Spiral waves and turbulence in mathematical models of oncolytic virotherapy induced by Hopf bifurcation

Oncolytic virotherapy is a promising targeted cancer treatment that employs viruses, which selectively infect tumor cells. Although its clinical efficiency has remained limited and it is often used in conjunction with other therapies, advances in genetic engineering have produced stronger and more selective viral strains, prompting continued interest in their dynamics. In particular, previous studies have noted that viruses with sufficiently high replication rates can induce oscillations reminiscent of predator-prey systems. Here, we extend this analysis to the spatial domain by starting from an established tumor-virus reaction-diffusion model, performing a center-manifold reduction that incorporates nonlinear terms to derive a complex Ginzburg-Landau amplitude equation, and estimating its parameters directly from the original kinetics. This reduced normal form equation explains the emergence of experimentally observed patterns—such as hollow rings and target waves—and shows that, at longer timescales, these patterns naturally evolve toward spiral waves and a turbulent regime. Our work provides a mechanistic link between the kinetic Hopf bifurcation and the rich spatiotemporal structures observed in oncolytic virotherapy models, suggesting that these patterns are not numerical artifacts but an intrinsic feature of the system.