The spread of infectious diseases in spatially structured populations: an invasory pair approximation

Data-driven optimal closures for mean-cluster models: Beyond the classical pair approximation

This study concerns the mean-clustering approach to modeling the evolution of lattice dynamics. Instead of tracking the state of individual lattice sites, this approach describes the time evolution of the concentrations of different cluster types. It leads to an infinite hierarchy of ordinary differential equations which must be closed by truncation using a so-called closure condition. This condition approximates the concentrations of higher-order clusters in terms of the concentrations of lower-order ones. The pair approximation is the most common form of closure. Here, we consider its generalization, termed the "optimal approximation," which we calibrate using a robust data-driven strategy. To fix attention, we focus on a recently proposed structured lattice model for a nickel-based oxide, similar to that used as cathode material in modern commercial Li-ion batteries. The form of the obtained optimal approximation allows us to deduce a simple sparse closure model. In addition to being more accurate than the classical pair approximation, this "sparse approximation" is also physically interpretable which allows us to a posteriori refine the hypotheses underlying construction of this class of closure models. Moreover, the mean-cluster model closed with this sparse approximation is linear and hence analytically solvable such that its parametrization is straightforward, although it offers a good approximation of the actual time evolution of the cluster concentrations on short timescales only. On the other hand, parametrization of the mean-cluster model closed with the pair approximation is shown to lead to an ill-posed inverse problem.

Epidemic dynamics on higher-dimensional small world networks

Dimension governs dynamical processes on networks. The social and technological networks which we encounter in everyday life span a wide range of dimensions, but studies of spreading on finite-dimensional networks are usually restricted to one or two dimensions. To facilitate investigation of the impact of dimension on spreading processes, we define a flexible higher-dimensional small world network model and characterize the dependence of its structural properties on dimension. Subsequently, we derive mean field, pair approximation, intertwined continuous Markov chain and probabilistic discrete Markov chain models of a COVID-19-inspired susceptible-exposed-infected-removed (SEIR) epidemic process with quarantine and isolation strategies, and for each model identify the basic reproduction number R 0 , which determines whether an introduced infinitesimal level of infection in an initially susceptible population will shrink or grow. We apply these four continuous state models, together with discrete state Monte Carlo simulations, to analyse how spreading varies with model parameters. Both network properties and the outcome of Monte Carlo simulations vary substantially with dimension or rewiring rate, but predictions of continuous state models change only slightly. A different trend appears for epidemic model parameters: as these vary, the outcomes of Monte Carlo change less than those of continuous state methods. Furthermore, under a wide range of conditions, the four continuous state approximations present similar deviations from the outcome of Monte Carlo simulations. This bias is usually least when using the pair approximation model, varies only slightly with network size, and decreases with dimension or rewiring rate. Finally, we characterize the discrepancies between Monte Carlo and continuous state models by simultaneously considering network efficiency and network size.

Effects of void nodes on epidemic spreads in networks

We present the pair approximation models for susceptible–infected–recovered (SIR) epidemic dynamics in a sparse network based on a regular network. Two processes are considered, namely, a Markovian process with a constant recovery rate and a non-Markovian process with a fixed recovery time. We derive the implicit analytical expression for the final epidemic size and explicitly show the epidemic threshold in both Markovian and non-Markovian processes. As the connection rate decreases from the original network connection, the epidemic threshold in which epidemic phase transits from disease-free to endemic increases, and the final epidemic size decreases. Additionally, for comparison with sparse and heterogeneous networks, the pair approximation models were applied to a heterogeneous network with a degree distribution. The obtained phase diagram reveals that, upon increasing the degree of the original random regular networks and decreasing the effective connections by introducing void nodes accordingly, the final epidemic size of the sparse network is close to that of the random network with average degree of 4. Thus, introducing the void nodes in the network leads to more heterogeneous network and reduces the final epidemic size.

