Non-linear transmission rates and the dynamics of infectious disease

Viral transmission and infection prevalence in a cannibalistic host-pathogen system

Cannibalism, while prevalent in the natural world, is often viewed as detrimental to a cannibal's health, especially when they consume pathogen-infected conspecifics. The argument stems from the idea that cannibalizing infected individuals increases the chance of coming into contact with a pathogen and subsequently becoming infected. Using an insect pest, the fall armyworm (Spodoptera frugiperda), that readily cannibalizes at the larval stage and its lethal pathogen, we experimentally examined how cannibalism affects viral transmission at both an individual and population level. Prior to death, the pathogen in the system stops the larval host from growing, resulting in infected individuals being smaller than healthy individuals. This leads to size-structured cannibalism of infected individuals with the larger healthy larvae consuming the smaller infected larvae, which is commonly observed. At the individual level, we show that the probability of cannibalism is relatively high for both infected and uninfected individuals especially when the cannibal is larger than the victim. However, the probability of the cannibal becoming infected given that a pathogen-infected individual has been cannibalized is relatively low. On a population level, when cannibalism is allowed to occur transmission rates decline. Additionally, by cannibalizing infected larvae, cannibals lower the infection risk for non-cannibals. Thus, cannibalism can decrease infection prevalence and, therefore, may not be as deleterious as once thought. Under certain circumstances, cannibalizing infected individuals, from the uninfected host's perspective, may even be advantageous, as one obtains a meal and decreases competition for resources with little chance of becoming infected.

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and L e ´ vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold λ by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.

Heterozygosity of the major histocompatibility complex predicts later self-reported pubertal maturation in men

Individual variation in the age of pubertal onset is linked to physical and mental health, yet the factors underlying this variation are poorly understood. Life history theory predicts that individuals at higher risk of mortality due to extrinsic causes such as infectious disease should sexually mature and reproduce earlier, whereas those at lower risk can delay puberty and continue to invest resources in somatic growth. We examined relationships between a genetic predictor of infectious disease resistance, heterozygosity of the major histocompatibility complex (MHC), referred to as the human leukocyte antigen (HLA) gene in humans, and self-reported pubertal timing. In a combined sample of men from Canada (n = 137) and the United States (n = 43), MHC heterozygosity predicted later self-reported pubertal development. These findings suggest a genetic trade-off between immunocompetence and sexual maturation in human males.

Parasite evolution in an age-structured population

Although mortality increases with age in most organisms, senescence is missing from models of parasite evolution. Since virulence evolves according to the host's mortality, and since virulence influences the intensity of transmission, which determines the average age at infection and thus the mortality rate of a senescing host, we expected that epi-evolutionary feedbacks would underlie the evolution of virulence in a population of senescing hosts. We tested this idea by extending an age-structured model of epidemiological dynamics with the parasite's evolution. A straightforward prediction of our model is that stronger senescence forces the evolution of higher virulence. However, the model also reveals that the evolved virulence depends on the average age at infection, giving an evolutionary feedback with the epidemiological situation, a prediction not found when assuming a constant mortality rate with age. Additionally, and in contrast to most models of parasite evolution, we found that the virulence at the evolutionary equilibrium is influenced by whether the force of infection depends on the density or on the frequency of infected hosts, due to changes in the average age at infection. Our findings suggest that ignoring age-specific effects, and in particular senescence, can give misleading predictions about parasite evolution.

An epidemic model for non-first-order transmission kinetics

Compartmental models in epidemiology characterize the spread of an infectious disease by formulating ordinary differential equations to quantify the rate of disease progression through subpopulations defined by the Susceptible-Infectious-Removed (SIR) scheme. The classic rate law central to the SIR compartmental models assumes that the rate of transmission is first order regarding the infectious agent. The current study demonstrates that this assumption does not always hold and provides a theoretical rationale for a more general rate law, inspired by mixed-order chemical reaction kinetics, leading to a modified mathematical model for non-first-order kinetics. Using observed data from 127 countries during the initial phase of the COVID-19 pandemic, we demonstrated that the modified epidemic model is more realistic than the classic, first-order-kinetics based model. We discuss two coefficients associated with the modified epidemic model: transmission rate constant k and transmission reaction order n. While k finds utility in evaluating the effectiveness of control measures due to its responsiveness to external factors, n is more closely related to the intrinsic properties of the epidemic agent, including reproductive ability. The rate law for the modified compartmental SIR model is generally applicable to mixed-kinetics disease transmission with heterogeneous transmission mechanisms. By analyzing early-stage epidemic data, this modified epidemic model may be instrumental in providing timely insight into a new epidemic and developing control measures at the beginning of an outbreak.

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.

