Dynamics of a feline retrovirus (FeLV) in host populations with variable spatial structure.

The predictions of epidemic models are remarkably affected by the underlying assumptions concerning host population dynamics and the relation between host density and disease transmission. Furthermore, hypotheses underlying distinct models are rarely tested. Domestic cats (Felis catus) can be used to compare models and test their predictions, because cat populations show variable spatial structure that probably results in variability in the relation between density and disease transmission. Cat populations also exhibit various dynamics. We compare four epidemiological models of Feline Leukaemia Virus (FeLV). We use two different incidence terms, i.e. proportionate mixing and pseudo-mass action. Population dynamics are modelled as logistic or exponential growth. Compared with proportionate mixing, mass action incidence with logistic growth results in a threshold population size under which the virus cannot persist in the population. Exponential growth of host populations results in systems where FeLV persistence at a steady prevalence and depression of host population growth are biologically unlikely to occur. Predictions of our models account for presently available data on FeLV dynamics in various populations of cats. Thus, host population dynamics and spatial structure can be determinant parameters in parasite transmission, host population depression, and disease control.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

Evolving public perceptions and stability in vaccine uptake

Recent vaccine scares and sudden spikes in vaccine demand remind us that the effectiveness of mass vaccination programs is governed by the public perception of vaccination. Previous work has shown that the tendency of individuals to optimize self-interest can lead to vaccination levels that are suboptimal for a community. We use game theory to relate population-level demand for vaccines to decision-making by individuals with varied beliefs about the costs of infection and vaccination. In contrast to previous work proposing that universal vaccination is impossible in a game theoretic context, we show that optimal individual behavior can vary between universal vaccination and no vaccination, depending on the relative costs and benefits to individuals. By coupling game models and epidemic models, we demonstrate that the pursuit of self-interest often leads to stable dynamics but can lead to oscillations in vaccine uptake over time. The instability is exacerbated in populations that are more homogeneous with respect to their perceptions of vaccine and infection risks. This research illustrates the importance of applying temporal models to an inherently temporal situation, namely, the time evolution of vaccine coverage in an informed population with a voluntary vaccination policy.     

Empirical determinants of measles metapopulation dynamics in England and Wales.

A key issue in metapopulation dynamics is the relative impact of internal patch dynamics and coupling between patches. This problem can be addressed by analysing large spatiotemporal data sets, recording the local and global dynamics of metapopulations. In this paper, we analyse the dynamics of measles meta-populations in a large spatiotemporal case notification data set, collected during the pre-vaccination era in England and Wales. Specifically, we use generalized linear statistical models to quantify the relative importance of local influences (birth rate and population size) and regional coupling on local epidemic dynamics. Apart from the proportional effect of local population size on case totals, the models indicate patterns of local and regional dynamic influences which depend on the current state of epidemics. Birth rate and geographic coupling are not associated with the size of major epidemics. By contrast, minor epidemics--and especially the incidence of local extinction of infection--are influenced both by birth rate and geographical coupling. Birth rate at a lag of four years provides the best fit, reflecting the delayed recruitment of susceptibles to school cohorts. A hierarchical index of spatial coupling to large centres provides the best spatial model. The model also indicates that minor epidemics and extinction patterns are more strongly influenced by this regional effect than the local impact of birth rate.     

Phase-Coherence Transitions and Communication in the Gamma Range between Delay-Coupled Neuronal Populations

Synchronization between neuronal populations plays an important role in information transmission between brain areas. In particular, collective oscillations emerging from the synchronized activity of thousands of neurons can increase the functional connectivity between neural assemblies by coherently coordinating their phases. This synchrony of neuronal activity can take place within a cortical patch or between different cortical regions. While short-range interactions between neurons involve just a few milliseconds, communication through long-range projections between different regions could take up to tens of milliseconds. How these heterogeneous transmission delays affect communication between neuronal populations is not well known. To address this question, we have studied the dynamics of two bidirectionally delayed-coupled neuronal populations using conductance-based spiking models, examining how different synaptic delays give rise to in-phase/anti-phase transitions at particular frequencies within the gamma range, and how this behavior is related to the phase coherence between the two populations at different frequencies. We have used spectral analysis and information theory to quantify the information exchanged between the two networks. For different transmission delays between the two coupled populations, we analyze how the local field potential and multi-unit activity calculated from one population convey information in response to a set of external inputs applied to the other population. The results confirm that zero-lag synchronization maximizes information transmission, although out-of-phase synchronization allows for efficient communication provided the coupling delay, the phase lag between the populations, and the frequency of the oscillations are properly matched.     