Targeted pandemic containment through identifying local contact network bottlenecks

Decision-making about pandemic mitigation often relies upon simulation modelling. Models of disease transmission through networks of contacts–between individuals or between population centres–are increasingly used for these purposes. Real-world contact networks are rich in structural features that influence infection transmission, such as tightly-knit local communities that are weakly connected to one another. In this paper, we propose a new flow-based edge-betweenness centrality method for detecting bottleneck edges that connect nodes in contact networks. In particular, we utilize convex optimization formulations based on the idea of diffusion with p-norm network flow. Using simulation models of COVID-19 transmission through real network data at both individual and county levels, we demonstrate that targeting bottleneck edges identified by the proposed method reduces the number of infected cases by up to 10% more than state-of-the-art edge-betweenness methods. Furthermore, the proposed method is orders of magnitude faster than existing methods.

During the COVID-19 pandemic decision makers frequently face questions like where to impose a lockdown, which traffic to close, and whom to quarantine, all required to be carried out at minimal costs. Establishing cost-effective pandemic control policies requires identifying good targets. New computational models from network theory and epidemic simulations over real contact networks provide a valuable tool for finding the right bottlenecks to target upon. Here we study a computationally efficient network centrality measure that enables us to detect local transmission bottlenecks, i.e., contact edges that are especially important for the spread of disease among small communities or local network structures inside large networks. We find that pandemic intervention strategies that target at local network structures significantly outperform interventions that solely focus on the entire network structure as a whole, which are traditionally believed to be the most effective.

Pair approximation model for the vaccination game: predicting the dynamic process of epidemic spread and individual actions against contagion

We successfully establish a theoretical framework of pairwise approximation for the vaccination game in which both the dynamic process of epidemic spread and individual actions in helping prevent social behaviours are quantitatively evaluated. In contrast with mean-field approximation, our model captures higher-order effects from neighbours by using an underlying network that shows how the disease spreads and how individual decisions evolve over time. This model considers not only imperfect vaccination but also intermediate protective measures other than vaccines. Our analytical predictions are validated by multi-agent simulation results that estimate random regular graphs at varying degrees.

Beyond the pair approximation: Modeling colonization population dynamics

The process of range expansion (colonization) is one of the basic types of biological dynamics, whereby a species grows and spreads outwards, occupying new territories. Spatial modeling of this process is naturally implemented as a stochastic cellular automaton, with individuals occupying nodes on a rectangular grid, births and deaths occurring probabilistically, and individuals only reproducing onto unoccupied neighboring spots. In this paper we derive several approximations that allow prediction of the expected range expansion dynamics, based on the reproduction and death rates. We derive several approximations, where the cellular automaton is described by a system of ordinary differential equations that preserves correlations among neighboring spots (up to a distance). This methodology allows us to develop accurate approximations of the population size and the expected spatial shape, at a fraction of the computational time required to simulate the original stochastic system. In addition, we provide simple formulas for the steady-state population densities for von Neumann and Moore neighborhoods. Finally, we derive concise approximations for the speed of range expansion in terms of the reproduction and death rates, for both types of neighborhoods. The methodology is generalizable to more complex scenarios, such as different interaction ranges and multiple-species systems.

Algorithmic discovery of dynamic models from infectious disease data

Theoretical models are typically developed through a deductive process where a researcher formulates a system of dynamic equations from hypothesized mechanisms. Recent advances in algorithmic methods can discover dynamic models inductively-directly from data. Most previous research has tested these methods by rediscovering models from synthetic data generated by the already known model. Here we apply Sparse Identification of Nonlinear Dynamics (SINDy) to discover mechanistic equations for disease dynamics from case notification data for measles, chickenpox, and rubella. The discovered models provide a good qualitative fit to the observed dynamics for all three diseases, However, the SINDy chickenpox model appears to overfit the empirical data, and recovering qualitatively correct rubella dynamics requires using power spectral density in the goodness-of-fit criterion. When SINDy uses a library of second-order functions, the discovered models tend to include mass action incidence and a seasonally varying transmission rate-a common feature of existing epidemiological models for childhood infectious diseases. We also find that the SINDy measles model is capable of out-of-sample prediction of a dynamical regime shift in measles case notification data. These results demonstrate the potential for algorithmic model discovery to enrich scientific understanding by providing a complementary approach to developing theoretical models.