Comparing the Indirect Effects between Exploiters in Predator-Prey and Host-Pathogen Systems

AbstractIn multipredator and multipathogen systems, exploiters interact indirectly via shared victim species. Interspecific prey competition and the degree of predator specialization are known to influence whether predators have competitive (i.e., (-,-)) or noncompetitive (i.e., (-,+) or (+,+)) indirect interactions. Much less is known about the population-level indirect interactions between pathogens that infect the same populations of host species. In this study, we use two-predator-two-prey and two-host-two-pathogen models to compare the indirect effects between predators with the indirect effects between pathogens. We focus on how the indirect interactions between pathogens are affected by the competitive abilities of susceptible and infected hosts, whether the pathogens are specialists or generalists, and the transmission pathway (direct vs. environmental transmission). In many cases, indirect effects between pathogens and predators follow similar patterns, for example, more positive indirect effects with increased interspecific competition between victim species. However, the indirect effects between pathogens can qualitatively differ, for example, more negative indirect effects with increased interspecific host competition. These contrasting patterns show that an important mechanistic difference between predatory and parasitic interactions (specifically, whether interactions are immediately lethal) can have important population-level effects on the indirect interactions between exploiters.

A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy

The ongoing Coronavirus Disease 2019 (COVID-19) pandemic threatens the health of humans and causes great economic losses. Predictive modeling and forecasting the epidemic trends are essential for developing countermeasures to mitigate this pandemic. We develop a network model, where each node represents an individual and the edges represent contacts between individuals where the infection can spread. The individuals are classified based on the number of contacts they have each day (their node degrees) and their infection status. The transmission network model was respectively fitted to the reported data for the COVID-19 epidemic in Wuhan (China), Toronto (Canada), and the Italian Republic using a Markov Chain Monte Carlo (MCMC) optimization algorithm. Our model fits all three regions well with narrow confidence intervals and could be adapted to simulate other megacities or regions. The model projections on the role of containment strategies can help inform public health authorities to plan control measures.

Intraguild predation decreases predator fitness with potentially varying effects on pathogen transmission in a herbivore host

Predators and pathogens often regulate the population dynamics of their prey or hosts. When species interact with both their predators and their pathogens, understanding each interaction in isolation may not capture the system's dynamics. For instance, predators can influence pathogen transmission via consumptive effects, such as feeding on infected prey, or non-consumptive effects, such as changing the prey's susceptibility to infection. A prey species' infection status can, in turn, influence predator's choice of prey and have negative fitness consequences for the predator. To test how intraguild predation (IGP), when predator and pathogen share the same prey/host, affects pathogen transmission, predator preference, and predator fitness, we conducted a series of experiments using a crop pest (Pseudoplusia includens), a generalist predator (Podisus maculiventris), and a generalist pathogen (Autographa californica multicapsid nuclear polyhedrovirus, AcMNPV). Using a field experiment, we quantified the effects of consumptive and non-consumptive predators on pathogen transmission. We found that a number of models provided similar fits to the data. These models included null models showing no effects of predation and models that included a predation effect. We also found that predators consumed infected prey more often when choosing between live infected or live healthy prey. Infected prey also reduced predator fitness. Developmental times of predators fed infected prey increased by 20% and longevity decreased by 45%, compared with those that consumed an equivalent number of non-infected prey. While this research shows an effect of the pathogen on intraguild predator fitness, we found no support that predators affected pathogen transmission.

Separate seasons of infection and reproduction can lead to multi-year population cycles

Many host-pathogen systems are characterized by a temporal order of disease transmission and host reproduction. For example, this can be due to pathogens infecting certain life cycle stages of insect hosts; transmission occurring during the aggregation of migratory birds; or plant diseases spreading between planting seasons. We develop a simple discrete-time epidemic model with density-dependent transmission and disease affecting host fecundity and survival. The model shows sustained multi-annual cycles in host population abundance and disease prevalence, both in the presence and absence of density dependence in host reproduction, for large horizontal transmissibility, imperfect vertical transmission, high virulence, and high reproductive capability. The multi-annual cycles emerge as invariant curves in a Neimark-Sacker bifurcation. They are caused by a carry-over effect, because the reproductive fitness of an individual can be reduced by virulent effects due to infection in an earlier season. As the infection process is density-dependent but shows an effect only in a later season, this produces delayed density dependence typical for second-order oscillations. The temporal separation between the infection and reproduction season is crucial in driving the cycles; if these processes occur simultaneously as in differential equation models, there are no sustained oscillations. Our model highlights the destabilizing effects of inter-seasonal feedbacks and is one of the simplest epidemic models that can generate population cycles.

A stochastic switched SIRS epidemic model with nonlinear incidence and vaccination: Stationary distribution and extinction

In this paper, a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases. Nonlinear incidence and vaccination strategies are also considered in the proposed model. By using the method of stochastic analysis, we point out the key parameters that determine the persistence and extinction of the diseases. Specifically, if [Formula: see text] is greater than 0, the stochastic system has a unique ergodic stationary distribution; while if [Formula: see text] is less than 0, the diseases will be extinct at an exponential rate.