Invasion and persistence of plant parasites in a spatially structured host population

Spatial structure is of central importance in the dynamics of plant-parasite interactions and is imposed by the growth habit and distribution of host plants and by parasite dispersal which is frequently restricted. To investigate the effects of spatial heterogeneity on the dynamics of plant parasites we introduce a simple model for epidemic development within a spatially structured host population. Here the host population is subdivided into a number of patches which are linked to allow for transmission from one patch to another with the connections defining the spatial structure of the host population. Three key parameters are identified that play a critical role in the ability of the parasite to invade and persist within the host population: the within-patch parasite basic reproductive number which characterises the infection dynamics at the local spatial scale; and the neighbourhood of interaction which describes which patches interact with which and the strength of coupling between patches within the neighbourhood which together characterise the spread of the parasite over larger spatial scales. Using both deterministic and stochastic formulations of the model, we investigate how the thresholds and probabilities of invasion and persistence are affected by these parameters, by demographic stochasticity and by differences in the initial level of infection.     

Spatial models of virus-immune dynamics

To date, the majority of theoretical models describing the dynamics of infectious diseases in vivo are based on the assumption of well-mixed virus and cell populations. Because many infections take place in solid tissues, spatially structured models represent an important step forward in understanding what happens when the assumption of well-mixed populations is relaxed. Here, we explore models of virus and virus-immune dynamics where dispersal of virus and immune effector cells was constrained to occur locally. The stability properties of our spatial virus-immune dynamics models remained robust under almost all biologically plausible dispersal schemes, regardless of their complexity. The various spatial dynamics were compared to the basic non-spatial dynamics and important differences were identified: When space was assumed to be homogeneous, the dynamics generated by non-spatial and spatially structured models differed substantially at the peak of the infection. Thus, non-spatial models may lead to systematic errors in the estimates of parameters underlying acute infection dynamics. When space was assumed to be heterogeneous, spatial coupling not only changed the equilibrium properties of the uncoupled populations but also equalized the dynamics and thereby reduced the likelihood of dynamic elimination of the infection. In line with experimental and clinical observations, long-lasting oscillation periods were virtually absent. When source-sink dynamics were considered, the long-term outcome of the infection depended critically on the degree of spatial coupling. The infection collapsed when emigration from source sites became too large. Finally, we discuss the implications of spatially structured models on medical treatment of infectious diseases, and note that a huge gap exists in data accurately describing infection dynamics in solid tissues.     

Morphologically-structured models of growing budding yeast populations

It has been well recognized that many key aspects of cell cycle regulation are encoded into the size distributions of growing budding yeast populations due to the tight coupling between cell growth and cell division present in this organism. Several attempts have been made to model the cell size distribution of growing yeast populations in order to obtain insight on the underlying control mechanisms, but most were based on the age structure of asymmetrically dividing populations. Here we propose a new framework that couples a morphologically-structured representation of the population with population balance theory to formulate a dynamic model for the size distribution of growing yeast populations. An advantage of the presented framework is that it allows derivation of simpler models that are directly identifiable from experiments. We show how such models can be derived from the general framework and demonstrate their utility in analyzing yeast population data. Finally, by employing a recently proposed numerical scheme, we proceed to integrate numerically the full distributed model to provide predictions of dynamics of the cell size structure of growing yeast populations.     

A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration

The drive to understand the invasion, spread and fade out of infectious disease in structured populations has produced a variety of mathematical models for pathogen dynamics in metapopulations. Very rarely are these models fully coupled, by which we mean that the spread of an infection within a subpopulation affects the transmission between subpopulations and vice versa. It is also rare that these models are accessible to biologists, in the sense that all parameters have a clear biological meaning and the biological assumptions are explained. Here we present an accessible model that is fully coupled without being an individual-based model. We use the model to show that the duration of an epidemic has a highly non-linear relationship with the movement rate between subpopulations, with a peak in epidemic duration appearing at small movement rates and a global maximum at large movement rates. Intuitively, the first peak is due to asynchrony in the dynamics of infection between subpopulations; we confirm this intuition and also show the peak coincides with successful invasion of the infection into most subpopulations. The global maximum at relatively large movement rates occurs because then the infectious agent perceives the metapopulation as if it is a single well-mixed population wherein the effective population size is greater than the critical community size.     

