RESUMEN
The dynamics that govern disease spread are hard to model because infections are functions of both the underlying pathogen as well as human or animal behavior. This challenge is increased when modeling how diseases spread between different spatial locations. Many proposed spatial epidemiological models require trade-offs to fit, either by abstracting away theoretical spread dynamics, fitting a deterministic model, or by requiring large computational resources for many simulations. We propose an approach that approximates the complex spatial spread dynamics with a Gaussian process. We first propose a flexible spatial extension to the well-known SIR stochastic process, and then we derive a moment-closure approximation to this stochastic process. This moment-closure approximation yields ordinary differential equations for the evolution of the means and covariances of the susceptibles and infectious through time. Because these ODEs are a bottleneck to fitting our model by MCMC, we approximate them using a low-rank emulator. This approximation serves as the basis for our hierarchical model for noisy, underreported counts of new infections by spatial location and time. We demonstrate using our model to conduct inference on simulated infections from the underlying, true spatial SIR jump process. We then apply our method to model counts of new Zika infections in Brazil from late 2015 through early 2016.
Asunto(s)
Simulación por Computador , Procesos Estocásticos , Infección por el Virus Zika , Humanos , Distribución Normal , Infección por el Virus Zika/epidemiología , Infección por el Virus Zika/transmisión , Modelos Epidemiológicos , Modelos Estadísticos , Cadenas de MarkovRESUMEN
AbstractInfection intensity can dictate disease outcomes but is typically ignored when modeling infection dynamics of microparasites (e.g., bacteria, virus, and fungi). However, for a number of pathogens of wildlife typically categorized as microparasites, accounting for infection intensity and within-host infection processes is critical for predicting population-level responses to pathogen invasion. Here, we develop a modeling framework we refer to as reduced-dimension host-parasite integral projection models (reduced IPMs) that we use to explore how within-host infection processes affect the dynamics of pathogen invasion and virulence evolution. We find that individual-level heterogeneity in pathogen load-a nearly ubiquitous characteristic of host-parasite interactions that is rarely considered in models of microparasites-generally reduces pathogen invasion probability and dampens virulence-transmission trade-offs in host-parasite systems. The latter effect likely contributes to widely predicted virulence-transmission trade-offs being difficult to observe empirically. Moreover, our analyses show that intensity-dependent host mortality does not always induce a virulence-transmission trade-off, and systems with steeper than linear relationships between pathogen intensity and host mortality rate are significantly more likely to exhibit these trade-offs. Overall, reduced IPMs provide a useful framework to expand our theoretical and data-driven understanding of how within-host processes affect population-level disease dynamics.
Asunto(s)
Interacciones Huésped-Patógeno , Parásitos , Animales , Interacciones Huésped-Parásitos , Dinámica Poblacional , VirulenciaRESUMEN
Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here, we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.
Asunto(s)
Enfermedades Transmisibles , Epidemias , Enfermedades Transmisibles/epidemiología , Humanos , Conceptos Matemáticos , Modelos Biológicos , Procesos EstocásticosRESUMEN
Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the k-regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for k-regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation.
Asunto(s)
Enfermedades Transmisibles , Epidemias , Enfermedades de Transmisión Sexual , Enfermedades Transmisibles/epidemiología , Simulación por Computador , Susceptibilidad a Enfermedades , Humanos , Modelos Biológicos , Enfermedades de Transmisión Sexual/epidemiología , Procesos EstocásticosRESUMEN
Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.
Asunto(s)
Evolución Biológica , Modelos Biológicos , Mutación/genética , Probabilidad , Procesos EstocásticosRESUMEN
Heterogeneity plays an important role in the emergence, persistence and control of infectious diseases. Metapopulation models are often used to describe spatial heterogeneity, and the transition from random- to heterogeneous-mixing is made by incorporating the interaction, or coupling, within and between subpopulations. However, such couplings are difficult to measure explicitly; instead, their action through the correlations between subpopulations is often all that can be observed. We use moment-closure methods to investigate how the coupling and resulting correlation are related, considering systems of multiple identical interacting populations on highly symmetric complex networks: the complete network, the k-regular tree network, and the star network. We show that the correlation between the prevalence of infection takes a relatively simple form and can be written in terms of the coupling, network parameters and epidemiological parameters only. These results provide insight into the effect of metapopulation network structure on endemic disease dynamics, and suggest that detailed case-reporting data alone may be sufficient to infer the strength of between population interaction and hence lead to more accurate mathematical descriptions of infectious disease behaviour.