Modelling evolution of virulence in populations with a distributed parasite load

Modelling evolution of virulence in host-parasite systems is an actively developing area of research with ever-growing literature. However, most of the existing studies overlook the fact that individuals within an infected population may have a variable infection load, i.e. infected populations are naturally structured with respect to the parasite burden. Empirical data suggests that the mortality and infectiousness of individuals can strongly depend on their infection load; moreover, the shape of distribution of infection load may vary on ecological and evolutionary time scales. Here we show that distributed infection load may have important consequences for the eventual evolution of virulence as compared to a similar model without structuring. Mathematically, we consider an SI model, where the dynamics of the infected subpopulation is described by a von Förster-type equation, in which the infection load plays the role of age. We implement the adaptive dynamics framework to predict evolutionary outcomes in this model. We demonstrate that for simple trade-off functions between virulence, disease transmission and parasite growth rates, multiple evolutionary attractors are possible. Interestingly, unlike in the case of unstructured models, achieving an evolutionary stable strategy becomes possible even for a variation of a single ecological parameter (the parasite growth rate) and keeping the other parameters constant. We conclude that evolution in disease-structured populations is strongly mediated by alterations in the overall shape of the parasite load distribution.

Genotypic variation in parasite avoidance behaviour and other mechanistic, nonlinear components of transmission

Traditional epidemiological models assume that transmission increases proportionally to the density of parasites. However, empirical data frequently contradict this assumption. General yet mechanistic models can explain why transmission depends nonlinearly on parasite density and thereby identify potential defensive strategies of hosts. For example, hosts could decrease their exposure rates at higher parasite densities (via behavioural avoidance) or decrease their per-parasite susceptibility when encountering more parasites (e.g. via stronger immune responses). To illustrate, we fitted mechanistic transmission models to 19 genotypes of Daphnia dentifera hosts over gradients of the trophically acquired parasite, Metschnikowia bicuspidata. Exposure rate (foraging, F) frequently decreased with parasite density (Z), and per-parasite susceptibility (U) frequently decreased with parasite encounters (F × Z). Consequently, infection rates (F × U × Z) often peaked at intermediate parasite densities. Moreover, host genotypes varied substantially in these responses. Exposure rates remained constant for some genotypes but decreased sensitively with parasite density for others (up to 78%). Furthermore, genotypes with more sensitive foraging/exposure also foraged faster in the absence of parasites (suggesting 'fast and sensitive' versus 'slow and steady' strategies). These relationships suggest that high densities of parasites can inhibit transmission by decreasing exposure rates and/or per-parasite susceptibility, and identify several intriguing axes for the evolution of host defence.

The human-snail transmission environment shapes long term schistosomiasis control outcomes: Implications for improving the accuracy of predictive modeling

INTRODUCTION

Schistosomiasis is a chronic parasitic trematode disease that affects over 240 million people worldwide. The Schistosoma lifecycle is complex, involving transmission via specific intermediate-host freshwater snails. Predictive mathematical models of Schistosoma transmission have often chosen to simplify or ignore the details of environmental human-snail interaction in their analyses. Schistosome transmission models now aim to provide better precision for policy planning of elimination of transmission. This heightens the importance of including the environmental complexity of vector-pathogen interaction in order to make more accurate projections.

METHODOLOGY AND PRINCIPAL FINDINGS

We propose a nonlinear snail force of infection (FOI) that takes into account an intermediate larval stage (miracidium) and snail biology. We focused, in particular, on the effects of snail force of infection (FOI) on the impact of mass drug administration (MDA) in human communities. The proposed (modified) model was compared to a conventional model in terms of their predictions. A longitudinal dataset generated in Kenya field studies was used for model calibration and validation. For each sample community, we calibrated modified and conventional model systems, then used them to model outcomes for a range of MDA regimens. In most cases, the modified model predicted more vigorous post-MDA rebound, with faster relapse to baseline levels of infection. The effect was pronounced in higher risk communities. When compared to observed data, only the modified system was able to successfully predict persistent rebound of Schistosoma infection.

CONCLUSION AND SIGNIFICANCE

The observed impact of varying location-specific snail inputs sheds light on the diverse MDA response patterns noted in operational research on schistosomiasis control, such as the recent SCORE project. Efficiency of human-to-snail transmission is likely to be much higher than predicted by standard models, which, in practice, will make local elimination by implementation of MDA alone highly unlikely, even over a multi-decade period.

Experimental investigation of alternative transmission functions: Quantitative evidence for the importance of nonlinear transmission dynamics in host-parasite systems

Understanding pathogen transmission is crucial for predicting and managing disease. Nonetheless, experimental comparisons of alternative functional forms of transmission remain rare, and those experiments that are conducted are often not designed to test the full range of possible forms.

To differentiate among 10 candidate transmission functions, we used a novel experimental design in which we independently varied four factors—duration of exposure, numbers of parasites, numbers of hosts and parasite density—in laboratory infection experiments.

We used interactions between amphibian hosts and trematode parasites as a model system and all candidate models incorporated parasite depletion. An additional manipulation involving anaesthesia addressed the effects of host behaviour on transmission form.

Across all experiments, nonlinear transmission forms involving either a power law or a negative binomial function were the best‐fitting models and consistently outperformed the linear density‐dependent and density‐independent functions. By testing previously published data for two other host–macroparasite systems, we also found support for the same nonlinear transmission forms.

