Multiple sources and routes of information transmission: Implications for epidemic dynamics

In a recent paper [20], we proposed and analyzed a compartmental ODE-based model describing the dynamics of an infectious disease where the presence of the pathogen also triggers the diffusion of information about the disease. In this paper, we extend this previous work by presenting results based on pairwise and simulation models that are better suited for capturing the population contact structure at a local level. We use the pairwise model to examine the potential of different information generating mechanisms and routes of information transmission to stop disease spread or to minimize the impact of an epidemic. The individual-based simulation is used to better differentiate between the networks of disease and information transmission and to investigate the impact of different basic network topologies and network overlap on epidemic dynamics. The paper concludes with an individual-based semi-analytic calculation of R₀ at the non-trivial disease free equilibrium.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

Inferring Epidemiological Dynamics with Bayesian Coalescent Inference: The Merits of Deterministic and Stochastic Models

Estimation of epidemiological and population parameters from molecular sequence data has become central to the understanding of infectious disease dynamics. Various models have been proposed to infer details of the dynamics that describe epidemic progression. These include inference approaches derived from Kingman’s coalescent theory. Here, we use recently described coalescent theory for epidemic dynamics to develop stochastic and deterministic coalescent susceptible–infected–removed (SIR) tree priors. We implement these in a Bayesian phylogenetic inference framework to permit joint estimation of SIR epidemic parameters and the sample genealogy. We assess the performance of the two coalescent models and also juxtapose results obtained with a recently published birth–death-sampling model for epidemic inference. Comparisons are made by analyzing sets of genealogies simulated under precisely known epidemiological parameters. Additionally, we analyze influenza A (H1N1) sequence data sampled in the Canterbury region of New Zealand and HIV-1 sequence data obtained from known United Kingdom infection clusters. We show that both coalescent SIR models are effective at estimating epidemiological parameters from data with large fundamental reproductive number R₀ and large population size S₀. Furthermore, we find that the stochastic variant generally outperforms its deterministic counterpart in terms of error, bias, and highest posterior density coverage, particularly for smaller R₀ and S₀. However, each of these inference models is shown to have undesirable properties in certain circumstances, especially for epidemic outbreaks with R₀ close to one or with small effective susceptible populations.     

A Class of Pairwise Models for Epidemic Dynamics on Weighted Networks

In this paper, we study the SIS (susceptible–infected–susceptible) and SIR (susceptible–infected–removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with stochastic network simulations to evaluate the impact of different weight distributions on epidemic thresholds and dynamics in general. For the SIR model, the basic reproductive ratio R ₀ is computed, and we show that (i) for both network models R ₀ is maximised if all weights are equal, and (ii) when the two models are ‘equally-matched’, the networks with a random weight distribution give rise to a higher R ₀ value. The models with different weight distributions are also used to explore the agreement between the pairwise and simulation models for different parameter combinations.     

The impact of network clustering and assortativity on epidemic behaviour

Epidemic models have successfully included many aspects of the complex contact structure apparent in real-world populations. However, it is difficult to accommodate variations in the number of contacts, clustering coefficient and assortativity. Investigations of the relationship between these properties and epidemic behaviour have led to inconsistent conclusions and have not accounted for their interrelationship. In this study, simulation is used to estimate the impact of social network structure on the probability of an SIR (susceptible-infective-removed) epidemic occurring and, if it does, the final size. Increases in assortativity and clustering coefficient are associated with smaller epidemics and the impact is cumulative. Derived values of the basic reproduction ratio (R₀) over networks with the highest property values are more than 20% lower than those derived from simulations with zero values of these network properties.     

The basic reproduction number and the probability of extinction for a dynamic epidemic model

We consider the spread of an epidemic through a population divided into n sub-populations, in which individuals move between populations according to a Markov transition matrix Σ and infectives can only make infectious contacts with members of their current population. Expressions for the basic reproduction number, R₀, and the probability of extinction of the epidemic are derived. It is shown that in contrast to contact distribution models, the distribution of the infectious period effects both the basic reproduction number and the probability of extinction of the epidemic in the limit as the total population size N → ∞. The interactions between the infectious period distribution and the transition matrix Σ mean that it is not possible to draw general conclusions about the effects on R₀ and the probability of extinction. However, it is shown that for n = 2, the basic reproduction number, R₀, is maximised by a constant length infectious period and is decreasing in ?, the speed of movement between the two populations.     

