Dynamics of epidemics from cavity master equations: Susceptible-infectious-susceptible models

Susceptible-infected-susceptible process on Erdős-Rényi graphs: Determining the infected fraction

There are many methods to estimate the quasistationary infected fraction of the SIS process on (random) graphs. A challenge is to adequately incorporate correlations, which is especially important in sparse graphs. Methods typically are either significantly biased in sparse graphs, or computationally very demanding already for small network sizes. The former applies to heterogeneous mean field and to the N-intertwined mean field approximation, the latter to most higher order approximations. In this paper we introduce a method to determine the infected fraction in sparse graphs, which we test on Erdős-Rényi graphs. Our method is based on degree pairs, does take into account correlations, and gives accurate estimates. At the same time, computations are very feasible and can easily be done even for large networks.

Spurious self-feedback of mean-field predictions inflates infection curves

The susceptible-infected-recovered (SIR) model and its variants form the foundation of our understanding of the spread of diseases. Here, each agent can be in one of three states (susceptible, infected, or recovered), and transitions between these states follow a stochastic process. The probability of an agent becoming infected depends on the number of its infected neighbors, hence all agents are correlated. The simplest mean-field theory of the same stochastic process, however, assumes that the agents are statistically independent. This leads to a self-feedback effect in the approximation: when an agent infects its neighbors, this infection may subsequently travel back to the original agent at a later time, leading to a self-infection of the agent which is not present in the underlying stochastic process. We here compute the first-order correction to the mean-field assumption from a systematic expansion, called dynamical TAP theory. This correction, which takes fluctuations up to second order in the interaction strength into account, cancels the self-feedback effect, leading to smaller infection rates. The correction significantly improves predictions compared to mean-field theory. In particular, it captures how sparsity dampens the spread of the disease: this indicates that reducing the number of contacts is more effective than predicted by mean-field models. We further apply the expansion to variants of the SIR model, such as the SIRS model, in which the immunity of an individual to the disease wanes over time. We find that up to the second order, the correction terms in the SIR and SIRS model are equivalent, meaning that fluctuations partially cancel the self-feedback effect even when self-feedback is in principle allowed.

Matrix Product Belief Propagation for reweighted stochastic dynamics over graphs

Stochastic processes on graphs can describe a great variety of phenomena ranging from neural activity to epidemic spreading. While many existing methods can accurately describe typical realizations of such processes, computing properties of extremely rare events is a hard task, particularly so in the case of recurrent models, in which variables may return to a previously visited state. Here, we build on the matrix product cavity method, extending it fundamentally in two directions: First, we show how it can be applied to Markov processes biased by arbitrary reweighting factors that concentrate most of the probability mass on rare events. Second, we introduce an efficient scheme to reduce the computational cost of a single node update from exponential to polynomial in the node degree. Two applications are considered: inference of infection probabilities from sparse observations within the SIRS epidemic model and the computation of both typical observables and large deviations of several kinetic Ising models.

Scalable parallel and distributed simulation of an epidemic on a graph

We propose an algorithm to simulate Markovian SIS epidemics with homogeneous rates and pairwise interactions on a fixed undirected graph, assuming a distributed memory model of parallel programming and limited bandwidth. This setup can represent a broad class of simulation tasks with compartmental models. Existing solutions for such tasks are sequential by nature. We provide an innovative solution that makes trade-offs between statistical faithfulness and parallelism possible. We offer an implementation of the algorithm in the form of pseudocode in the Appendix. Also, we analyze its algorithmic complexity and its induced dynamical system. Finally, we design experiments to show its scalability and faithfulness. In our experiments, we discover that graph structures that admit good partitioning schemes, such as the ones with clear community structures, together with the correct application of a graph partitioning method, can lead to better scalability and faithfulness. We believe this algorithm offers a way of scaling out, allowing researchers to run simulation tasks at a scale that was not accessible before. Furthermore, we believe this algorithm lays a solid foundation for extensions to more advanced epidemic simulations and graph dynamics in other fields.

Inference in conditioned dynamics through causality restoration

Estimating observables from conditioned dynamics is typically computationally hard. While obtaining independent samples efficiently from unconditioned dynamics is usually feasible, most of them do not satisfy the imposed conditions and must be discarded. On the other hand, conditioning breaks the causal properties of the dynamics, which ultimately renders the sampling of the conditioned dynamics non-trivial and inefficient. In this work, a Causal Variational Approach is proposed, as an approximate method to generate independent samples from a conditioned distribution. The procedure relies on learning the parameters of a generalized dynamical model that optimally describes the conditioned distribution in a variational sense. The outcome is an effective and unconditioned dynamical model from which one can trivially obtain independent samples, effectively restoring the causality of the conditioned dynamics. The consequences are twofold: the method allows one to efficiently compute observables from the conditioned dynamics by averaging over independent samples; moreover, it provides an effective unconditioned distribution that is easy to interpret. This approximation can be applied virtually to any dynamics. The application of the method to epidemic inference is discussed in detail. The results of direct comparison with state-of-the-art inference methods, including the soft-margin approach and mean-field methods, are promising.