Although manipulations of parasite density are common in transmission studies, the comprehensive set of variables tested in our experiments revealed that variation in density alone was least likely to differentiate among competing transmission functions. Across host–pathogen systems, nonlinear functions may often more accurately represent transmission dynamics and thus provide more realistic predictions for infection.

Breaking beta: deconstructing the parasite transmission function

Transmission is a fundamental step in the life cycle of every parasite but it is also one of the most challenging processes to model and quantify. In most host–parasite models, the transmission process is encapsulated by a single parameter β. Many different biological processes and interactions, acting on both hosts and infectious organisms, are subsumed in this single term. There are, however, at least two undesirable consequences of this high level of abstraction. First, nonlinearities and heterogeneities that can be critical to the dynamic behaviour of infections are poorly represented; second, estimating the transmission coefficient β from field data is often very difficult. In this paper, we present a conceptual model, which breaks the transmission process into its component parts. This deconstruction enables us to identify circumstances that generate nonlinearities in transmission, with potential implications for emergent transmission behaviour at individual and population scales. Such behaviour cannot be explained by the traditional linear transmission frameworks. The deconstruction also provides a clearer link to the empirical estimation of key components of transmission and enables the construction of flexible models that produce a unified understanding of the spread of both micro- and macro-parasite infectious disease agents.

This article is part of the themed issue ‘Opening the black box: re-examining the ecology and evolution of parasite transmission’.

Mate Limitation in Fungal Plant Parasites Can Lead to Cyclic Epidemics in Perennial Host Populations

Fungal plant parasites represent a growing concern for biodiversity and food security. Most ascomycete species are capable of producing different types of infectious spores both asexually and sexually. Yet the contributions of both types of spores to epidemiological dynamics have still to been fully researched. Here we studied the effect of mate limitation in parasites which perform both sexual and asexual reproduction in the same host. Since mate limitation implies positive density dependence at low population density, we modeled the dynamics of such species with both density-dependent (sexual) and density-independent (asexual) transmission rates. A first simple SIR model incorporating these two types of transmission from the infected compartment, suggested that combining sexual and asexual spore production can generate persistently cyclic epidemics in a significant part of the parameter space. It was then confirmed that cyclic persistence could occur in realistic situations by parameterizing a more detailed model fitting the biology of the Black Sigatoka disease of banana, for which literature data are available. We discuss the implications of these results for research on and management of Sigatoka diseases of banana.

Recalibrating disease parameters for increasing realism in modeling epidemics in closed settings

Background

The homogeneous mixing assumption is widely adopted in epidemic modelling for its parsimony and represents the building block of more complex approaches, including very detailed agent-based models. The latter assume homogeneous mixing within schools, workplaces and households, mostly for the lack of detailed information on human contact behaviour within these settings. The recent data availability on high-resolution face-to-face interactions makes it now possible to assess the goodness of this simplified scheme in reproducing relevant aspects of the infection dynamics.

Methods

We consider empirical contact networks gathered in different contexts, as well as synthetic data obtained through realistic models of contacts in structured populations. We perform stochastic spreading simulations on these contact networks and in populations of the same size under a homogeneous mixing hypothesis. We adjust the epidemiological parameters of the latter in order to fit the prevalence curve of the contact epidemic model. We quantify the agreement by comparing epidemic peak times, peak values, and epidemic sizes.

Results

Good approximations of the peak times and peak values are obtained with the homogeneous mixing approach, with a median relative difference smaller than 20 % in all cases investigated. Accuracy in reproducing the peak time depends on the setting under study, while for the peak value it is independent of the setting. Recalibration is found to be linear in the epidemic parameters used in the contact data simulations, showing changes across empirical settings but robustness across groups and population sizes.

Conclusions

An adequate rescaling of the epidemiological parameters can yield a good agreement between the epidemic curves obtained with a real contact network and a homogeneous mixing approach in a population of the same size. The use of such recalibrated homogeneous mixing approximations would enhance the accuracy and realism of agent-based simulations and limit the intrinsic biases of the homogeneous mixing.

Electronic supplementary material

The online version of this article (doi:10.1186/s12879-016-2003-3) contains supplementary material, which is available to authorized users.

Mathematical models to characterize early epidemic growth: A review

There is a long tradition of using mathematical models to generate insights into the transmission dynamics of infectious diseases and assess the potential impact of different intervention strategies. The increasing use of mathematical models for epidemic forecasting has highlighted the importance of designing reliable models that capture the baseline transmission characteristics of specific pathogens and social contexts. More refined models are needed however, in particular to account for variation in the early growth dynamics of real epidemics and to gain a better understanding of the mechanisms at play. Here, we review recent progress on modeling and characterizing early epidemic growth patterns from infectious disease outbreak data, and survey the types of mathematical formulations that are most useful for capturing a diversity of early epidemic growth profiles, ranging from sub-exponential to exponential growth dynamics. Specifically, we review mathematical models that incorporate spatial details or realistic population mixing structures, including meta-population models, individual-based network models, and simple SIR-type models that incorporate the effects of reactive behavior changes or inhomogeneous mixing. In this process, we also analyze simulation data stemming from detailed large-scale agent-based models previously designed and calibrated to study how realistic social networks and disease transmission characteristics shape early epidemic growth patterns, general transmission dynamics, and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014-2015 Ebola epidemic in West Africa.