Susceptible-infected-removed epidemic models with dynamic partnerships

The author extends the classical, stochastic, Susceptible-Infected-Removed (SIR) epidemic model to allow for disease transmission through a dynamic network of partnerships. A new method of analysis allows for a fairly complete understanding of the dynamics of the system for small and large time. The key insight is to analyze the model by tracking the configurations of all possible dyads, rather than individuals. For large populations, the initial dynamics are approximated by a branching process whose threshold for growth determines the epidemic threshold, R ₀, and whose growth rate, , determines the rate at which the number of cases increases. The fraction of the population that is ever infected, , is shown to bear the same relationship to R ₀ as in models without partnerships. Explicit formulas for these three fundamental quantities are obtained for the simplest version of the model, in which the population is treated as homogeneous, and all transitions are Markov. The formulas allow a modeler to determine the error introduced by the usual assumption of instantaneous contacts for any particular set of biological and sociological parameters. The model and the formulas are then generalized to allow for non-Markov partnership dynamics, non-uniform contact rates within partnerships, and variable infectivity. The model and the method of analysis could also be further generalized to allow for demographic effects, recurrent susceptibility and heterogeneous populations, using the same strategies that have been developed for models without partnerships.     

Contact-based transmission models in terrestrial gastropod populations infected with parasitic mites

Parasite transmission fundamentally affects the epidemiology of host-parasite systems, and is considered to be a key element in the epidemiological modelling of infectious diseases. Recent research has stressed the importance of detailed disease-specific variables involved in the transmission process. Riccardoella limacum is a hematophagous mite living in the mantle cavity of terrestrial gastropods. In this study, we experimentally examined whether the transmission success of R. limacum is affected by the contact frequency, parasite load and/or behaviour of the land snail Arianta arbustorum, a common host of R. limacum. In the experiment the transmission success was mainly affected by physical contacts among snails and slightly influenced by parasite intensity of the infected snail. Using these results we developed two different transmission models based on contact frequencies and transmission probability among host snails. As parameters for the models we used life-history data from three natural A. arbustorum populations with different population densities. Data on contact frequencies of video-recorded snail groups were used to fit the density response of the contact function, assuming either a linear relationship (model 1) or a second-degree polynomial relationship based on the ideal gas model of animal encounter (model 2). We calculated transmission coefficients (β), basic reproductive ratios (R₀) and host threshold population densities for parasite persistence in the three A. arbustorum populations. We found higher transmission coefficients (β) and larger R₀-values in model 1 than in model 2. Furthermore, the host population with the highest density showed larger R₀-values (16.47-22.59) compared to populations with intermediate (2.71-7.45) or low population density (0.75-4.10). Host threshold population density for parasite persistence ranged from 0.35 to 2.72 snails per m². Our results show that the integration of the disease-relevant biology of the organisms concerned may improve models of host-parasite dynamics.     

Modeling influenza seasonality in the tropics and subtropics

Climate drivers such as humidity and temperature may play a key role in influenza seasonal transmission dynamics. Such a relationship has been well defined for temperate regions. However, to date no models capable of capturing the diverse seasonal pattern in tropical and subtropical climates exist. In addition, multiple influenza viruses could cocirculate and shape epidemic dynamics. Here we construct seven mechanistic epidemic models to test the effect of two major climate drivers (humidity and temperature) and multi-strain co-circulation on influenza transmission in Hong Kong, an influenza epidemic center located in the subtropics. Based on model fit to long-term influenza surveillance data from 1998 to 2018, we found that a simple model incorporating the effect of both humidity and temperature best recreated the influenza epidemic patterns observed in Hong Kong. The model quantifies a bimodal effect of absolute humidity on influenza transmission where both low and very high humidity levels facilitate transmission quadratically; the model also quantifies the monotonic but nonlinear relationship with temperature. In addition, model results suggest that, at the population level, a shorter immunity period can approximate the co-circulation of influenza virus (sub)types. The basic reproductive number R₀ estimated by the best-fit model is also consistent with laboratory influenza survival and transmission studies under various combinations of humidity and temperature levels. Overall, our study has developed a simple mechanistic model capable of quantifying the impact of climate drivers on influenza transmission in (sub)tropical regions. This model can be applied to improve influenza forecasting in the (sub)tropics in the future.     

Physiological analysis on oscillatory behavior of glucose-insulin regulation by model with delays

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R₀, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R₀ is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.     