Asunto(s)
Enfermedades Transmisibles/epidemiología , Enfermedades Endémicas , Dinámica Poblacional , Humanos , Cadenas de Markov , Modelos Biológicos , Análisis Numérico Asistido por Computador , Procesos EstocásticosRESUMEN
Many chemical reaction networks in biological systems present complex oscillatory dynamics. In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Although stochastic models based on the probabilistic approach of the Chemical Master Equation (CME) have been proposed, studies in the literature have been limited by the challenges of solving the CME and the lack of computational power to perform large-scale stochastic simulations. In this paper, we show that the infinite set of stationary moment equations describing the stochastic Brusselator and Schnakenberg oscillatory reactions networks can be truncated and solved using maximization of the entropy of the distributions. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods and suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system. We conclude that maximum entropy models can be used as an efficient closure scheme alternative for moment equations to predict the non-equilibrium stationary distributions of stochastic chemical reactions with oscillatory dynamics. This prediction is accomplished without any prior knowledge of the system dynamics and without imposing any biased assumptions on the mathematical relations among species involved.
RESUMEN
Gastrointestinal nematodes represent important sources of economic losses in farmed ruminants, and the increasing frequency of anthelmintic resistance requires an increased ability to explore alternative strategies. Theoretical approaches at the crossroads of immunology and epidemiology are valuable tools in that context. In the case of Teladorsagia circumcincta in sheep, the immunological mechanisms important for resistance are increasingly well-characterized. However, despite the existence of a wide range of theoretical models, there is no framework integrating the characteristic features of this immune response into a tractable phenomenological model. Here, we propose to bridge that gap by developing a flexible modelling framework that allows for variability in nematode larval intake which can be used to track the variations in worm burdens. We parameterize this model using data from trickle infection of sheep and show that using simple immunological assumptions, our model can capture the dynamics of both adult worm burdens and nematode fecal egg counts. In addition, our analysis reveals interesting dose-dependent effects on the immune response. Finally, we discuss potential developments of this model and highlight how an improved cross-talk between empiricists and theoreticians would facilitate important advances in the study of infectious diseases.
Asunto(s)
Modelos Biológicos , Infecciones por Nematodos/veterinaria , Ostertagia/inmunología , Ostertagia/parasitología , Enfermedades de las Ovejas/inmunología , Animales , Heces/parasitología , Infecciones por Nematodos/epidemiología , Infecciones por Nematodos/inmunología , Infecciones por Nematodos/parasitología , Recuento de Huevos de Parásitos , Carga de Parásitos , Ovinos , Enfermedades de las Ovejas/epidemiología , Enfermedades de las Ovejas/parasitologíaRESUMEN
Mathematical models describing the movement of multiple interacting subpopulations are relevant to many biological and ecological processes. Standard mean-field partial differential equation descriptions of these processes suffer from the limitation that they implicitly neglect to incorporate the impact of spatial correlations and clustering. To overcome this, we derive a moment dynamics description of a discrete stochastic process which describes the spreading of distinct interacting subpopulations. In particular, we motivate our model by mimicking the geometry of two typical cell biology experiments. Comparing the performance of the moment dynamics model with a traditional mean-field model confirms that the moment dynamics approach always outperforms the traditional mean-field approach. To provide more general insight we summarise the performance of the moment dynamics model and the traditional mean-field model over a wide range of parameter regimes. These results help distinguish between those situations where spatial correlation effects are sufficiently strong, such that a moment dynamics model is required, from other situations where spatial correlation effects are sufficiently weak, such that a traditional mean-field model is adequate.