Equilibrium properties of the spatial SIS model as a point pattern dynamics - How is infection distributed over space?

We revisit the classical epidemiological SIS model as a stochastic point pattern dynamics with special focus on its spatial distribution at equilibrium. In this model, each point on a continuous space is either susceptible S or infectious I, and infection occurs with an infection kernel as a function of distance from I to S. This stochastic process has been mathematically described by the hierarchical dynamics of the probabilities that a point, a pair made by two points, and a triplet made by three points, etc., is in a specific configuration of status. Using a simple closure thereby triplet probabilities that appear in the dynamics are approximated, we show that the average singlet probabilities and the pair probabilities that describe spatial distributions of Ss and Is at equilibrium can be explicitly derived using the infection kernel; Is are spatially clustered in the same order of the infection kernel. The results highlight the advantage of point pattern approach to model spatial population dynamics in general ecology where local interactions among individuals likely depend on distance between them.

Understanding the temporal pattern of spreading in heterogeneous networks: Theory of the mean infection time

For a reliable prediction of an epidemic or information spreading pattern in complex systems, well-defined measures are essential. In the susceptible-infected model on heterogeneous networks, the cluster of infected nodes in the intermediate-time regime exhibits too large fluctuation in size to use its mean size as a representative value. The cluster size follows quite a broad distribution, which is shown to be derived from the variation of the cluster size with the time when a hub node was first infected. On the contrary, the distribution of the time taken to infect a given number of nodes is well concentrated at its mean, suggesting the mean infection time is a better measure. We show that the mean infection time can be evaluated by using the scaling behaviors of the boundary area of the infected cluster and use it to find a nonexponential but algebraic spreading phase in the intermediate stage on strongly heterogeneous networks. Such slow spreading originates in only small-degree nodes left susceptible, while most hub nodes are already infected in the early exponential-spreading stage. Our results offer a way to detour around large statistical fluctuations and quantify reliably the temporal pattern of spread under structural heterogeneity.

SIR dynamics in random networks with communities

This paper investigates the effects of the community structure of a network on the spread of an epidemic. To this end, we first establish a susceptible-infected-recovered (SIR) model in a two-community network with an arbitrary joint degree distribution. The network is formulated as a probability generating function. We also obtain the sufficient conditions for disease outbreak and extinction, which involve the first-order and second-order moments of the degree distribution. As an example, we then study the effect of community structure on epidemic spread in a complex network with a Poisson joint degree distribution. The numerical solutions of the SIR model well agree with stochastic simulations based on the Monte Carlo method, confirming that the model is reliable and accurate. Finally, by strengthening the community structure in the simulation, i.e. fixing the total degree distribution and reducing the number ratio of the external edges, we can increase or decrease the final cumulative epidemic incidence depending on the transmissibility of the virus between humans and the community structure at that point. Why community structure can affect disease dynamics in a complicated way is also discussed. In any case, for large-scale epidemics, strengthening the community structure to reduce the size of disease is undoubtedly an effective way.

Epidemics on plants: Modeling long-range dispersal on spatially embedded networks

Here we develop an epidemic model that accounts for long-range dispersal of pathogens between plants. This model generalizes the classical compartmental models-Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR)-to take into account those factors that are key to understand epidemics in real plant populations. These ingredients are the spatial characteristics of the plots and fields in which plants are embedded and the effect of long-range dispersal of pathogens. The spatial characteristics are included through the use of random rectangular graphs which allow to consider the effects of the elongation of plots and fields, while the long-range dispersal is implemented by considering transformations, such as the Mellin and Laplace transforms, of a generalization of the adjacency matrix of the geometric graph. Our results point out that long-range dispersal favors the propagation of pathogens while the elongation of plant plots increases the epidemic threshold and decreases dramatically the number of affected plants. Interestingly, our model is able of reproducing the existence of patchy regions of infected plants and the absence of a clear propagation front centered in the initial infected plants, as it is observed in real plant epidemics.

