C4 $ \begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0, \\ r\eta<n{\sigma}<-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array} $ LAS LAS US
C5 $ \begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0,\\ n{\sigma}=-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array} $ LAS SN
C6 $ {\mu}>0, n{\sigma}\ge r\eta $ GAS-2
C7 $ \begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0, \\n{\sigma}>-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array} $ GAS-2
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