Impact of periodic nutrient input rate on trophic chain properties

Marine ecosystems are characterized by a strong influence of hydrodynamics on biological processes. The associated models involve the coupling of physical to biological models and therefore require a large number of state variables. The consequent high complexity limits our capacity to perform a complete and detailed study and even prevents any complete mathematical study of these models. It is also difficult to disentangle among all the processes involved, which ones actually drive the system at any moment. Hydrodynamics, among other consequences, affect the way under which the nutrients are supplied to marine ecosystems. The variability of nutrient input rate in marine systems generally results from runs-off in coastal systems and from physical processes (wind forcing and hydrodynamics) in open ocean. This paper is devoted to the study of the effects of the nutrient input rate variability on the dynamics and the functioning of trophic chains. In this context, we aim to provide an understandable study, based on simplified system models. We consider a periodic nutrient input rate and analyze how this variability modifies some system properties: its dynamics, its functioning and its structure. The dynamics is obtained by numerical simulations and when possible, enlighten by already published mathematical results. The functioning is measured by the time averaged state variables during the simulation period, and their variability. The structure concerns the number of surviving populations, a proxy of specific biodiversity. We show how these properties can be affected and provide some conditions under which the modifications can occur. We also highlight that, even if the physical process is the main driving force in the global dynamics, the choice of the biological model is important to understand the biological response of the system to physical forcing.     

Dynamics of stochastic epidemics on heterogeneous networks

Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease prevalence fully, meaning that the skewness of the degree distribution affects the variance of the prevalence of infection. We compare these asymptotic results to simulation and find a close agreement for city-sized populations.     

Measles on the edge: coastal heterogeneities and infection dynamics

Mathematical models can help elucidate the spatio-temporal dynamics of epidemics as well as the impact of control measures. The gravity model for directly transmitted diseases is currently one of the most parsimonious models for spatial epidemic spread. This model uses distance-weighted, population size-dependent coupling to estimate host movement and disease incidence in metapopulations. The model captures overall measles dynamics in terms of underlying human movement in pre-vaccination England and Wales (previously established). In spatial models, edges often present a special challenge. Therefore, to test the model's robustness, we analyzed gravity model incidence predictions for coastal cities in England and Wales. Results show that, although predictions are accurate for inland towns, they significantly underestimate coastal persistence. We examine incidence, outbreak seasonality, and public transportation records, to show that the model's inaccuracies stem from an underestimation of total contacts per individual along the coast. We rescue this predicted 'edge effect' by increasing coastal contacts to approximate the number of per capita inland contacts. These results illustrate the impact of 'edge effects' on epidemic metapopulations in general and illustrate directions for the refinement of spatiotemporal epidemic models.     

Impact of the Infection Period Distribution on the Epidemic Spread in a Metapopulation Model

Epidemic models usually rely on the assumption of exponentially distributed sojourn times in infectious states. This is sometimes an acceptable approximation, but it is generally not realistic and it may influence the epidemic dynamics as it has already been shown in one population. Here, we explore the consequences of choosing constant or gamma-distributed infectious periods in a metapopulation context. For two coupled populations, we show that the probability of generating no secondary infections is the largest for most parameter values if the infectious period follows an exponential distribution, and we identify special cases where, inversely, the infection is more prone to extinction in early phases for constant infection durations. The impact of the infection duration distribution on the epidemic dynamics of many connected populations is studied by simulation and sensitivity analysis, taking into account the potential interactions with other factors. The analysis based on the average nonextinct epidemic trajectories shows that their sensitivity to the assumption on the infectious period distribution mostly depends on , the mean infection duration and the network structure. This study shows that the effect of assuming exponential distribution for infection periods instead of more realistic distributions varies with respect to the output of interest and to other factors. Ultimately it highlights the risk of misleading recommendations based on modelling results when models including exponential infection durations are used for practical purposes.     

Epidemic growth rate and household reproduction number in communities of households,schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.     

Asymmetry in the Presence of Migration Stabilizes Multistrain Disease Outbreaks

We study the effect of migration between coupled populations, or patches, on the stability properties of multistrain disease dynamics. The epidemic model used in this work displays a Hopf bifurcation to oscillations in a single, well-mixed population. It is shown numerically that migration between two non-identical patches stabilizes the endemic steady state, delaying the onset of large amplitude outbreaks and reducing the total number of infections. This result is motivated by analyzing generic Hopf bifurcations with different frequencies and with diffusive coupling between them. Stabilization of the steady state is again seen, indicating that our observation in the full multistrain model is based on qualitative characteristics of the dynamics rather than on details of the disease model.     

