# American Institute of Mathematical Sciences

## Diffusion modeling of tumor-CD4$^+$-cytokine interactions with treatments: asymptotic behavior and stationary patterns

 1 School of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi 710121, China 2 Department of Gastroenterology, Xi'an Honghui Hospital, Xi'an, Shaanxi 710000, China

* Corresponding author: Wenbin Yang

Received  October 2020 Revised  February 2021 Published  March 2021

Fund Project: The first author is supported by NSF of China grant 12001425

In this work, we consider a diffusive tumor-CD4$^+$-cytokine interactions model with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria, including the system persistence and the stability conditions. We also give the existence of nonconstant positive steady states (i.e., a stationary pattern), which indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems, linearization method, the method of energy integral and the Leray-Schauder degree.

Citation: Wenbin Yang, Yujing Gao, Xiaojuan Wang. Diffusion modeling of tumor-CD4$^+$-cytokine interactions with treatments: asymptotic behavior and stationary patterns. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021090
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar [2] L. Anderson, S. Jang and J. L. Yu, Qualitative behavior of systems of tumor-${\rm{CD}}4^+$-cytokine interactions with treatments, Math. Methods Appl. Sci., 38 (2015), 4330-4344.  doi: 10.1002/mma.3370.  Google Scholar [3] F. Ansarizadeh, M. Singh and D. Richards, Modelling of tumor cells regression in response to chemotherapeutic treatment, Appl. Math. Modelling, 48 (2017), 96-112.  doi: 10.1016/j.apm.2017.03.045.  Google Scholar [4] M. A. Brown and J. Hural, Functions of IL-4 and control of its expression, Critical Reviews in Immunology, 17 (1997), 1-32.  doi: 10.1615/CritRevImmunol.v17.i1.10.  Google Scholar [5] F. Dai and B. Liu, Optimal control problem for a general reaction-diffusion tumor-immune system with chemotherapy, J. Franklin Inst., 358 (2021), 448-473.  doi: 10.1016/j.jfranklin.2020.10.032.  Google Scholar [6] A. D'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Modelling, 47 (2008), 614-637.  doi: 10.1016/j.mcm.2007.02.032.  Google Scholar [7] A. D'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena, 208 (2005), 220-235.  doi: 10.1016/j.physd.2005.06.032.  Google Scholar [8] A. Ducrot and J. Guo, Asymptotic behavior of solutions to a class of diffusive predator-prey systems, J. Evol. Equ., 18 (2018), 755-775.  doi: 10.1007/s00028-017-0418-y.  Google Scholar [9] S. Habib, M. P. Carmen and S. D. Thomas, Complex dynamics of tumors: Modeling an emerging brain tumor system with coupled reaction-diffusion equations, Physica A: Statistical Mechanics and its Applications, 327 (2003), 501-524.  doi: 10.1016/S0378-4371(03)00391-1.  Google Scholar [10] L. E. Harrington, R. D. Hatton, P. R. Mangan, H. Turner, T. L. Murphy, K. M. Murphy and C. T. Weaver, Interleukin 17-producing cd4+ effector t cells develop via a lineage distinct from the t helper type 1 and 2 lineages, Nature Immunology, 6 (2005), 1123-1132.  doi: 10.1038/ni1254.  Google Scholar [11] C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [12] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar [13] J. Manimaran and L. Shangerganesh, Solvability and numerical simulations for tumor invasion model with nonlinear diffusion, Computational and Mathematical Methods, 2 (2020), e1068, 20pp. doi: 10.1002/cmm4.1068.  Google Scholar [14] C.-V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar [15] W. -E. Paul, Fundamental Immunology, 6$^nd$ edition, Lippincott Williams & Wilkins, Philadelphia, 2008. Google Scholar [16] W. Raymond and M.-D. Ruddon, Cancer Biology, 4\begin{document}$^nd$\end{document} edition, Oxford University Press, Oxford, 2007.   Google Scholar [17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^nd$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [18] J. P. Tripathi, S. Abbas and M. Thakur, Dynamical analysis of a prey-predator model with Beddington-Deangelis type function response incorporating a prey refuge, Nonlinear Dyn., 80 (2015), 177-196.  doi: 10.1007/s11071-014-1859-2.  Google Scholar [19] W. Yang, Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior, Math. Methods Appl. Sci., 43 (2020), 5629-5644.  doi: 10.1002/mma.6301.  Google Scholar [20] L. Yang and S. Zhong, Dynamics of a diffusive predator-prey model with modified Leslie-Gower schemes and additive allee effect, Comput. Appl. Math., 34 (2015), 671-690.  doi: 10.1007/s40314-014-0131-1.  Google Scholar [21] R. Zeng, Qualitative analysis of a strongly coupled predator-prey system with modified Holling-Tnner functional response, Bound. Value Probl., 2018 (2018), Paper No. 98, 21 pp. doi: 10.1186/s13661-018-1015-x.  Google Scholar [22] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, 1986.  Google Scholar