Asunto(s)
Comunicación Celular , Movimiento Celular , Modelos Biológicos , Técnicas de Cocultivo , Dermis/citología , Células Endoteliales/citología , Fibroblastos/citología , Humanos , Reproducibilidad de los ResultadosRESUMEN
Typical mutation-selection models assume well-mixed populations, but dispersal and migration within many natural populations is spatially limited. Such limitations can lead to enhanced variation among locations as different types become clustered in different places. Such clustering weakens competition between unlike types relative to competition between like types; thus, the rate by which a fitter type displaces an inferior competitor can be affected by the spatial scale of movement. In this paper, we use a birth-death model to show that limited migration can affect asexual populations by creating competitive refugia. We use a moment closure approach to show that as population structure is introduced by limiting migration, the equilibrial frequency of deleterious mutants increases. We support and extend the model through stochastic simulation, and we use a spatially explicit cellular automaton approach to corroborate the results. We discuss the implications of these results for standing variation in structured populations and adaptive valley crossing in Wright's "shifting balance" process.
Asunto(s)
Mutación , Selección Genética , Modelos Teóricos , Procesos EstocásticosRESUMEN
In this paper we elucidate how small-scale movements, such as those associated with searching for food and avoiding predators, affect the stability of predator-prey dynamics. We investigate an individual-based Lotka-Volterra model with density-dependent movement, in which the predator and prey populations live in a very large number of coupled patches. The rates at which individuals leave patches depend on the local densities of heterospecifics, giving rise to one reaction norm for each of the two species. Movement rates are assumed to be much faster than demographics rates. A spatial structure of predators and prey emerges which affects the global population dynamics. We derive a criterion which reveals how demographic stability depends on the relationships between the per capita covariance and densities of predators and prey. Specifically, we establish that a positive relationship with prey density and a negative relationship with predator density tend to be stabilizing. On a more mechanistic level we show how these relationships are linked to the movement reaction norms of predators and prey. Numerical results show that these findings hold both for local and global movements, i.e., both when migration is biased towards neighbouring patches and when all patches are reached with equal probability.
Asunto(s)
Migración Animal/fisiología , Conducta Predatoria/fisiología , Animales , Densidad de Población , Especificidad de la EspecieRESUMEN
Previous mathematical models of spatial farm-to-farm transmission of foot and mouth disease (FMD) have explored the impacts of control measures such as culling and vaccination during a single outbreak in a country normally free of FMD. As a result, these models do not include factors that are relevant to countries where FMD is endemic in some regions, like long-term waning natural and vaccine immunity, use of prophylactic vaccination and disease re-importations. These factors may have implications for disease dynamics and control, yet few models have been developed for FMD-endemic settings. Here we develop and study an SEIRV (susceptible-exposed-infectious-recovered-vaccinated) pair approximation model of FMD. We focus on long term dynamics by exploring characteristics of repeated outbreaks of FMD and their dependence on disease re-importation, loss of natural immunity, and vaccine waning. We find that the effectiveness of ring and prophylactic vaccination strongly depends on duration of natural immunity, rate of vaccine waning, and disease re-introduction rate. However, the number and magnitude of FMD outbreaks are generally more sensitive to the duration of natural immunity than the duration of vaccine immunity. If loss of natural immunity and/or vaccine waning happen rapidly, then multiple epidemic outbreaks result, making it difficult to eliminate the disease. Prophylactic vaccination is more effective than ring vaccination, at the same per capita vaccination rate. Finally, more frequent disease re-importation causes a higher cumulative number of infections, although a lower average epidemic peak. Our analysis demonstrates significant differences between dynamics in FMD-free settings versus FMD-endemic settings, and that dynamics in FMD-endemic settings can vary widely depending on factors such as the duration of natural and vaccine immunity and the rate of disease re-importations. We conclude that more mathematical models tailored to FMD-endemic countries should be developed that include these factors.
Asunto(s)
Control de Enfermedades Transmisibles , Transmisión de Enfermedad Infecciosa , Enfermedades Endémicas , Fiebre Aftosa/epidemiología , Modelos Biológicos , AnimalesRESUMEN
Population growth is a fundamental process in ecology and evolution. The population size dynamics during growth are often described by deterministic equations derived from kinetic models. Here, we simulate several population growth models and compare the size averaged over many stochastic realizations with the deterministic predictions. We show that these deterministic equations are generically bad predictors of the average stochastic population dynamics. Specifically, deterministic predictions overestimate the simulated population sizes, especially those of populations starting with a small number of individuals. Describing population growth as a stochastic birth process, we prove that the discrepancy between deterministic predictions and simulated data is due to unclosed-moment dynamics. In other words, the deterministic approach does not consider the variability of birth times, which is particularly important with small population sizes. We show that some moment-closure approximations describe the growth dynamics better than the deterministic prediction. However, they do not reduce the error satisfactorily and only apply to some population growth models. We explicitly solve the stochastic growth dynamics, and our solution applies to any population growth model. We show that our solution exactly quantifies the dynamics of a community composed of different strains and correctly predicts the fixation probability of a strain in a serial dilution experiment. Our work sets the foundations for a more faithful modeling of community and population dynamics. It will allow the development of new tools for a more accurate analysis of experimental and empirical results, including the inference of important growth parameters.