Scaling properties and symmetrical patterns in the epidemiology of rotavirus infection

The rich epidemiological database of the incidence of rotavirus, as a cause of severe diarrhoea in young children, coupled with knowledge of the natural history of the infection, can make this virus a paradigm for studies of epidemic dynamics. The cyclic recurrence of childhood rotavirus epidemics in unvaccinated populations provides one of the best documented phenomena in population dynamics. This paper makes use of epidemiological data on rotavirus infection in young children admitted to hospital in Melbourne, Australia from 1977 to 2000. Several mathematical methods were used to characterize the overall dynamics of rotavirus infections as a whole and individually as serotypes G1, G2, G3, G4 and G9. These mathematical methods are as follows: seasonal autoregressive integrated moving-average (SARIMA) models, power spectral density (PSD), higher-order spectral analysis (HOSA) (bispectrum estimation and quadratic phase coupling (QPC)), detrended fluctuation analysis (DFA), wavelet analysis (WA) and a surrogate data analysis technique. Each of these techniques revealed different dynamic aspects of rotavirus epidemiology. In particular, we confirm the existence of an annual, biannual and a quinquennial period but additionally we found other embedded cycles (e.g. ca. 3 years). There seems to be an overall unique geometric and dynamic structure of the data despite the apparent changes in the dynamics of the last years. The inherent dynamics seems to be conserved regardless of the emergence of new serotypes, the re-emergence of old serotypes or the transient disappearance of a particular serotype. More importantly, the dynamics of all serotypes is multiple synchronized so that they behave as a single entity at the epidemic level. Overall, the whole dynamics follow a scale-free power-law fractal scaling behaviour. We found that there are three different scaling regions in the time-series, suggesting that processes influencing the epidemic dynamics of rotavirus over less than 12 months differ from those that operate between 1 and ca. 3 years, as well as those between 3 and ca. 5 years. To discard the possibility that the observed patterns could be due to artefacts, we applied a surrogate data analysis technique which enabled us to discern if only random components or linear features of the incidence of rotavirus contribute to its dynamics. The global dynamics of the epidemic is portrayed by wavelet-based incidence analysis. The resulting wavelet transform of the incidence of rotavirus crisply reveals a repeating pattern over time that looks similar on many scales (a property called self-similarity). Both the self-similar behaviour and the absence of a single characteristic scale of the power-law fractal-like scaling of the incidence of rotavirus infection imply that there is not a universal inherently more virulent serotype to which severe gastroenteritis can uniquely be ascribed.     

Exact epidemic models on graphs using graph-automorphism driven lumping

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.     

Effects of tick population dynamics and host densities on the persistence of tick-borne infections

The transmission and the persistence of tick-borne infections are strongly influenced by the densities and the structure of host populations. By extending previous models and analysis, in this paper we analyse how the persistence of ticks and pathogens, is affected by the dynamics of tick populations, and by their host densities. The effect of host densities on infection persistence is explored through the analysis and simulation of a series of models that include different assumptions on tick-host dynamics and consider different routes of infection transmission. Ticks are assumed to feed on two types of host species which vary in their reservoir competence. Too low densities of competent hosts (i.e., hosts where transmission can occur) do not sustain the infection cycle, while too high densities of incompetent hosts may dilute the competent hosts so much to make infection persistence impossible. A dilution effect may occur also for competent hosts as a consequence of reduced tick to host ratio; this is possible only if the regulation of tick populations is such that tick density does not increase linearly with host densities.     

High host density favors greater virulence: a model of parasite–host dynamics based on multi-type branching processes

We use a multitype continuous time Markov branching process model to describe the dynamics of the spread of parasites of two types that can mutate into each other in a common host population. While most mathematical models for the virulence of infectious diseases focus on the interplay between the dynamics of host populations and the optimal characteristics for the success of the pathogen, our model focuses on how pathogen characteristics may change at the start of an epidemic, before the density of susceptible hosts decline. We envisage animal husbandry situations where hosts are at very high density and epidemics are curtailed before host densities are much reduced. The use of three pathogen characteristics: lethality, transmissibility and mutability allows us to investigate the interplay of these in relation to host density. We provide some numerical illustrations and discuss the effects of the size of the enclosure containing the host population on the encounter rate in our model that plays the key role in determining what pathogen type will eventually prevail. We also present a multistage extension of the model to situations where there are several populations and parasites can be transmitted from one of them to another. We conclude that animal husbandry situations with high stock densities will lead to very rapid increases in virulence, where virulent strains are either more transmissible or favoured by mutation. Further the process is affected by the nature of the farm enclosures.