# American Institute of Mathematical Sciences

August  2018, 23(6): 2393-2414. doi: 10.3934/dcdsb.2018060

## Qualitative analysis of kinetic-based models for tumor-immune system interaction

 1 BCAM -Basque Center for Applied Mathematics, azarredo, 14, E-48009 Bilbao, Basque Country -Spain 2 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

* Corresponding author: Maria Groppi

Received  April 2017 Revised  September 2017 Published  February 2018

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting components are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation.

Citation: Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060
##### References:

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##### References:
Phase portrait of system (15) for $A>1/X$
Comparison of the time evolution of solutions to system (14) and (15) (thickest curves) for $G = 5$ (left) and $G = 50$ (right)
Nullcline surfaces of system (15)
Comparison between the trajectories of the nonautonomous system (18) (solid curves) and of the limit system (19) (dashed curves) respectively; the dotted line represents the intersection of the tangent plane to the stable manifold in $E_2$ with the plane $Y_3 = 0$, that can be considered an approximation of the right boundary of $R$; the curve $\gamma$ is dash-dotted
Solutions for increasing values of D
Qualitative bifurcation diagram versus $A$ for $C^*<BG/F+G/(FX)$ : forward bifurcation of equilibria (parameter values used: $B = 1, C^* = 4.5, F = 1, G = 1, X = 1/5$)
Qualitative bifurcation diagram versus $A$ for $C^*>BG/F+G/(FX)$ : backward bifurcation of equilibria (parameter values used: $B = 1, C^* = 9, F = 1, G = 1, X = 1/5$)
Phase portrait, representative of the case $C^*>BG/F+G/(FX)$ and $A^*<A<1/X$
Comparison of the time evolution of solutions to system (14) and (21) (thickest curves) for $D = 1$ (left) and $D = 10$ (right), with $D/E = 1.5$
Threshold values $D^*$ versus initial data $Y_{10}$
 $Y_{10}$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$ $0.7$ $0.8$ $1.0$ $2.0$ $D^*$ $1.43$ $2.7$ $4.08$ $5.52$ $7.02$ $8.54$ $10.1$ $13.26$ $29.67$
 $Y_{10}$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$ $0.7$ $0.8$ $1.0$ $2.0$ $D^*$ $1.43$ $2.7$ $4.08$ $5.52$ $7.02$ $8.54$ $10.1$ $13.26$ $29.67$
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