RESUMEN
Many pathogens spread via environmental transmission, without requiring host-to-host direct contact. While models for environmental transmission exist, many are simply constructed intuitively with structures analogous to standard models for direct transmission. As model insights are generally sensitive to the underlying model assumptions, it is important that we are able understand the details and consequences of these assumptions. We construct a simple network model for an environmentally-transmitted pathogen and rigorously derive systems of ordinary differential equations (ODEs) based on different assumptions. We explore two key assumptions, namely homogeneity and independence, and demonstrate that relaxing these assumptions can lead to more accurate ODE approximations. We compare these ODE models to a stochastic implementation of the network model over a variety of parameters and network structures, demonstrating that with fewer restrictive assumptions we are able to achieve higher accuracy in our approximations and highlighting more precisely the errors produced by each assumption. We show that less restrictive assumptions lead to more complicated systems of ODEs and the potential for unstable solutions. Due to the rigour of our derivation, we are able to identify the reason behind these errors and propose potential resolutions.
Asunto(s)
Enfermedades Transmisibles , Microbiología Ambiental , Modelos Biológicos , Enfermedades Transmisibles/transmisiónRESUMEN
Stochastic chemical kinetics at the single-cell level give rise to heterogeneous populations of cells even when all individuals are genetically identical. This heterogeneity can lead to nonuniform behaviour within populations, including different growth characteristics, cell-fate dynamics, and response to stimuli. Ultimately, these diverse behaviours lead to intricate population dynamics that are inherently multiscale: the population composition evolves based on population-level processes that interact with stochastically distributed single-cell states. Therefore, descriptions that account for this heterogeneity are essential to accurately model and control chemical processes. However, for real-world systems such models are computationally expensive to simulate, which can make optimisation problems, such as optimal control or parameter inference, prohibitively challenging. Here, we consider a class of multiscale population models that incorporate population-level mechanisms while remaining faithful to the underlying stochasticity at the single-cell level and the interplay between these two scales. To address the complexity, we study an order-reduction approximations based on the distribution moments. Since previous moment-closure work has focused on the single-cell kinetics, extending these techniques to populations models prompts us to revisit old observations as well as tackle new challenges. In this extended multiscale context, we encounter the previously established observation that the simplest closure techniques can lead to non-physical system trajectories. Despite their poor performance in some systems, we provide an example where these simple closures outperform more sophisticated closure methods in accurately, efficiently, and robustly solving the problem of optimal control of bioproduction in a microbial consortium model.
Asunto(s)
Modelos Biológicos , Simulación por Computador , Humanos , Dinámica Poblacional , Procesos EstocásticosRESUMEN
Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.
Asunto(s)
Algoritmos , Simulación por Computador , Método de Montecarlo , Procesos Estocásticos , IncertidumbreRESUMEN
During infection, the rates of pathogen replication, death, and migration affect disease progression, dissemination, transmission, and resistance evolution. Here, we follow the population dynamics of Vibrio cholerae in a mouse model by labeling individual bacteria with one of >500 unique, fitness-neutral genomic tags. Using the changes in tag frequencies and CFU numbers, we inform a mathematical model that describes the within-host spatiotemporal bacterial dynamics. This allows us to disentangle growth, death, forward, and retrograde migration rates continuously during infection. Our model has robust predictive power across various experimental setups. The population dynamics of V. cholerae shows substantial spatiotemporal heterogeneity in replication, death, and migration. Importantly, we find that the niche available to V. cholerae in the host increases with inoculum size, suggesting cooperative effects during infection. Therefore, it is not enough to consider just the likelihood of exposure (50% infectious dose) but rather the magnitude of exposure to predict outbreaks. IMPORTANCE Determining the rates of bacterial migration, replication, and death during infection is important for understanding how infections progress. Separately measuring these rates is often difficult in systems where multiple processes happen simultaneously. Here, we use next-generation sequencing to measure V. cholerae migration, replication, death, and niche size along the mouse gastrointestinal tract. We show that the small intestine of the mouse is a heterogeneous environment, and the population dynamic characteristics change substantially between adjacent gut sections. Our approach also allows us to characterize the effect of inoculum size on these processes. We find that the niche size in mice increases with the infectious dose, hinting at cooperative effects in larger inocula. The dose-response relationship between inoculum size and final pathogen burden is important for the infected individual and is thought to influence the progression of V. cholerae epidemics.
