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# Nonlinear dynamics in tumor-immune system interaction models with delays

In memory of my Ph.D. thesis supervisor Professor Herbert I. Freedman (1940 - 2017)

Research was partially supported by National Science Foundation grant (DMS-1853622)

• In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

Mathematics Subject Classification: Primary: 34K18, 92C37; Secondary: 37N25.

 Citation:

• Figure 1.  Three processes in cancer immunoediting: (a) Elimination corresponds to immunosurveillance; (b) Equilibrium represents the process by which the immune system iteratively selects and/or promotes the generation of tumor cell variants with increasing capacities to survive immune attack; (c) Escape is the process wherein the immunologically sculpted tumor expands in an uncontrolled manner in the immunocompetent host. Adapted from Dunn [44]

Figure 2.  Scheme of the essential mechanisms of interaction between the tumor cells and immune effector cells

Figure 3.  (a) Solution trajectories converge to the stable equilibrium $P_2 = (92.1911, 1.3344);$ (b) Periodic solutions bifurcated from the positive equilibrium when $\tau = 2.0>\tau_0$

Figure 4.  The stability regions of the equilibria (a) $P^1_2(8.18971, 1.6092)$ and (b) $P^3_2(447.134, 0.17298)$ with blue dashed lines in the $(B_2, \tau)$-parameter plane

Figure 5.  The bifurcation diagram for system (40). (a) $\alpha_2>0.$ (b) $\alpha_2<0.$

Figure 6.  Phase portraits for the case VIa in Table 7.5.2 of [58]: (a) Bifurcation diagram in $(\mu_1,\mu_2)$; (b) Phase portraits of (81)

Figure 7.  The bifurcation diagram of (81) on the $(\mu_1, \mu_2)$-plane

Figure 8.  (a) The periodic solution $(x(t), y(t))$ bifurcated from the microscopic equilibrium $(8.18971, 1.6092)$ with $\tau = 0.333814.$ (b) The corresponding solution $x(t)$ in terms of time $t.$ (c) The periodic solution $(x(t), y(t))$ bifurcated from the macroscopic equilibrium $(447.134, 0.172977)$ as $\tau = 2.08803.$ (d) The corresponding solution $x(t)$ in terms of time $t.$

Figure 9.  Graphs of $f(T)$ for three different parameter sets: (a) $u = v = 1$; (b) $u = v = 5>1$; (c) $u = \frac{1}{2}<1<v = 2.$

Figure 10.  The phase portraits of system (87): (a) with parameter set (I); (b) with parameter set (II)

Figure 11.  (a) The converging solutions of system (87) in terms of $t$ when $\tau = 4, \delta = 20;$ (b) The solution trajectories of system (87) spiral toward the positive equilibrium in the $(T, E)$-plane when $\tau = 4, \delta = 20$; (c) The periodic solutions of system (87) in terms of $t$ when $\tau = \tau_0$; (d) The periodic trajectories of system (87) in the $(T, E)$-plane

Figure 12.  Stability diagram of system (87) on the ($\tau, \delta$)-delay parameter space

Figure 13.  (a) The locations of $f(\omega)$ and $h(\omega)$ when $a = 0.1, b = 0.1$. (a) $0.2 < \omega < 1.5;$ (b) $0.26 < \omega < 0.3.$

Figure 14.  (a) The stable solutions of system (87) when $\tau = 3.8, \delta = 13.5,\ \Delta = 2;$ (b) The solution trajectory of system (87) converges to the positive equilibrium in the $(T,E)$ plane; (c) The periodic solutions $T(t)$ and $E(t)$ of system (87) in terms of $t$ when $\tau = \tau_0 = 3.8641,\ \delta = \delta_0 = 15.1378,\ \Delta = \Delta_0 = 2.0592$; (d) The periodic trajectories of system (87) in the $(T,E)$ plane

Figure 15.  The stability diagram of the positive equilibrium for system (87) in the ($\tau, \delta,\ \Delta$) parameter space

Figure 16.  (a)(b) The regular periodic oscillations in system (87) with $\tau = 0.5, \delta = 5, \Delta = 8;$ (c)(d) The irregular long periodic oscillations in system (87) with $\tau = 0.5, \delta = 15, \Delta = 8;$ (e)(f) The chaotic solutions in system (87) with $\tau = 0.5, \delta = 50, \Delta = 38.$

Table 1.  Existence and stability chart for the ODE model (Gałach [56])

 Sign of $\zeta$ Conditions $P_0$ $P_1$ $P_2$ $\zeta > 0$ $\alpha \delta< \sigma$ stable – – $\alpha \delta> \sigma$ unstable – stable $\zeta< 0$ $\alpha(\beta\delta-\zeta)^2+4\beta\zeta\sigma<0$ stable – – $\alpha(\beta\delta-\zeta)^2+4\beta\zeta\sigma>0$ $\alpha \delta> \sigma$$\zeta + \beta \delta<0$ unstable – stable $\alpha \delta< \sigma$} stable unstable stable
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