Dynamics analysis of SIR epidemic model with correlation coefficients and clustering coefficient in networks

In this paper, the correlation coefficients between nodes in states are used as dynamic variables, and we construct SIR epidemic dynamic models with correlation coefficients by using the pair approximation method in static networks and dynamic networks, respectively. Considering the clustering coefficient of the network, we analytically investigate the existence and the local asymptotic stability of each equilibrium of these models and derive threshold values for the prevalence of diseases. Additionally, we obtain two equivalent epidemic thresholds in dynamic networks, which are compared with the results of the mean field equations.

Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions

Pair approximation models have been used to study the spread of infectious diseases in spatially distributed host populations, and to explore disease control strategies such as vaccination and case isolation. Here we introduce a pair approximation model of individual uptake of non-pharmaceutical interventions (NPIs) for an acute self-limiting infection, where susceptible individuals can learn the NPIs either from other susceptible individuals who are already practicing NPIs ("social learning"), or their uptake of NPIs can be stimulated by being neighbours of an infectious person ("exposure learning"). NPIs include individual measures such as hand-washing and respiratory etiquette. Individuals can also drop the habit of using NPIs at a certain rate. We derive a spatially defined expression of the basic reproduction number R0 and we also numerically simulate the model equations. We find that exposure learning is generally more efficient than social learning, since exposure learning generates NPI uptake in the individuals at immediate risk of infection. However, if social learning is pre-emptive, beginning a sufficient amount of time before the epidemic, then it can be more effective than exposure learning. Interestingly, varying the initial number of individuals practicing NPIs does not significantly impact the epidemic final size. Also, if initial source infections are surrounded by protective individuals, there are parameter regimes where increasing the initial number of source infections actually decreases the infection peak (instead of increasing it) and makes it occur sooner. The peak prevalence increases with the rate at which individuals drop the habit of using NPIs, but the response of peak prevalence to changes in the forgetting rate are qualitatively different for the two forms of learning. The pair approximation methodology developed here illustrates how analytical approaches for studying interactions between social processes and disease dynamics in a spatially structured population should be further pursued.

The epidemic model based on the approximation for third-order motifs on networks

The spread of an infectious disease may depend on the structure of the network. To study the influence of the structure parameters of the network on the spread of the epidemic, we need to put these parameters into the epidemic model. The method of moment closure introduces structure parameters into the epidemic model. In this paper, we present a new moment closure epidemic model based on the approximation of third-order motifs in networks. The order of a motif defined in this paper is determined by the number of the edges in the motif, rather than by the number of nodes in the motif as defined in the literature. We provide a general approach to deriving a set of ordinary differential equations that describes, to a high degree of accuracy, the spread of an infectious disease. Using this method, we establish a susceptible-infected-recovered (SIR) model. We then calculate the basic reproduction number of the SIR model, and find that it decreases as the clustering coefficient increases. Finally, we perform some simulations using the proposed model to study the influence of the clustering coefficient on the final epidemic size, the maximum number of infected, and the peak time of the disease. The numerical simulations based on the SIR model in this paper fit the stochastic simulations based on the Monte Carlo method well at different levels of clustering. Our results show that the clustering coefficient poses impediments to the spread of disease under an SIR model.

