Message passing approach for general epidemic models

Susceptible-infectious-recovered models revisited: From the individual level to the population level

A Susceptible Infective Recovered (SIR) model is usually unable to mimic the actual epidemiological system exactly. The reasons for this inaccuracy include observation errors and model discrepancies due to assumptions and simplifications made by the SIR model. Hence, this work proposes calibration and prediction methods for the SIR model with a one-time reported number of infected cases. Given that the observation errors of the reported data are assumed to be heteroscedastic, we propose two predictors to predict the actual epidemiological system by modeling the model discrepancy through a Gaussian Process model. One is the calibrated SIR model, and the other one is the discrepancy-corrected predictor, which integrates the calibrated SIR model with the Gaussian Process predictor to solve the model discrepancy. A wild bootstrap method quantifies the two predictors' uncertainty, while two numerical studies assess the performance of the proposed method. The numerical results show that, the proposed predictors outperform the existing ones and the prediction accuracy of the discrepancy-corrected predictor is improved by at least 49.95%.

The classic susceptible-infectious-recovered (SIR) model, has been used extensively to study the dynamical evolution of an infectious disease in a large population. The SIR-susceptible (SIRS) model is an extension of the SIR model to allow modeling imperfect immunity (those who have recovered might become susceptible again). SIR(S) models assume observed counts are “mass balanced.” Here, mass balance means that total count equals the sum of counts of the individual components of the model. However, since the observed counts have errors, we propose a model that assigns the mass balance to the hidden process of a (Bayesian) hierarchical SIRS (HSIRS) model. Another challenge is to capture the stochastic or random nature of an epidemic process in a SIRS. The HSIRS model accomplishes this through modeling the dynamical evolution on a transformed scale. Through simulation, we compare the HSIRS model to the classic SIRS model, a model where it is assumed that the observed counts are mass balanced and the dynamical evolution is deterministic.

Biological systems are remarkably susceptible to relatively small temperature changes. The most obvious example is fever, when a modest rise in body temperature of only few Kelvin has strong effects on our immune system and how it fights pathogens. Another very important example is climate change, when even smaller temperature changes lead to dramatic shifts in ecosystems. Although it is generally accepted that the main effect of an increase in temperature is the acceleration of biochemical reactions according to the Arrhenius equation, it is not clear how it affects large biochemical networks with complicated architectures. For developmental systems such as fly and frog, it has been shown that the system response to temperature deviates in a characteristic manner from the linear Arrhenius plot of single reactions, but a rigorous explanation has not been given yet. Here we use a graph-theoretical interpretation of the mean first-passage times of a biochemical master equation to give a statistical description. We find that in the limit of large system size and if the network has a bias towards a target state, then the Arrhenius plot is generically quadratic, in excellent agreement with numerical simulations for large networks as well as with experimental data for developmental times in fly. We also discuss under which conditions this generic response can be violated, for example for linear chains, which have only one spanning tree.

We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of$N$nodes, there are$3^{N}$configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

In recent analytical work, Biskup et al. (Europhys. Lett., 60 (2002) 21) studied the behaviour of d-dimensional finite-volume liquid-vapour systems at a fixed excess δN of particles above the ambient gas density. By identifying a dimensionless parameter Δ(δN) and a universal constant Δc(d), they showed in the limit of large system sizes that for Δ < Δc the excess is absorbed in the background (“evaporated” system), while for Δ > Δc a droplet of the dense phase occurs (“condensed” system). Also the fraction λΔ of excess particles forming the droplet is given explicitly. Furthermore, they argue that the same is true for solid-gas systems. By making use of the well-known equivalence of the lattice-gas picture with the spin-(1/2) Ising model, we performed Monte Carlo simulations of the Ising model with nearest-neighbour couplings on a square lattice with periodic boundary conditions at fixed magnetisation, corresponding to a fixed particles excess. To test the applicability of the analytical results to much smaller, practically accessible system sizes, we measured the largest minority droplet, corresponding to the solid phase, at various system sizes (L = 40,…,640). Using analytic values for the spontaneous magnetisation m0, the susceptibility χ and the Wulff interfacial free energy density τW for the infinite system, we were able to determine λΔ numerically in very good agreement with the theoretical prediction.

We develop a formalism to describe the discrete-time dynamics of systems containing an arbitrary number of interacting species. The individual-based model, which forms our starting point, is described by a Markov chain, which in the limit of large system sizes is shown to be very well-approximated by a Fokker-Planck-like equation, or equivalently by a set of stochastic difference equations. This formalism is applied to the specific case of two species: one predator species and its prey species. Quasicycles, stochastic cycles sustained and amplified by the demographic noise, previously found in continuous-time predator-prey models are shown to exist, and their behavior predicted from a linear noise analysis is shown to be in very good agreement with simulations. The effects of the noise on other attractors in the corresponding deterministic map, such as periodic cycles, quasiperiodicity, and chaos, are also investigated.