Networks, epidemics and vaccination through contact tracing

We consider a (social) network whose structure can be represented by a simple random graph having a pre-specified degree distribution. A Markovian susceptible-infectious-removed (SIR) epidemic model is defined on such a social graph. We then consider two real-time vaccination models for contact tracing during the early stages of an epidemic outbreak. The first model considers vaccination of each friend of an infectious individual (once identified) independently with probability ρ. The second model is related to the first model but also sets a bound on the maximum number an infectious individual can infect before being identified. Expressions are derived for the influence on the reproduction number of these vaccination models. We give some numerical examples and simulation results based on the Poisson and heavy-tail degree distributions where it is shown that the second vaccination model has a bigger advantage compared to the first model for the heavy-tail degree distribution.     

From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province

The global rise in the use of methamphetamine has been documented to have reached epidemic proportions. Researchers have focussed on the social implications of the epidemic. A typical drug use cycle consists of concealed drugs use after initiation, addiction, treatment-recovery-relapse cycle, whose dynamics are not well understood. The model by White and Comiskey [41], on heroin epidemics, treatment and ODE modelling, is modified to model the dynamics of methamphetamine use in a South African province. The analysis of the model is presented in terms of the methamphetamine epidemic threshold R₀. It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation where a stable drug free equilibrium co-exists with a stable drug persistent equilibrium for a certain defined range of R₀. The stabilities of the model equilibria are ascertained and persistence conditions established. Furthermore, numerical simulations are performed; these include fitting the model to the available data on the number of patients with methamphetamine problems. The implications of the results to drug policy, treatment and prevention are discussed.     

A model of wild bee populations accounting for spatial heterogeneity and climate‐induced temporal variability of food resources at the landscape level

The viability of wild bee populations and the pollination services that they provide are driven by the availability of food resources during their activity period and within the surroundings of their nesting sites. Changes in climate and land use influence the availability of these resources and are major threats to declining bee populations. Because wild bees may be vulnerable to interactions between these threats, spatially explicit models of population dynamics that capture how bee populations jointly respond to land use at a landscape scale and weather are needed. Here, we developed a spatially and temporally explicit theoretical model of wild bee populations aiming for a middle ground between the existing mapping of visitation rates using foraging equations and more refined agent‐based modeling. The model is developed for Bombus sp. and captures within‐season colony dynamics. The model describes mechanistically foraging at the colony level and temporal population dynamics for an average colony at the landscape level. Stages in population dynamics are temperature‐dependent triggered by a theoretical generalized seasonal progression, which can be informed by growing degree days. The purpose of the LandscapePhenoBee model is to evaluate the impact of system changes and within‐season variability in resources on bee population sizes and crop visitation rates. In a simulation study, we used the model to evaluate the impact of the shortage of food resources in the landscape arising from extreme drought events in different types of landscapes (ranging from different proportions of semi‐natural habitats and early and late flowering crops) on bumblebee populations.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Using Multi-Compartment Ensemble Modeling As an Investigative Tool of Spatially Distributed Biophysical Balances: Application to Hippocampal Oriens-Lacunosum/Moleculare (O-LM) Cells

Multi-compartmental models of neurons provide insight into the complex, integrative properties of dendrites. Because it is not feasible to experimentally determine the exact density and kinetics of each channel type in every neuronal compartment, an essential goal in developing models is to help characterize these properties. To address biological variability inherent in a given neuronal type, there has been a shift away from using hand-tuned models towards using ensembles or populations of models. In collectively capturing a neuron''s output, ensemble modeling approaches uncover important conductance balances that control neuronal dynamics. However, conductances are never entirely known for a given neuron class in terms of its types, densities, kinetics and distributions. Thus, any multi-compartment model will always be incomplete. In this work, our main goal is to use ensemble modeling as an investigative tool of a neuron''s biophysical balances, where the cycling between experiment and model is a design criterion from the start. We consider oriens-lacunosum/moleculare (O-LM) interneurons, a prominent interneuron subtype that plays an essential gating role of information flow in hippocampus. O-LM cells express the hyperpolarization-activated current (I _h). Although dendritic I _h could have a major influence on the integrative properties of O-LM cells, the compartmental distribution of I _h on O-LM dendrites is not known. Using a high-performance computing cluster, we generated a database of models that included those with or without dendritic I _h. A range of conductance values for nine different conductance types were used, and different morphologies explored. Models were quantified and ranked based on minimal error compared to a dataset of O-LM cell electrophysiological properties. Co-regulatory balances between conductances were revealed, two of which were dependent on the presence of dendritic I _h. These findings inform future experiments that differentiate between somatic and dendritic I _h, thereby continuing a cycle between model and experiment.     

