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

Abstract

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.

A General Approach for the Modelling of Negative Feedback Physiological Control Systems

“…It is readily seen that, in (20), L = x plays the role of the activation species (i.e., the variation in L triggers the reflex arc), while f e = y is the inhibition variable (i.e., the variation in f e brings L back close to its starting value); with this notation, the model ( 20 (22b)…”

“…Most of the physiological systems, taken at their own different scales, prove to have analogous and overlapping control dynamics. It is possible to consider, as examples, at the nano-scale, the genetic circuits for biomolecular process control, described in [8][9][10][11][12][13][14] and in the relevant books [15][16][17] with their bibliographies, and at the micro-and milli-scale, as the cell population dynamics involved, for instance, in the generation and growth of tumours [18][19][20][21][22] and the regulation of sleep-wake cycles [23]. Finally, examples at the macro-scale level are the control systems for the regulation of cardiovascular processes, neuromuscular activities, respiration dynamics, etc.…”

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

“…In the general case u is vectorial but here for the sake of the notation simplicity we will suppose it scalar. Just to give some examples of activities we can mention: the level of proteins defining the degree of immunogenicity and 'abnormality' of a tumor cell [49] , the level of activation for an immune system effector [49], the 'level of effectiveness in performing the job that a species is expected to do' in a multi species environment [50], the pair (opinion, connectivity) in models of opinion formation in social networks [51], the viral load for a subject during an epidemics [20]. Of course, a very important class of activities is the couple position-velocity (x, v) [52].…”

A kinetic model with time‐dependent proliferative/destructive rates

“…For instance, in an ecological system, the reproduction rate depends on the seasons [28], then an independent‐time shape for the nonconservative parameters would not be realistic in that case. Also, when dealing with the immune system interactions, both the proliferation and the destruction of cells are relevant effects in the description that should not be neglected [29, 30]. Therefore, this new nonconservative framework can be applied in different contexts, beyond those already mentioned; these generality and versatility of the model represent one of the key highlights of the current work.…”