Plant Quantity Affects Development and Survival of a Gregarious Insect Herbivore and Its Endoparasitoid Wasp

Virtually all studies of plant-herbivore-natural enemy interactions focus on plant quality as the major constraint on development and survival. However, for many gregarious feeding insect herbivores that feed on small or ephemeral plants, the quantity of resources is much more limiting, yet this area has received virtually no attention. Here, in both lab and semi-field experiments using tents containing variably sized clusters of food plants, we studied the effects of periodic food deprivation in a tri-trophic system where quantitative constraints are profoundly important on insect performance. The large cabbage white Pieris brassicae, is a specialist herbivore of relatively small wild brassicaceous plants that grow in variable densities, with black mustard (Brassica nigra) being one of the most important. Larvae of P. brassicae are in turn attacked by a specialist endoparasitoid wasp, Cotesia glomerata. Increasing the length of food deprivation of newly molted final instar caterpillars significantly decreased herbivore and parasitoid survival and biomass, but shortened their development time. Moreover, the ability of caterpillars to recover when provided with food again was correlated with the length of the food deprivation period. In outdoor tents with natural vegetation, we created conditions similar to those faced by P. brassicae in nature by manipulating plant density. Low densities of B. nigra lead to potential starvation of P. brassicae broods and their parasitoids, replicating nutritional conditions of the lab experiments. The ability of both unparasitized and parasitized caterpillars to find corner plants was similar but decreased with central plant density. Survival of both the herbivore and parasitoid increased with plant density and was higher for unparasitized than for parasitized caterpillars. Our results, in comparison with previous studies, reveal that quantitative constraints are far more important that qualitative constraints on the performance of gregarious insect herbivores and their gregarious parasitoids in nature.

Modeling Heterogeneity in Direct Infectious Disease Transmission in a Compartmental Model

Mathematical models have been used to understand the transmission dynamics of infectious diseases and to assess the impact of intervention strategies. Traditional mathematical models usually assume a homogeneous mixing in the population, which is rarely the case in reality. Here, we construct a new transmission function by using as the probability density function a negative binomial distribution, and we develop a compartmental model using it to model the heterogeneity of contact rates in the population. We explore the transmission dynamics of the developed model using numerical simulations with different parameter settings, which characterize different levels of heterogeneity. The results show that when the reproductive number, R0, is larger than one, a low level of heterogeneity results in dynamics similar to those predicted by the homogeneous mixing model. As the level of heterogeneity increases, the dynamics become more different. As a test case, we calibrated the model with the case incidence data for severe acute respiratory syndrome (SARS) in Beijing in 2003, and the estimated parameters demonstrated the effectiveness of the control measures taken during that period.

The scaling of contact rates with population density for the infectious disease models

Contact rates and patterns among individuals in a geographic area drive transmission of directly-transmitted pathogens, making it essential to understand and estimate contacts for simulation of disease dynamics. Under the uniform mixing assumption, one of two mechanisms is typically used to describe the relation between contact rate and population density: density-dependent or frequency-dependent. Based on existing evidence of population threshold and human mobility patterns, we formulated a spatial contact model to describe the appropriate form of transmission with initial growth at low density and saturation at higher density. We show that the two mechanisms are extreme cases that do not capture real population movement across all scales. Empirical data of human and wildlife diseases indicate that a nonlinear function may work better when looking at the full spectrum of densities. This estimation can be applied to large areas with population mixing in general activities. For crowds with unusually large densities (e.g., transportation terminals, stadiums, or mass gatherings), the lack of organized social contact structure deviates the physical contacts towards a special case of the spatial contact model - the dynamics of kinetic gas molecule collision. In this case, an ideal gas model with van der Waals correction fits well; existing movement observation data and the contact rate between individuals is estimated using kinetic theory. A complete picture of contact rate scaling with population density may help clarify the definition of transmission rates in heterogeneous, large-scale spatial systems.

Distinguishing between indirect and direct modes of transmission using epidemiological time series

Pathogen transmission can involve direct and/or indirect pathways. Using theoretical models, in this study we ask, "do directly and indirectly transmitted pathogens yield different population-level epidemiological dynamics?" and "can the transmission pathway be inferred from population-level epidemiological data?" Our approach involves comparing the continuous-time dynamics of a class of compartmental epidemiological models with direct versus environmentally mediated indirect transmission pathways. Combing analytical theory and numerical simulations we show that models with direct and indirect transmission can produce quantitatively similar time series when the pathogen cannot reproduce in the environment, particularly when the environmental pathogen dynamics are fast. We apply these results to a previous study on chronic wasting disease and show that identifying the transmission pathway is more difficult than previously acknowledged. Our analysis and simulations also yield conditions under which numerical differences can potentially identify the transmission route in oscillating endemic systems and systems where the environmental pathogen dynamics are not fast. This work begins to identify how differences in the transmission pathway can result in quantitatively different epidemiological dynamics and how those differences can be used to identify the transmission pathway from population level time series.