RESUMEN
In this paper, we establish the explicit connection between deterministic trait-based population-level models (in the form of partial differential equations) and species-level models (in the form of ordinary differential equations), in the context of eco-evolutionary systems. In particular, by starting from a population-level model of density distributions in trait space, we derive what amounts to an extension of the typical models at the species level known from adaptive dynamics literature, to account not only for abundance and mean trait values, but also explicitly for trait variances. Thus, we arrive at an explicitly polymorphic model at the species level. The derivations make precise the relationship between the parameters in the two classes of models and allow us to distinguish between notions of fitness on the population and species levels. Through a formal stability analysis, we see that exponential growth of an eigenvalue in the trait covariance matrix corresponds to a breakdown of the underlying assumptions of the species-level model. In biological terms, this may be interpreted as a speciation event: that is, we obtain an explicit notion of the blow-up of the variance of (possibly a linear combination of) traits as a precursor to speciation. Moreover, since evolutionary volatility of the mean trait value is proportional to trait variance, this provides a notion that species at the cusp of speciation are also the most adaptive. We illustrate these concepts and considerations using a numerical simulation.
RESUMEN
It is increasingly apparent that heterogeneity in the interaction between individuals plays an important role in the dynamics, persistence, evolution and control of infectious diseases. In epidemic modelling two main forms of heterogeneity are commonly considered: spatial heterogeneity due to the segregation of populations and heterogeneity in risk at the same location. The transition from random-mixing to heterogeneous-mixing models is made by incorporating the interaction, or coupling, within and between subpopulations. However, such couplings are difficult to measure explicitly; instead, their action through the correlations between subpopulations is often all that can be observed. Here, using moment-closure methodology supported by stochastic simulation, we investigate how the coupling and resulting correlation are related. We focus on the simplest case of interactions, two identical coupled populations, and show that for a wide range of parameters the correlation between the prevalence of infection takes a relatively simple form. In particular, the correlation can be approximated by a logistic function of the between population coupling, with the free parameter determined analytically from the epidemiological parameters. These results suggest that detailed case-reporting data alone may be sufficient to infer the strength of between population interaction and hence lead to more accurate mathematical descriptions of infectious disease behaviour.
Asunto(s)
Enfermedades Transmisibles/epidemiología , Modelos Biológicos , Epidemias , Humanos , Cadenas de Markov , Procesos EstocásticosRESUMEN
The spread of an infectious disease may depend on the structure of the network. To study the influence of the structure parameters of the network on the spread of the epidemic, we need to put these parameters into the epidemic model. The method of moment closure introduces structure parameters into the epidemic model. In this paper, we present a new moment closure epidemic model based on the approximation of third-order motifs in networks. The order of a motif defined in this paper is determined by the number of the edges in the motif, rather than by the number of nodes in the motif as defined in the literature. We provide a general approach to deriving a set of ordinary differential equations that describes, to a high degree of accuracy, the spread of an infectious disease. Using this method, we establish a susceptible-infected-recovered (SIR) model. We then calculate the basic reproduction number of the SIR model, and find that it decreases as the clustering coefficient increases. Finally, we perform some simulations using the proposed model to study the influence of the clustering coefficient on the final epidemic size, the maximum number of infected, and the peak time of the disease. The numerical simulations based on the SIR model in this paper fit the stochastic simulations based on the Monte Carlo method well at different levels of clustering. Our results show that the clustering coefficient poses impediments to the spread of disease under an SIR model.