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##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar [2] L. Anderson, S. Jang and J. L. Yu, Qualitative behavior of systems of tumor-${\rm{CD}}4^+$-cytokine interactions with treatments, Math. Methods Appl. Sci., 38 (2015), 4330-4344.  doi: 10.1002/mma.3370.  Google Scholar [3] F. Ansarizadeh, M. Singh and D. Richards, Modelling of tumor cells regression in response to chemotherapeutic treatment, Appl. Math. Modelling, 48 (2017), 96-112.  doi: 10.1016/j.apm.2017.03.045.  Google Scholar [4] M. A. Brown and J. Hural, Functions of IL-4 and control of its expression, Critical Reviews in Immunology, 17 (1997), 1-32.  doi: 10.1615/CritRevImmunol.v17.i1.10.  Google Scholar [5] F. Dai and B. Liu, Optimal control problem for a general reaction-diffusion tumor-immune system with chemotherapy, J. Franklin Inst., 358 (2021), 448-473.  doi: 10.1016/j.jfranklin.2020.10.032.  Google Scholar [6] A. D'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Modelling, 47 (2008), 614-637.  doi: 10.1016/j.mcm.2007.02.032.  Google Scholar [7] A. D'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena, 208 (2005), 220-235.  doi: 10.1016/j.physd.2005.06.032.  Google Scholar [8] A. Ducrot and J. Guo, Asymptotic behavior of solutions to a class of diffusive predator-prey systems, J. Evol. Equ., 18 (2018), 755-775.  doi: 10.1007/s00028-017-0418-y.  Google Scholar [9] S. Habib, M. P. Carmen and S. D. Thomas, Complex dynamics of tumors: Modeling an emerging brain tumor system with coupled reaction-diffusion equations, Physica A: Statistical Mechanics and its Applications, 327 (2003), 501-524.  doi: 10.1016/S0378-4371(03)00391-1.  Google Scholar [10] L. E. Harrington, R. D. Hatton, P. R. Mangan, H. Turner, T. L. Murphy, K. M. Murphy and C. T. Weaver, Interleukin 17-producing cd4+ effector t cells develop via a lineage distinct from the t helper type 1 and 2 lineages, Nature Immunology, 6 (2005), 1123-1132.  doi: 10.1038/ni1254.  Google Scholar [11] C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [12] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar [13] J. Manimaran and L. Shangerganesh, Solvability and numerical simulations for tumor invasion model with nonlinear diffusion, Computational and Mathematical Methods, 2 (2020), e1068, 20pp. doi: 10.1002/cmm4.1068.  Google Scholar [14] C.-V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar [15] W. -E. Paul, Fundamental Immunology, 6$^nd$ edition, Lippincott Williams & Wilkins, Philadelphia, 2008. Google Scholar [16] W. Raymond and M.-D. Ruddon, Cancer Biology, 4\begin{document}$^nd$\end{document} edition, Oxford University Press, Oxford, 2007.   Google Scholar [17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^nd$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [18] J. P. Tripathi, S. Abbas and M. Thakur, Dynamical analysis of a prey-predator model with Beddington-Deangelis type function response incorporating a prey refuge, Nonlinear Dyn., 80 (2015), 177-196.  doi: 10.1007/s11071-014-1859-2.  Google Scholar [19] W. Yang, Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior, Math. Methods Appl. Sci., 43 (2020), 5629-5644.  doi: 10.1002/mma.6301.  Google Scholar [20] L. Yang and S. Zhong, Dynamics of a diffusive predator-prey model with modified Leslie-Gower schemes and additive allee effect, Comput. Appl. Math., 34 (2015), 671-690.  doi: 10.1007/s40314-014-0131-1.  Google Scholar [21] R. Zeng, Qualitative analysis of a strongly coupled predator-prey system with modified Holling-Tnner functional response, Bound. Value Probl., 2018 (2018), Paper No. 98, 21 pp. doi: 10.1186/s13661-018-1015-x.  Google Scholar [22] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, 1986.  Google Scholar
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