Modelling and inference for epidemic models featuring non-linear infection pressure

We consider a Susceptible-Infective-Removed (SIR) stochastic epidemic model in which the infection rate is of the form βN⁻¹X(t)Y(t)(α). It is demonstrated that both the threshold behaviour of this model and the behaviour of the corresponding deterministic model differ markedly from the standard SIR model (i.e. α=1). Methods of statistical inference for this model are described, given outbreak data, and the extent to which all three model parameters can be estimated is considered.

Escherichia coli O157 infection on cattle farms: the formulation of the force of infection and its effect on control effectiveness

The kernel of modelling transmission dynamics of infectious diseases lies in constructing the force of infection (FOI). Traditionally, it was based on mass-action law. In this paper, we show, based on survey data of Escherichia coli O157 infection on Scottish cattle farms, that the actual form of FOI deviates greatly from mass-action law. Further, control effectiveness deviates qualitatively: the epidemic of mass-action FOI can be controlled with a control effort larger than the so-called herd immunity, while the epidemic inferred from the survey data of E. coli O157 infection was shown to be difficult to control. This indicates that, at least for E. coli O157 infection on cattle farms, it is risky to rely on models of transmission dynamics that were based on mass-action law to design the optimal intervention programme for infectious diseases. This suggests the importance of collecting epidemic data and model selection from data-driven models to infer the most likely model of transmission dynamics.

Simple models for complex systems: exploiting the relationship between local and global densities

Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.

A rigorous approach to investigating common assumptions about disease transmission: Process algebra as an emerging modelling methodology for epidemiology

Changing scale, for example, the ability to move seamlessly from an individual-based model to a population-based model, is an important problem in many fields. In this paper, we introduce process algebra as a novel solution to this problem in the context of models of infectious disease spread. Process algebra allows us to describe a system in terms of the stochastic behaviour of individuals, and is a technique from computer science. We review the use of process algebra in biological systems, and the variety of quantitative and qualitative analysis techniques available. The analysis illustrated here solves the changing scale problem: from the individual behaviour we can rigorously derive equations to describe the mean behaviour of the system at the level of the population. The biological problem investigated is the transmission of infection, and how this relates to individual interactions.

Random, top-down, or bottom-up coexistence of parasites: malaria population dynamics in multi-parasitic settings

Epidemiological models concerned with the control of malaria using interventions such as bed nets and vaccines increasingly incorporate realistic aspects of malaria biology. The increasing complexity of these models limits their ability to abstract ecological processes and to address questions on the regulation of population dynamics using time-series data, particularly in regards to interactions between different pathogens and the regulatory role of innate (bottom-up) and acquired (top-down) immunity. We use a theoretical framework to test hypotheses on the importance of population-level immunity and parasite abundance in regulating the population dynamics of malaria. We use qualitative loop analyses to examine the sign of the interaction between Plasmodium falciparum and P. vivax at the population level, and we discuss implications of this sign for the within-host regulation of parasites. Our analyses of monthly malaria time-series data from the island of Espirito Santo, Vanuatu (1983-1997), show that the dynamics of P. falciparum are not sensitive to P. vivax, whereas infections by the latter increase in response to those of the former. These results support a differential use of resources inside the hosts, a resource-consumer interaction between hosts and their immune system, and within-host regulation of parasites. Finally, our results emphasize the need to better understand factors regulating malaria dynamics before developing control strategies and call for the use of control strategies directed at the interruption of transmission, such as vector control and the use of bed nets.

Host-pathogen time series data in wildlife support a transmission function between density and frequency dependence

A key aim in epidemiology is to understand how pathogens spread within their host populations. Central to this is an elucidation of a pathogen's transmission dynamics. Mathematical models have generally assumed that either contact rate between hosts is linearly related to host density (density-dependent) or that contact rate is independent of density (frequency-dependent), but attempts to confirm either these or alternative transmission functions have been rare. Here, we fit infection equations to 6 years of data on cowpox virus infection (a zoonotic pathogen) for 4 natural populations to investigate which of these transmission functions is best supported by the data. We utilize a simple reformulation of the traditional transmission equations that greatly aids the estimation of the relationship between density and host contact rate. Our results provide support for an infection rate that is a saturating function of host density. Moreover, we find strong support for seasonality in both the transmission coefficient and the relationship between host contact rate and host density, probably reflecting seasonal variations in social behavior and/or host susceptibility to infection. We find, too, that the identification of an appropriate loss term is a key component in inferring the transmission mechanism. Our study illustrates how time series data of the host-pathogen dynamics, especially of the number of susceptible individuals, can greatly facilitate the fitting of mechanistic disease models.

