Systematic Approximations to Susceptible-Infectious-Susceptible Dynamics on Networks

Heterogeneous pair-approximation analysis for susceptible-infectious-susceptible epidemics on networks

The pair heterogeneous mean-field (PHMF) model has been used extensively in previous studies to investigate the dynamics of susceptible-infectious-susceptible epidemics on complex networks. However, the approximate treatment of the classical or reduced PHMF models lacks a rigorous theoretical analysis. By means of the standard and full PHMF models, we first derived the equivalent conditions for the approximate model treatment. Furthermore, we analytically derived a novel epidemic threshold for the PHMF model, and we demonstrated via numerical simulations that this threshold condition differs from all those reported in earlier studies. Our findings indicate that both the reduced and full PHMF models agree well with continuous-time stochastic simulations, especially when infection is spreading at considerably higher rates.

Extensions of mean-field approximations for environmentally-transmitted pathogen networks

Many pathogens spread via environmental transmission, without requiring host-to-host direct contact. While models for environmental transmission exist, many are simply constructed intuitively with structures analogous to standard models for direct transmission. As model insights are generally sensitive to the underlying model assumptions, it is important that we are able understand the details and consequences of these assumptions. We construct a simple network model for an environmentally-transmitted pathogen and rigorously derive systems of ordinary differential equations (ODEs) based on different assumptions. We explore two key assumptions, namely homogeneity and independence, and demonstrate that relaxing these assumptions can lead to more accurate ODE approximations. We compare these ODE models to a stochastic implementation of the network model over a variety of parameters and network structures, demonstrating that with fewer restrictive assumptions we are able to achieve higher accuracy in our approximations and highlighting more precisely the errors produced by each assumption. We show that less restrictive assumptions lead to more complicated systems of ODEs and the potential for unstable solutions. Due to the rigour of our derivation, we are able to identify the reason behind these errors and propose potential resolutions.

Fitting the reproduction number from UK coronavirus case data and why it is close to 1

We present a method for rapid calculation of coronavirus growth rates and [Formula: see text]-numbers tailored to publicly available UK data. We assume that the case data comprise a smooth, underlying trend which is differentiable, plus systematic errors and a non-differentiable noise term, and use bespoke data processing to remove systematic errors and noise. The approach is designed to prioritize up-to-date estimates. Our method is validated against published consensus [Formula: see text]-numbers from the UK government and is shown to produce comparable results two weeks earlier. The case-driven approach is combined with weight-shift-scale methods to monitor trends in the epidemic and for medium-term predictions. Using case-fatality ratios, we create a narrative for trends in the UK epidemic: increased infectiousness of the B1.117 (Alpha) variant, and the effectiveness of vaccination in reducing severity of infection. For longer-term future scenarios, we base future [Formula: see text] on insight from localized spread models, which show [Formula: see text] going asymptotically to 1 after a transient, regardless of how large the [Formula: see text] transient is. This accords with short-lived peaks observed in case data. These cannot be explained by a well-mixed model and are suggestive of spread on a localized network. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

A novel hybrid SEIQR model incorporating the effect of quarantine and lockdown regulations for COVID-19

Mitigating the devastating effect of COVID-19 is necessary to control the infectivity and mortality rates. Hence, several strategies such as quarantine of exposed and infected individuals and restricting movement through lockdown of geographical regions have been implemented in most countries. On the other hand, standard SEIR based mathematical models have been developed to understand the disease dynamics of COVID-19, and the proper inclusion of these restrictions is the rate-limiting step for the success of these models. In this work, we have developed a hybrid Susceptible-Exposed-Infected-Quarantined-Removed (SEIQR) model to explore the influence of quarantine and lockdown on disease propagation dynamics. The model is multi-compartmental, and it considers everyday variations in lockdown regulations, testing rate and quarantine individuals. Our model predicts a considerable difference in reported and actual recovered and deceased cases in qualitative agreement with recent reports.

