Timothy Earl Figueroa Lapuz

arXiv2026

Escape, Oscillatory Relapse or Elimination: A Geometric Blow-up Analysis of a Tumour Model with Singular Bifurcations

The five-compartment tumour-immune model introduced by Kuznetsov et al. captures critical clinical phenomena, including tumour dormancy, immune evasion (`sneaking through'), and immunostimulation. Recent studies by Osojnik et al. highlighted the model's excitable and oscillatory dynamics, providing a mechanism for tumour relapse. In this paper, we recast these findings through the lens of geometric singular perturbation theory (GSPT) and the blow-up method, establishing a formal mathematical foundation for these phenomena. Because the model features a multiple-time-scale structure with dynamics separated by up to three orders of magnitude, standard Tikhonov-Fenichel theory is insufficient. To resolve this, we employ the parametrisation method to systematically compute higher-order approximations of the slow vector fields. We introduce a methodological novelty by demonstrating that the parametrisation method is indispensable within the geometric blow-up analysis. Specifically, we show that higher-order correction terms to the local centre manifolds are required to match the vector field on the corresponding slow manifold. Using these combined tools, we rigorously prove the existence of stable relaxation oscillations and transient large excursions. We uncover a rich collection of singular bifurcations, including a saddle-node on invariant circle and two distinct singular Andronov-Hopf bifurcations that drive complete and incomplete canard explosions. At the highly degenerate origin, our blow-up analysis reveals the spatial collision of a singular transcritical bifurcation and a nilpotent pseudo-singularity. We conjecture that this collision forms a novel singular transcritical Bogdanov-Takens organising centre. Ultimately, these geometric findings yield a precise bifurcation diagram in the parameter space of effector cell supply and death rates.

math.DS↗

arXiv2026

Coordinate-independent model reductions of chemical reaction networks based on geometric singular perturbation theory

The quasi-steady-state approximation (QSSA) is a standard technique for reducing the complexity of chemical reaction networks (CRNs). The validity of any QSSA-based model is restricted to specific parameter regimes. Selecting the appropriate reduction is not always straightforward. At times, QSSAs are misused outside of their validity regions and, even when a particular QSSA is considered valid in a given parameter regime, other QSSAs may be simultaneously valid, creating ambiguity. Here, we employ a more powerful alternative: a constructive model reduction framework based on coordinate-independent geometric singular perturbation theory (ci-GSPT) and the parametrization method. A key advantage of this approach is its ability to derive reduced models independent of a clear timescale separation in the variables for a specific parameter configuration. We demonstrate our approach on two benchmark systems. For the Michaelis-Menten (MM) reaction, we show that the framework provides a systematic approach by exploring parameter configurations across three orders of magnitude: asymptotically large, small, and `order one'. A consequence of this systematic analysis is a geometric classification that categorizes the resulting model reductions and provides a point of comparison between our approach and common QSSA variants in the literature. For the more complex Kim-Forger model, we show that this approach successfully produces a reduction without the need for a coordinate transformation, showcasing its applicability to larger systems.

math.DS↗