Spread of competing viruses on heterogeneous networks

In this paper, we propose a model where two strains compete with each other at the expense of common susceptible individuals on heterogeneous networks by using pair-wise approximation closed by the probability-generating function (PGF). All of the strains obey the susceptible-infected-recovered (SIR) mechanism. From a special perspective, we first study the dynamical behaviour of an SIR model closed by the PGF, and obtain the basic reproduction number via two methods. Then we build a model to study the spreading dynamics of competing viruses and discuss the conditions for the local stability of equilibria, which is different from the condition obtained by using the heterogeneous mean-field approach. Finally, we perform numerical simulations on Barabási-Albert networks to complement our theoretical research, and show some dynamical properties of the model with competing viruses.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

DEMOGRAPHICS INDUCE EXTINCTION OF DISEASE IN AN SIS MODEL BASED ON CONDITIONAL MARKOV CHAIN

Demographics have significant effects on disease spread in populations and the topological evolution of the underlying networks that represent the populations. In the context of network-based epidemic modeling, Markov chain-based approach and pairwise approximation are two powerful tools — the former can capture stochastic effects of disease transmission dynamics and the latter can characterize the dynamical correlations in each pair of connected individuals. However, to our best knowledge, the study on epidemic spreading in networks relying on these two techniques is still lacking. To fill this gap, in this paper, a deterministic pairwise susceptible–infected–susceptible (SIS) epidemic model with demographics on complex networks with arbitrary degree distributions is studied based on a continuous time conditional Markov chain. This deterministic model is rigorously derived — using the moment generating function — from the Kolmogorov differential equations for the evolution of individuals and pairs. It is found that demographics will induce the extinction of the disease by reducing the basic reproduction number or lowering the epidemic prevalence after the disease prevails. Moreover, due to the demographical effects, the resulting network tends to a homogeneous network with a degree distribution similar to Poisson distribution, irrespective of the initial network structure. Additionally, we find excellent agreement between numerical solutions and individual-based stochastic simulations using both Erdös–Renyi (ER) random and Barabási–Albert (BA) scale-free initial networks. Our results may provide new insights on the understanding of the influence of demographics on epidemic dynamics and network evolution.

Dynamics of epidemic diseases on a growing adaptive network

The study of epidemics on static networks has revealed important effects on disease prevalence of network topological features such as the variance of the degree distribution, i.e. the distribution of the number of neighbors of nodes, and the maximum degree. Here, we analyze an adaptive network where the degree distribution is not independent of epidemics but is shaped through disease-induced dynamics and mortality in a complex interplay. We study the dynamics of a network that grows according to a preferential attachment rule, while nodes are simultaneously removed from the network due to disease-induced mortality. We investigate the prevalence of the disease using individual-based simulations and a heterogeneous node approximation. Our results suggest that in this system in the thermodynamic limit no epidemic thresholds exist, while the interplay between network growth and epidemic spreading leads to exponential networks for any finite rate of infectiousness when the disease persists.

Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure

We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem for generating exact closures (here expressing joint probabilities in terms of their constituent marginal probabilities) for susceptible-infectious-removed (SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and removal processes, this enables us to obtain a systematic reduction in the number of differential equations needed for an exact 'moment closure' representation of the underlying stochastic model. We define 'transmission blocks' as a possible extension of the block concept in graph theory and show that the order at which the exact moment closure representation is curtailed is the size of the largest transmission block. More generally, approximate closures of the hierarchy of moment equations for these dynamics are typically defined for the first and second order yielding mean-field and pairwise models respectively. It is frequently implied that, in principle, closed models can be written down at arbitrary order if only we had the time and patience to do this. However, for epidemic dynamics on networks, these higher-order models have not been defined explicitly. Here we unambiguously define hierarchies of approximate closed models that can utilise subsystem states of any order, and show how well-known models are special cases of these hierarchies.