A Low-Dimensional Network Model for an SIS Epidemic: Analysis of the Super Compact Pairwise Model

Network-based models of epidemic spread have become increasingly popular in recent decades. Despite a rich foundation of such models, few low-dimensional systems for modeling SIS-type diseases have been proposed that manage to capture the complex dynamics induced by the network structure. We analyze one recently introduced model and derive important epidemiological quantities for the system. We derive the epidemic threshold and analyze the bifurcation that occurs, and we use asymptotic techniques to derive an approximation for the endemic equilibrium when it exists. We consider the sensitivity of this approximation to network parameters, and the implications for disease control measures are found to be in line with the results of existing studies.

Not One, but Many Critical States: A Dynamical Systems Perspective

The past decade has seen growing support for the critical brain hypothesis, i.e., the possibility that the brain could operate at or very near a critical state between two different dynamical regimes. Such critical states are well-studied in different disciplines, therefore there is potential for a continued transfer of knowledge. Here, I revisit foundations of bifurcation theory, the mathematical theory of transitions. While the mathematics is well-known it's transfer to neural dynamics leads to new insights and hypothesis.

Uniformly accurate nonlinear transmission rate models arising from disease spread through pair contacts

We derive and asymptotically analyze mass-action models for disease spread that include transient pair formation and dissociation. Populations of unpaired susceptible individuals and infected individuals are distinguished from the population of three types of pairs of individuals: both susceptible, one susceptible and one infected, and both infected. Disease transmission can occur only within a pair consisting of one susceptible individual and one infected individual. We use perturbation expansion to formally derive uniformly valid approximations for the dynamics of the total infected and susceptible populations under different conditions including combinations of fast association, fast transmission, and fast dissociation limits. The effective equations are derived from the fundamental mass-action system without implicitly imposing transmission mechanisms, such as those used in frequency-dependent models. Our results represent submodels that show how effective nonlinear transmission can arise from pairing dynamics and are juxtaposed with density-based mass-action and frequency-based models.

Improving pairwise approximations for network models with susceptible-infected-susceptible dynamics

Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the k-regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for k-regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation.

A colored mean-field model for analyzing the effects of awareness on epidemic spreading in multiplex networks

We study the impact of susceptible nodes' awareness on epidemic spreading in social systems, where the systems are modeled as multiplex networks coupled with an information layer and a contact layer. We develop a colored heterogeneous mean-field model taking into account the portion of the overlapping neighbors in the two layers. With theoretical analysis and numerical simulations, we derive the epidemic threshold which determines whether the epidemic can prevail in the population and find that the impacts of awareness on threshold value only depend on epidemic information being available in network nodes' overlapping neighborhood. When there is no link overlap between the two network layers, the awareness cannot help one to raise the epidemic threshold. Such an observation is different from that in a single-layer network, where the existence of awareness almost always helps.

Susceptible-infected-recovered epidemics in random networks with population awareness

The influence of epidemic information-based awareness on the spread of infectious diseases on networks cannot be ignored. Within the effective degree modeling framework, we discuss the susceptible-infected-recovered model in complex networks with general awareness and general degree distribution. By performing the linear stability analysis, the conditions of epidemic outbreak can be deduced and the results of the previous research can be further expanded. Results show that the local awareness can suppress significantly the epidemic spreading on complex networks via raising the epidemic threshold and such effects are closely related to the formulation of awareness functions. In addition, our results suggest that the recovered information-based awareness has no effect on the critical condition of epidemic outbreak.

Pair formation models for sexually transmitted infections: A primer

For modelling sexually transmitted infections, duration of partnerships can strongly influence the transmission dynamics of the infection. If partnerships are monogamous, pairs of susceptible individuals are protected from becoming infected, while pairs of infected individuals delay onward transmission of the infection as long as they persist. In addition, for curable infections re-infection from an infected partner may occur. Furthermore, interventions based on contact tracing rely on the possibility of identifying and treating partners of infected individuals. To reflect these features in a mathematical model, pair formation models were introduced to mathematical epidemiology in the 1980's. They have since been developed into a widely used tool in modelling sexually transmitted infections and the impact of interventions. Here we give a basic introduction to the concepts of pair formation models for a susceptible-infected-susceptible (SIS) epidemic. We review some results and applications of pair formation models mainly in the context of chlamydia infection.