Climate and recruitment limitation of hosts: the dynamics of American cutaneous leishmaniasis seen through semi-mechanistic seasonal models

Diseases cycle as a response to endogenous and exogenous factors. For infectious diseases caused by vector-transmitted pathogens, the exogenous factors are commonly equated to climatic forces and the endogenous factors to the recruitment of new susceptible individuals. Mathematical models that explicitly (parametrically) consider both types of factor are, however, very rare. An approach is presented to model the effects of endogenous and exogenous factors parametrically, using a time series for American cutaneous leishmaniasis (ACL) from Costa Rica. The seasonality of the disease is modelled using a seasonal autoregressive approach. The latter has the advantage of allowing the use of semi-mechanistic frameworks that consider infection clearance, while explicitly introducing the feedbacks produced by the transition between immune classes, as well as climatic forcing. It also uses a relatively small number of degrees of freedom (compared with the numbers involved in semi-parametric approaches), making it useful for relatively short time series and series with abrupt changes. Compared with non-mechanistic models built for prediction purposes, this way of modelling seems to increase the likelihood of the data being explained by a plausible mechanism. The approach used in this study of ACL could be useful in investigating the changes that occur in other diseases that show non-stationary seasonal dynamics, and can be easily adapted to model the dynamics of other infectious diseases that show trends or breakpoints. The present results support the view that humans affected by ACL are mostly incidental hosts, and indicate that, at the population level, there is a delay of about 5 months between human infection with the causative parasites and the onset of clinical symptoms. They encourage the development of surveillance systems, for monitoring the prevalence of infection in the sandflies that act as vectors, and the use of sentinel hosts, so that control measures can be rapidly applied or strengthened before a serious outbreak occurs. The development of more accurate mathematical models of ACL will depend largely on advances in the ecology of the disease and of all the hosts of the causative parasites.

Host heterogeneity in susceptibility and disease dynamics: tests of a mathematical model

Most mathematical models of disease assume that transmission is linearly dependent on the densities of host and pathogen. Recent data for animal diseases, however, have cast doubt on this assumption, without assessing the usefulness of alternative models. In this article, we use a combination of laboratory dose-response experiments, field transmission experiments, and observations of naturally occurring populations to show that virus transmission in gypsy moths is a nonlinear function of virus density, apparently because of heterogeneity among individual gypsy moth larvae in their susceptibility to the virus. Dose-response experiments showed that larvae from a laboratory colony of gypsy moths are substantially less heterogeneous in their susceptibility to the virus than are larvae from feral populations, and field experiments showed that there is a more strongly nonlinear relationship between transmission and virus density for feral larvae than for lab larvae. This nonlinearity in transmission changes the dynamics of the virus in natural populations so that a model incorporating host heterogeneity in susceptibility to the virus gives a much better fit to data on virus dynamics from large-scale field plots than does a classical model that ignores host heterogeneity. Our results suggest that heterogeneity among individuals has important effects on the dynamics of disease in insects at several spatial and temporal scales and that heterogeneity in susceptibility may be of general importance in the ecology of disease.

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.

Influence of the transmission function on a simulated pathogen spread within a population

The mathematical function for the horizontal transmission of a pathogen is a driving force of epidemiological models. This paper aims at studying the influence of different transmission functions on a simulated pathogen spread. These functions were chosen in the literature and their biological relevance is discussed. A theoretical SIR (Susceptible-Infectious-Recovered) model was used to study the effect of the function used on simulated results. With a constant total population size, different equilibrium values for the number of infectious (NI) were reached, depending on the transmission function used. With an increasing population size, the transmission functions could be assimilated to either density-dependent (DD), where an equilibrium was obtained, or frequency-dependent (FD), with an exponential increase in NI. An analytical study corroborated the simulated results. As a conclusion, the choice between the different transmission functions, particularly between DD and FD, must be carefully considered for a varying population size.

When individual behaviour matters: homogeneous and network models in epidemiology

Heterogeneity in host contact patterns profoundly shapes population-level disease dynamics. Many epidemiological models make simplifying assumptions about the patterns of disease-causing interactions among hosts. In particular, homogeneous-mixing models assume that all hosts have identical rates of disease-causing contacts. In recent years, several network-based approaches have been developed to explicitly model heterogeneity in host contact patterns. Here, we use a network perspective to quantify the extent to which real populations depart from the homogeneous-mixing assumption, in terms of both the underlying network structure and the resulting epidemiological dynamics. We find that human contact patterns are indeed more heterogeneous than assumed by homogeneous-mixing models, but are not as variable as some have speculated. We then evaluate a variety of methodologies for incorporating contact heterogeneity, including network-based models and several modifications to the simple SIR compartmental model. We conclude that the homogeneous-mixing compartmental model is appropriate when host populations are nearly homogeneous, and can be modified effectively for a few classes of non-homogeneous networks. In general, however, network models are more intuitive and accurate for predicting disease spread through heterogeneous host populations.

Density-dependent resistance of the gypsy moth Lymantria dispar to its nucleopolyhedrovirus, and the consequences for population dynamics

The processes controlling disease resistance can strongly influence the population dynamics of insect outbreaks. Evidence that disease resistance is density-dependent is accumulating, but the exact form of this relationship is highly variable from species to species. It has been hypothesized that insects experiencing high population densities might allocate more energy to disease resistance than those at lower densities, because they are more likely to encounter density-dependent pathogens. In contrast, the increased stress of high-density conditions might leave insects more vulnerable to disease. Both scenarios have been reported for various outbreak Lepidoptera in the literature. We tested the relationship between larval density and disease resistance with the gypsy moth (Lymantria dispar) and one of its most important density-dependent mortality factors, the nucleopolyhedrovirus (NPV) LdMNPV, in a series of bioassays. Larvae were reared in groups at different densities, fed the virus individually, and then reared individually to evaluate response to infection. In this system, resistance to the virus decreased with increasing larval density. Similarly, time to death was faster at high densities than at lower densities. Implications of density-resistance relationships for insect-pathogen population dynamics were explored in a mathematical model. In general, an inverse relationship between rearing density and disease resistance has a stabilizing effect on population dynamics.