The search for a unified formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for the master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that (1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; (2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; (3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that general Markov processes admit a unified and self-consistent non-equilibrium steady-state thermodynamic formalism, regardless of underlying detailed models.

We investigate numerically temporal correlations in a one-dimensional critical-slope sandpile model with rules that on average conserve the number of particles. Our work is motivated by the existence of two well-separated time scales in self-organized sandpile models, one related to the spreading of avalanches and the other imposed by the external driving. We assume that avalanches are instantaneous events on the time scale imposed by the external deposition and study the autocorrelation function of the series of successive avalanche amplitudes. We find that the autocorrelation function has a log-normal form and for large system sizes tends to a constant, implying that the temporal correlations become stronger in the limit of large system size. We independently test this result by calculating the power spectrum of the series of successive avalanche lifetimes and sizes. For large system sizes L there is a frequency regime where the power spectrum tends to a 1/f type of noise, in agreement with the tendency of the autocorrelation function to approach a constant in large systems. @S1063-651X~96!07212-1#

Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.

We discuss statics and dynamics of condensation in a zero-range process with compartments of limited sizes. For the symmetric dynamics the stationary state has a factorized form. For the asymmetric dynamics the steady state factorizes only for special hopping rules which allow for overjumps of fully occupied compartments. In the limit of large system size the grand canonical analysis is exact also in a condensed phase, and for a broader class of hopping rates as compared to the previously studied systems with infinite compartments. The dynamics of condensation exhibits dynamical self-blocking, which significantly prolongs relaxation times. These general features are illustrated with a concrete example: an inhomogeneous system with hopping rates that result in Bose-Einstein-like condensations.

In recent years, small-world graphs have gained considerable interest as models of real-world systems, which often display features residing between regularity and randomness. The most notable of these models is the Watts-Strogatz graph [1], though alternatives have been proposed [2]. The unifying characteristics of these models are that any two nodes are joined with a small number of links between them (i.e. short path length), while at the same time connected node pairs exhibit an abundance of triangular relations resulting in a high degree of local redundancy (i.e. high clustering). Theoretical investigations of small-world graph models have generally applied asymptotic evaluations in the limit of large system size [3] or the continuum approximation [4] to the algorithmic definition of the graph, in the absence of an analytic representation. In this study, we introduce a generative model of directed small-world graphs, a canonical model of Watts-Strogatz digraphs, and propose an approach that yields the graph’s defining adjacency matrix in algebraic terms, with the goal to provide mathematically rigorous access to the study of finite-size small-world graphs [5]. The proposed approach makes use of random annihilation operators whose algebraic properties can be utilized to assess algebraically well-defined graph-theoretic measures in an analytically exact framework, valid nonasymptotically for all graph sizes. We demonstrated the application of our approach by calculating, for the first time, the asymmetry index and total clustering coefficient of small worlds in an exact fashion. We then utilize the exact nonasymptotic expression for the clustering coefficient in order to assess the small-worldness of structural brain networks in an analytic setting. Using the number of nodes and edges of the given brain networks to construct the equivalent small-world network, we observe that a significant edge rewiring of at least 20% up to 60% is required to produce the small-worldness indices observed in these networks. Importantly, the maximum of the small-worldness index however occurs in all cases at one order of magnitude lower than the required rewiring found. This result suggests that neural graphs reside far away from the small-world regime of the Watts-Strogatz model.

SZNAJD MODEL WITH SYNCHRONOUS UPDATING ON COMPLEX NETWORKS

Recently, Sá, Ribeiro, and Prosen [Phys. Rev. X 10, 021019 (2020)10.1103/PhysRevX.10.021019] introduced complex spacing ratios to analyze eigenvalue correlations in non-Hermitian systems. At present there are no analytical results for the probability distribution of these ratios in the limit of large system size. We derive an approximation formula for the Ginibre universality class of random matrix theory which converges exponentially fast to the limit of infinite matrix size. We also give results for moments of the distribution in this limit.

The presence of a global internal symmetry in a quantum many-body system is reflected in the fact that the entanglement between its subparts is endowed with an internal structure, namely it can be decomposed as a sum of contributions associated to each symmetry sector. The symmetry resolution of entanglement measures provides a formidable tool to probe the out-of-equilibrium dynamics of quantum systems. Here, we study the time evolution of charge-imbalance-resolved negativity after a global quench in the context of free-fermion systems, complementing former works for the symmetry-resolved entanglement entropy. We find that the charge-imbalance-resolved logarithmic negativity shows an effective equipartition in the scaling limit of large times and system size, with a perfect equipartition for early and infinite times. We also derive and conjecture a formula for the dynamics of the charged Rényi logarithmic negativities. We argue that our results can be understood in the framework of the quasiparticle picture for the entanglement dynamics, and provide a conjecture that we expect to be valid for generic integrable models.