Impacts of constrained culling and vaccination on control of foot and mouth disease in near-endemic settings: a pair approximation model

Many countries have eliminated foot and mouth disease (FMD), but outbreaks remain common in other countries. Rapid development of international trade in animals and animal products has increased the risk of disease introduction to FMD-free countries. Most mathematical models of FMD are tailored to settings that are normally disease-free, and few models have explored the impact of constrained control measures in a 'near-endemic' spatially distributed host population subject to frequent FMD re-introductions from nearby endemic wild populations, as characterizes many low-income, resource-limited countries. Here we construct a pair approximation model of FMD and investigate the impact of constraints on total vaccine supply for prophylactic and ring vaccination, and constraints on culling rates and cumulative culls. We incorporate natural immunity waning and vaccine waning, which are important factors for near-endemic populations. We find that, when vaccine supply is sufficiently limited, the optimal approach for minimizing cumulative infections combines rapid deployment of ring vaccination during outbreaks with a contrasting approach of careful rationing of prophylactic vaccination over the year, such that supplies last as long as possible (and with the bulk of vaccines dedicated toward prophylactic vaccination). Thus, for optimal long-term control of the disease by vaccination in near-endemic settings when vaccine supply is limited, it is best to spread out prophylactic vaccination as much as possible. Regardless of culling constraints, the optimal culling strategy is rapid identification of infected premises and their immediate contacts at the initial stages of an outbreak, and rapid culling of infected premises and farms deemed to be at high risk of infection (as opposed to culling only the infected farms). Optimal culling strategies are similar when social impact is the outcome of interest. We conclude that more FMD transmission models should be developed that are specific to the challenges of FMD control in near-endemic, low-income countries.

Dynamics and control of foot-and-mouth disease in endemic countries: a pair approximation model

Previous mathematical models of spatial farm-to-farm transmission of foot and mouth disease (FMD) have explored the impacts of control measures such as culling and vaccination during a single outbreak in a country normally free of FMD. As a result, these models do not include factors that are relevant to countries where FMD is endemic in some regions, like long-term waning natural and vaccine immunity, use of prophylactic vaccination and disease re-importations. These factors may have implications for disease dynamics and control, yet few models have been developed for FMD-endemic settings. Here we develop and study an SEIRV (susceptible-exposed-infectious-recovered-vaccinated) pair approximation model of FMD. We focus on long term dynamics by exploring characteristics of repeated outbreaks of FMD and their dependence on disease re-importation, loss of natural immunity, and vaccine waning. We find that the effectiveness of ring and prophylactic vaccination strongly depends on duration of natural immunity, rate of vaccine waning, and disease re-introduction rate. However, the number and magnitude of FMD outbreaks are generally more sensitive to the duration of natural immunity than the duration of vaccine immunity. If loss of natural immunity and/or vaccine waning happen rapidly, then multiple epidemic outbreaks result, making it difficult to eliminate the disease. Prophylactic vaccination is more effective than ring vaccination, at the same per capita vaccination rate. Finally, more frequent disease re-importation causes a higher cumulative number of infections, although a lower average epidemic peak. Our analysis demonstrates significant differences between dynamics in FMD-free settings versus FMD-endemic settings, and that dynamics in FMD-endemic settings can vary widely depending on factors such as the duration of natural and vaccine immunity and the rate of disease re-importations. We conclude that more mathematical models tailored to FMD-endemic countries should be developed that include these factors.

Analytical investigation of self-organized criticality in neural networks

Dynamical criticality has been shown to enhance information processing in dynamical systems, and there is evidence for self-organized criticality in neural networks. A plausible mechanism for such self-organization is activity-dependent synaptic plasticity. Here, we model neurons as discrete-state nodes on an adaptive network following stochastic dynamics. At a threshold connectivity, this system undergoes a dynamical phase transition at which persistent activity sets in. In a low-dimensional representation of the macroscopic dynamics, this corresponds to a transcritical bifurcation. We show analytically that adding activity-dependent rewiring rules, inspired by homeostatic plasticity, leads to the emergence of an attractive steady state at criticality and present numerical evidence for the system's evolution to such a state.