On the evolution of demographic strategies in populations with equilibrium and cyclic densities

Optimal demographic strategies (life histories of l_x-m_x schedules) for populations with stable and cylicly varying densities are predicted on the basis of a density dependent population dynamics model; the cyclic density pattern—often seen in lemmings and voles —is assumed to have relatively slow density increase followed by an abrupt density crash. Analysis shows that low reproductive rates for all ages and a Type I or II survival schedule will be favored by natural selection in stable populations. High reproductive rates for all ages and a Type III survival schedule will be favored by natural selection in cyclic populations. These strategies may be regarded as the endpoints of a continuum. Features predicted to be optimal for opportunistic species are expected to be more pronounced the more hostile the environment is. Since all quantities used in formulating these predictions are operationally defined, the predicted patterns are testable. Predictions derived from the model are compared, in a general way, with available data. This comparison suggests the validity of the theory.     

Forecasting the 2001 Foot-and-Mouth Disease Epidemic in the UK

Near real-time epidemic forecasting approaches are needed to respond to the increasing number of infectious disease outbreaks. In this paper, we retrospectively assess the performance of simple phenomenological models that incorporate early sub-exponential growth dynamics to generate short-term forecasts of the 2001 foot-and-mouth disease epidemic in the UK. For this purpose, we employed the generalized-growth model (GGM) for pre-peak predictions and the generalized-Richards model (GRM) for post-peak predictions. The epidemic exhibits a growth-decelerating pattern as the relative growth rate declines inversely with time. The uncertainty of the parameter estimates \( (r{\text{ and }}p) \) narrows down and becomes more precise using an increasing amount of data of the epidemic growth phase. Indeed, using only the first 10–15 days of the epidemic, the scaling of growth parameter (p) displays wide uncertainty with the confidence interval for p ranging from values ~ 0.5 to 1.0, indicating that less than 15 epidemic days of data are not sufficient to discriminate between sub-exponential (i.e., p < 1) and exponential growth dynamics (i.e., p = 1). By contrast, using 20, 25, or 30 days of epidemic data, it is possible to recover estimates of p around 0.6 and the confidence interval is substantially below the exponential growth regime. Local and national bans on the movement of livestock and a nationwide cull of infected and contiguous premises likely contributed to the decelerating trajectory of the epidemic. The GGM and GRM provided useful 10-day forecasts of the epidemic before and after the peak of the epidemic, respectively. Short-term forecasts improved as the model was calibrated with an increasing length of the epidemic growth phase. Phenomenological models incorporating generalized-growth dynamics are useful tools to generate short-term forecasts of epidemic growth in near real time, particularly in the context of limited epidemiological data as well as information about transmission mechanisms and the effects of control interventions.     

Effects of kin-structure on disease dynamics in raccoons (Procyon lotor) inhabiting a fragmented landscape

The spatial aggregation of related individuals can affect disease dynamics by altering contact rates between individuals and/or groups of individuals. However, kin-structure likely affects contact rates in a scale-dependent manner: increasing contact rates within, but decreasing contact rates amongst spatial groups. Thus, the effect of kin-structure on disease exposure risk should depend upon the spatial scale relevant for pathogen transmission. This study was undertaken to test the effects of kin-structure on disease dynamics in raccoons (Procyon lotor) inhabiting a highly fragmented agricultural ecosystem. Raccoons (n = 566) were trapped at 31 spatially discrete habitat patches (local populations). Trapped individuals were genotyped at 13 microsatellites and their serum screened for antibodies to canine distemper virus (CDV) and multiple Leptospira spp. serovars. Our analyses revealed that the effects of kin-structure on transmission dynamics are dependent on both spatial scale and the pathogen under consideration. Thus, in the directly transmitted pathogen (CDV), the probability of pathogen co-exposure (the proportion of individual pairs where both individuals were seropositive for a particular pathogen) was positively correlated with genealogical proximity within local populations. Conversely, in the environmentally transmitted leptospires co-exposure was homogeneous within local populations and was positively correlated with gene flow among these populations. Our analyses also revealed that, after controlling for confounding effects of landscape and demographic variables, the effects of kin-structure on disease exposure risk was pathogen-specific. In particular, increasing kin-structure was associated with increased disease exposure risk in the directly transmitted pathogen (CDV), and alternatively with decreased disease exposure risk in environmentally transmitted leptospires. The potential, scale-dependent, effects of kin-structure on disease exposure risk revealed by our analyses could play an important role in elucidating disease dynamics in natural populations and in planning effective disease mitigation strategies.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.