Multi-strain spreading dynamics under arbitrary transmission kernels

Compartmental models of epidemic dynamics have long described the propagation of a single, immutable transmissible state through a population via pairwise contact, and multi-strain generalizations have extended this framework to incorporate mutation, competition, and cross-immunity. Here we study a minimal generalization with no sink states or feedback, in which transmission acts through an arbitrary column-stochastic kernel $Q$ on a finite set of strains, encoding mutation during transmission with no further structural assumptions. We derive the mean-field approximation for the well-mixed regime and show that it admits an exact closed-form solution for any $Q$, expressible as a single matrix exponential applied to the initial condition. A spectral decomposition of this solution reveals that the location of the long-time attractor and the rate of approach are governed by the eigenstructure of $Q$. We extend the analysis to structured populations via a pairwise mean-field approximation on regular contact networks, and validate both approximations against stochastic simulations. The framework provides an entry into the analysis of dynamical systems in which mutation and transmission occur on the same time scale, drawing parallels to the propagation of discrete signals through populations under noisy communication.