| Citation: |
Haichao Xiong, Jun Zhang, Weinian Zhang. USE SLOW-SPREAD OF ONCOLYTIC VIRUS TO DEPRESS EXPONENTIAL GROWTH OF TUMOR CELLS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1158-1185. doi: 10.11948/20220168
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USE SLOW-SPREAD OF ONCOLYTIC VIRUS TO DEPRESS EXPONENTIAL GROWTH OF TUMOR CELLS
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Abstract
In this paper we analyze an ODE model for oncolytic dynamics of exponential growth of tumor cells with slow-spread of virus, which was modeled by Komarova and Wodarz but not discussed yet. The involved four parameters render finding equilibria to be a difficult problem of algebraic varieties. We discuss resultants of polynomials to give complete conditions for distribution and qualitative properties of equilibria. We prove that the degenerate equilibrium is either a saddle-node or a cusp, which is of codimension infinity. Moreover, we prove that the equilibrium of center type is either a rough center or a weak center of order 1. Furthermore, analyzing equilibria at infinity, showing existence of a homoclinic orbit and giving nonexistence of limit cycles, we exhibit global phase portraits, which suggest strategies of tumor control.
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Keywords:
- Tumor control /
- cusp /
- weak center /
- homoclinic orbit /
- resultant
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References
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Haichao Xiong, Jun Zhang, Weinian Zhang. USE SLOW-SPREAD OF ONCOLYTIC VIRUS TO DEPRESS EXPONENTIAL GROWTH OF TUMOR CELLS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1158-1185. doi: 10.11948/20220168
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Figure 1. (a) Phase portrait of system (5.1) near
$ O_1 $ . (b) Phase portrait of system (5.2) near the$ \omega $ -axis. (c) Phase portrait of system (5.3) near$ \widetilde{O}^*_1 $ . -
Figure 2. (a) Phase portrait of system (5.4) near
$ O_2 $ . (b) Phase portrait of system (5.5) near the$ \eta $ -axis. (c) Phase portrait of system (5.6) near$ \widetilde{O}^*_2 $ . - Figure 3. Global phase portraits of system (1.2).
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Figure 4. Simulations of system (1.2). (a) No cycles for
$ (\beta,a,\varepsilon_1,\varepsilon_2)=(26/5,16/9,1/2,1/4) $ . (b) No cycles for$ (\beta,a,\varepsilon_1,\varepsilon_2)=(315/100,4/9,1/2,1) $ . (c) No cycles for$ (\beta,a,\varepsilon_1,\varepsilon_2)=(14/5,25/36,1/2,1/16) $ . -
Figure 5. Simulations of system (1.2). (a) One saddle and one stable focus when
$ (\beta,a,\varepsilon_1,\varepsilon_2)=(13/2,21/20,1,1) $ . (b) Coexistence of homoclinic orbit and center when$ (\beta,a,\varepsilon_1,\varepsilon_2)=(6,1,1,1) $ .