Impact of imitation processes on the effectiveness of ring vaccination

Ring vaccination can be a highly effective control strategy for an emerging disease or in the final phase of disease eradication, as witnessed in the eradication of smallpox. However, the impact of behavioural dynamics on the effectiveness of ring vaccination has not been explored in mathematical models. Here, we analyze a series of stochastic models of voluntary ring vaccination. Contacts of an index case base vaccinating decisions on their own individual payoffs to vaccinate or not vaccinate, and they can also imitate the behaviour of other contacts of the index case. We find that including imitation changes the probability of containment through ring vaccination considerably. Imitation can cause a strong majority of contacts to choose vaccination in some cases, or to choose non-vaccination in other cases-even when the equivalent solution under perfectly rational (non-imitative) behaviour yields mixed choices. Moreover, imitation processes can result in very different outcomes in different stochastic realizations sampled from the same parameter distributions, by magnifying moderate tendencies toward one behaviour or the other: in some realizations, imitation causes a strong majority of contacts not to vaccinate, while in others, imitation promotes vaccination and reduces the number of secondary infections. Hence, the effectiveness of ring vaccination can depend significantly and unpredictably on imitation processes. Therefore, our results suggest that risk communication efforts should be initiated early in an outbreak when ring vaccination is to be applied, especially among subpopulations that are heavily influenced by peer opinions.

Networks and the epidemiology of infectious disease

The science of networks has revolutionised research into the dynamics of interacting elements. It could be argued that epidemiology in particular has embraced the potential of network theory more than any other discipline. Here we review the growing body of research concerning the spread of infectious diseases on networks, focusing on the interplay between network theory and epidemiology. The review is split into four main sections, which examine: the types of network relevant to epidemiology; the multitude of ways these networks can be characterised; the statistical methods that can be applied to infer the epidemiological parameters on a realised network; and finally simulation and analytical methods to determine epidemic dynamics on a given network. Given the breadth of areas covered and the ever-expanding number of publications, a comprehensive review of all work is impossible. Instead, we provide a personalised overview into the areas of network epidemiology that have seen the greatest progress in recent years or have the greatest potential to provide novel insights. As such, considerable importance is placed on analytical approaches and statistical methods which are both rapidly expanding fields. Throughout this review we restrict our attention to epidemiological issues.

Deterministic epidemic models on contact networks: correlations and unbiological terms

The relationship between system-level and subsystem-level master equations is investigated and then utilised for a systematic and potentially automated derivation of the hierarchy of moment equations in a susceptible-infectious-removed (SIR) epidemic model. In the context of epidemics on contact networks we use this to show that the approximate nature of some deterministic models such as mean-field and pair-approximation models can be partly understood by the identification of implicit anomalous terms. These terms describe unbiological processes which can be systematically removed up to and including the nth order by nth order moment closure approximations. These terms lead to a detailed understanding of the correlations in network-based epidemic models and contribute to understanding the connection between individual-level epidemic processes and population-level models. The connection with metapopulation models is also discussed. Our analysis is predominantly made at the individual level where the first and second order moment closure models correspond to what we term the individual-based and pair-based deterministic models, respectively. Matlab code is included as supplementary material for solving these models on transmission networks of arbitrary complexity.

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 quasistationary 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.

Reproduction numbers for epidemics on networks using pair approximation

One way to describe the spread of an infection on a network is by using the method of pair approximation. This method is a deterministic pair-based variant of the usual methods used to describe the progress of an epidemic in randomly mixing populations. However, although the ideas of pair approximation are intuitively clear, it is not straightforward to make all assumptions used explicit. Furthermore, in literature problems arise in defining basic quantities like the basic reproduction number R(0) and the real-time epidemic growth rate parameter r. We formulate the pair approximations and the needed assumptions explicitly. We discuss problems inherent to this method. Furthermore, we define a new reproduction number, similar to R(0) and a new real-time growth rate parameter similar to r. We illustrate the methods of the paper by an example for which we can compare the approximation of the reproduction number with exact results.