Spreading Properties for SIR Models on Homogeneous Trees

Epidemic Spreading on Complex Networks as Front Propagation into an Unstable State

We study epidemic arrival times in meta-population disease models through the lens of front propagation into unstable states. We demonstrate that several features of invasion fronts in the PDE context are also relevant to the network case. We show that the susceptible-infected-recovered model on a network is linearly determined in the sense that the arrival times in the nonlinear system are approximated by the arrival times of the instability in the system linearized near the disease-free state. Arrival time predictions are extended to general compartmental models with a susceptible-exposed-infected-recovered model as the primary example. We then study a recent model of social epidemics where higher-order interactions lead to faster invasion speeds. For these pushed fronts, we compute corrections to the estimated arrival time in this case. Finally, we show how inhomogeneities in local infection rates lead to faster average arrival times.

From random walks on networks to nonlinear diffusion

Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on the collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equation is, in general, density dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to the network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium-type equation on networks, which shows similar properties to its continuum equivalent. For this equation, self-similar solutions on a lattice and on homogeneous trees can be found, showing finite speed of propagation in contrast to commonly used linear diffusion equations. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.