Stochastic oscillations in models of epidemics on a network of cities

We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of disease spread on a network of $n$ cities. In the model a fraction $f_{jk}$ of individuals from city $k$ commute to city $j$, where they may infect, or be infected by, others. Starting from a continuous time Markov description of the model the deterministic equations, which are valid in the limit when the population of each city is infinite, are recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple: a unique non-trivial fixed point always exists and has the feature that the fraction of susceptible, infected and recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have an analogously simple dynamics: all oscillations have a single frequency, equal to that found in the one city case. We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation of the deterministic equations at the fixed point.Comment: 13 pages, 7 figure

Fluctuations and oscillations in a simple epidemic model

We show that the simplest stochastic epidemiological models with spatial correlations exhibit two types of oscillatory behaviour in the endemic phase. In a large parameter range, the oscillations are due to resonant amplification of stochastic fluctuations, a general mechanism first reported for predator-prey dynamics. In a narrow range of parameters that includes many infectious diseases which confer long lasting immunity the oscillations persist for infinite populations. This effect is apparent in simulations of the stochastic process in systems of variable size, and can be understood from the phase diagram of the deterministic pair approximation equations. The two mechanisms combined play a central role in explaining the ubiquity of oscillatory behaviour in real data and in simulation results of epidemic and other related models.Comment: acknowledgments added; a typo in the discussion that follows Eq. (3) is corrected

Modeling the long term dynamics of pre-vaccination pertussis

The dynamics of strongly immunizing childhood infections is still not well understood. Although reports of successful modeling of several incidence data records can be found in the literature, the key determinants of the observed temporal patterns have not been clearly identified. In particular, different models of immunity waning and degree of protection applied to disease and vaccine induced immunity have been debated in the literature on pertussis. Here we study the effect of disease acquired immunity on the long term patterns of pertussis prevalence. We compare five minimal models, all of which are stochastic, seasonally forced, well-mixed models of infection based on susceptible-infective-recovered dynamics in a closed population. These models reflect different assumptions about the immune response of naive hosts, namely total permanent immunity, immunity waning, immunity waning together with immunity boosting, reinfection of recovered, and repeat infection after partial immunity waning. The power spectra of the output prevalence time series characterize the long term dynamics of the models. For epidemiological parameters consistent with published data for pertussis, the power spectra show quantitative and even qualitative differences that can be used to test their assumptions by comparison with ensembles of several decades long pre-vaccination data records. We illustrate this strategy on two publicly available historical data sets.Comment: paper (31 pages, 11 figures, 1 table) and supplementary material (19 pages, 5 figures, 2 tables

Phase lag in epidemics on a network of cities

We study the synchronisation and phase-lag of fluctuations in the number of infected individuals in a network of cities between which individuals commute. The frequency and amplitude of these oscillations is known to be very well captured by the van Kampen system-size expansion, and we use this approximation to compute the complex coherence function that describes their correlation. We find that, if the infection rate differs from city to city and the coupling between them is not too strong, these oscillations are synchronised with a well defined phase lag between cities. The analytic description of the effect is shown to be in good agreement with the results of stochastic simulations for realistic population sizes.Comment: 10 pages, 6 figure

Cluster approximations for infection dynamics on random networks

In this paper, we consider a simple stochastic epidemic model on large regular random graphs and the stochastic process that corresponds to this dynamics in the standard pair approximation. Using the fact that the nodes of a pair are unlikely to share neighbors, we derive the master equation for this process and obtain from the system size expansion the power spectrum of the fluctuations in the quasi-stationary state. We show that whenever the pair approximation deterministic equations give an accurate description of the behavior of the system in the thermodynamic limit, the power spectrum of the fluctuations measured in long simulations is well approximated by the analytical power spectrum. If this assumption breaks down, then the cluster approximation must be carried out beyond the level of pairs. We construct an uncorrelated triplet approximation that captures the behavior of the system in a region of parameter space where the pair approximation fails to give a good quantitative or even qualitative agreement. For these parameter values, the power spectrum of the fluctuations in finite systems can be computed analytically from the master equation of the corresponding stochastic process.Comment: the notation has been changed; Ref. [26] and a new paragraph in Section IV have been adde

Assessing trigeminal microstructure changes in patients with classical trigeminal neuralgia

Introduction. The crucial role of neuro-vascular conflict (NVC) in trigeminal neuralgia (TN) is getting increasingly challenged. Microstructural changes can be assessed using fractional anisotropy (FA) in diffusion tensor images (DTI). Objective. To evaluate usefulness of FA in brain MRI with DTI for TN lateralization assessment. Materials and methods. The study included 51 patients with classical TN divided into two groups: neurosurgical intervention free, post radiofrequency ablation (RFA), and a control group (patients without facial pain). All the patients were tested for NVC with FIESTA (Fast Imaging Employing Steady State Acquisition) brain MRI at 3Т. Difference in thickness of trigeminal roots on the intact and symptomatic sides was assessed for each group. The findings were compared to those in the control group. The MRI protocol was supplemented with DTI. The FA difference in thickness of the intact and symptomatic roots (∆FA) was calculated for each study group to assess microstructural root changes. The results were compared to those in the control group. Results. In trigeminal root DTIs, ∆FA over 0.075 [0.029; 0.146] is statistically significant to establish NVC-associated microstructural changes on the symptomatic side in patients without any past surgeries (p = 0,030). In patients with a history of trigeminal ganglion RFA, statistically significant (p = 0.026) thinned symptomatic trigeminal root (difference in thickness of trigeminal roots over 0.45 cm [0.4; 0.6]) was found as compared to that of the control patients. Conclusion. FA may be used as a quantitative demyelination biomarker in clinical TN. Trigeminal ganglion RFA leads to hypotrophy throughout the trigeminal nerve root