Eating yourself sick: transmission of disease as a function of foraging ecology

Species interactions may profoundly influence disease outbreaks. However, disease ecology has only begun to integrate interactions between hosts and their food resources (foraging ecology) despite that hosts often encounter their parasites while feeding. A zooplankton-fungal system illustrated this central connection between foraging and transmission. Using experiments that varied food density for Daphnia hosts, density of fungal spores and body size of Daphnia, we produced mechanistic yet general models for disease transmission rate based on broadly applicable components of feeding biology. Best performing models could explain why prevalence of infection declined at high food density and rose sharply as host size increased (a pattern echoed in nature). In comparison, the classic mass-action model for transmission performed quite poorly. These foraging-based models should broadly apply to systems in which hosts encounter parasites while eating, and they will catalyse future integration of the roles of Daphnia as grazer and host.

Effects of species diversity on disease risk

The transmission of infectious diseases is an inherently ecological process involving interactions among at least two, and often many, species. Not surprisingly, then, the species diversity of ecological communities can potentially affect the prevalence of infectious diseases. Although a number of studies have now identified effects of diversity on disease prevalence, the mechanisms underlying these effects remain unclear in many cases. Starting with simple epidemiological models, we describe a suite of mechanisms through which diversity could increase or decrease disease risk, and illustrate the potential applicability of these mechanisms for both vector-borne and non-vector-borne diseases, and for both specialist and generalist pathogens. We review examples of how these mechanisms may operate in specific disease systems. Because the effects of diversity on multi-host disease systems have been the subject of much recent research and controversy, we describe several recent efforts to delineate under what general conditions host diversity should increase or decrease disease prevalence, and illustrate these with examples. Both models and literature reviews suggest that high host diversity is more likely to decrease than increase disease risk. Reduced disease risk with increasing host diversity is especially likely when pathogen transmission is frequency-dependent, and when pathogen transmission is greater within species than between species, particularly when the most competent hosts are also relatively abundant and widespread. We conclude by identifying focal areas for future research, including (1) describing patterns of change in disease risk with changing diversity; (2) identifying the mechanisms responsible for observed changes in risk; (3) clarifying additional mechanisms in a wider range of epidemiological models; and (4) experimentally manipulating disease systems to assess the impact of proposed mechanisms.

On representing network heterogeneities in the incidence rate of simple epidemic models

Mean-field ecological models ignore space and other forms of contact structure. At the opposite extreme, high-dimensional models that are both individual-based and stochastic incorporate the distributed nature of ecological interactions. In between, moment approximations have been proposed that represent the effect of correlations on the dynamics of mean quantities. As an alternative closer to the typical temporal models used in ecology, we present here results on “modified mean-field equations” for infectious disease dynamics, in which only mean quantities are followed and the effect of heterogeneous mixing is incorporated implicitly. We specifically investigate the previously proposed empirical parameterization of heterogeneous mixing in which the bilinear incidence rate SI is replaced by a nonlinear term kS^pI^q, for the case of stochastic SIRS dynamics on different contact networks, from a regular lattice to a random structure via small-world configurations. We show that, for two distinct dynamical cases involving a stable equilibrium and a noisy endemic steady state, the modified mean-field model approximates successfully the steady state dynamics as well as the respective short and long transients of decaying cycles. This result demonstrates that early on in the transients an approximate power-law relationship is established between global (mean) quantities and the covariance structure in the network. The approach fails in the more complex case of persistent cycles observed within the narrow range of small-world configurations.

Modelling pathogen transmission: the interrelationship between local and global approaches

We describe two spatial (cellular automaton) host-pathogen models with contrasting types of transmission, where the biologically realistic transmission mechanisms are based entirely on 'local' interactions. The two models, fixed contact area (FCA) and fixed contact number (FCN), may be viewed as local 'equivalents' of commonly used global density- (and frequency-) dependent models. Their outputs are compared with each other and with the patterns generated by these global terms. In the FCN model, unoccupied cells are bypassed, but in the FCA model these impede pathogen spread, extending the period of the epidemic and reducing the prevalence of infection when the pathogen persists. Crucially, generalized linear modelling reveals that the global transmission terms betaSI and beta'SI/N are equally good at describing transmission in both the FCA and FCN models when infected individuals are homogeneously distributed and N is approximately constant, as at the quasi-equilibrium. However, when N varies, the global frequency-dependent term beta'SI/N is better than the density-dependent one, betaSI, at describing transmission in both the FCA and FCN models. Our approach may be used more generally to compare different local contact structures and select the most appropriate global transmission term.