Inferring the dynamics of a spatial epidemic from time-series data

Estimating the delay between host infection and disease (incubation period) and assessing its significance to the epidemiology of plant diseases

Knowledge of the incubation period of infectious diseases (time between host infection and expression of disease symptoms) is crucial to our epidemiological understanding and the design of appropriate prevention and control policies. Plant diseases cause substantial damage to agricultural and arboricultural systems, but there is still very little information about how the incubation period varies within host populations. In this paper, we focus on the incubation period of soilborne plant pathogens, which are difficult to detect as they spread and infect the hosts underground and above-ground symptoms occur considerably later. We conducted experiments on Rhizoctonia solani in sugar beet, as an example patho-system, and used modelling approaches to estimate the incubation period distribution and demonstrate the impact of differing estimations on our epidemiological understanding of plant diseases. We present measurements of the incubation period obtained in field conditions, fit alternative probability models to the data, and show that the incubation period distribution changes with host age. By simulating spatially-explicit epidemiological models with different incubation-period distributions, we study the conditions for a significant time lag between epidemics of cryptic infection and the associated epidemics of symptomatic disease. We examine the sensitivity of this lag to differing distributional assumptions about the incubation period (i.e. exponential versus Gamma). We demonstrate that accurate information about the incubation period distribution of a pathosystem can be critical in assessing the true scale of pathogen invasion behind early disease symptoms in the field; likewise, it can be central to model-based prediction of epidemic risk and evaluation of disease management strategies. Our results highlight that reliance on observation of disease symptoms can cause significant delay in detection of soil-borne pathogen epidemics and mislead practitioners and epidemiologists about the timing, extent, and viability of disease control measures for limiting economic loss.

Host growth can cause invasive spread of crops by soilborne pathogens

Invasive soilborne plant pathogens cause substantial damage to crops and natural populations, but our understanding of how to prevent their epidemics or reduce their damage is limited. A key and experimentally-tested concept in the epidemiology of soilborne plant diseases is that of a threshold spacing between hosts below which epidemics (invasive spread) can occur. We extend this paradigm by examining how plant-root growth may alter the conditions for occurrence of soilborne pathogen epidemics in plant populations. We hypothesise that host-root growth can 1) increase the probability of pathogen transmission between neighbouring plants and, consequently, 2) decrease the threshold spacing for epidemics to occur. We predict that, in systems initially below their threshold conditions, root growth can trigger soilborne pathogen epidemics through a switch from non-invasive to invasive behaviour, while in systems above threshold conditions root growth can enhance epidemic development. As an example pathosystem, we studied the fungus Rhizoctonia solani on sugar beet in field experiments. To address hypothesis 1, we recorded infections within inoculum-donor and host-recipient pairs of plants with differing spacing. We translated these observations into the individual-level concept of pathozone, a host-centred form of dispersal kernel. To test hypothesis 2 and our prediction, we used the pathozone to parameterise a stochastic model of pathogen spread in a host population, contrasting scenarios of spread with and without host growth. Our results support our hypotheses and prediction. We suggest that practitioners of agriculture and arboriculture account for root system expansion in order to reduce the risk of soilborne-disease epidemics. We discuss changes in crop design, including increasing plant spacing and using crop mixtures, for boosting crop resilience to invasion and damage by soilborne pathogens. We speculate that the disease-induced root growth observed in some pathosystems could be a pathogen strategy to increase its population through host manipulation.

Moment approximation of infection dynamics in a population of moving hosts

The modelling of contact processes between hosts is of key importance in epidemiology. Current studies have mainly focused on networks with stationary structures, although we know these structures to be dynamic with continuous appearance and disappearance of links over time. In the case of moving individuals, the contact network cannot be established. Individual-based models (IBMs) can simulate the individual behaviours involved in the contact process. However, with very large populations, they can be hard to simulate and study due to the computational costs. We use the moment approximation (MA) method to approximate a stochastic IBM with an aggregated deterministic model. We illustrate the method with an application in animal epidemiology: the spread of the highly pathogenic virus H5N1 of avian influenza in a poultry flock. The MA method is explained in a didactic way so that it can be reused and extended. We compare the simulation results of three models: 1. an IBM, 2. a MA, and 3. a mean-field (MF). The results show a close agreement between the MA model and the IBM. They highlight the importance for the models to capture the displacement behaviours and the contact processes in the study of disease spread. We also illustrate an original way of using different models of the same system to learn more about the system itself, and about the representation we build of it.

A MATHEMATICAL MODEL FOR SPATIALLY EXPANDING INFECTED AREA OF EPIDEMICS TRANSMITTED THROUGH HETEROGENEOUSLY DISTRIBUTED SUSCEPTIBLE UNITS

Little is known about the effect of environmental heterogeneity on the spatial expansion of epidemics. In this work, to focus on the question of how the extent of epidemic damage depends on the spatial distribution of susceptible units, we develop a mathematical model with a simple stochastic process, and analyze it. We assume that the unit of infection is immobile, as town, plant, etc. and classify the units into three classes: susceptible, infective and recovered. We consider the range expanded by infected units, the infected rangeR, assuming a certain generalized relation between R and the total number of infected units k, making use of an index, a sort of fractal dimension, to characterize the spatial distribution of infected units. From the results of our modeling analysis, we show that the expected velocity of spatial expansion of infected range is significantly affected by the fractal nature of spatial distribution of immobile susceptible units, and is temporally variable. When the infection finally terminates at a moment, the infected range at the moment is closely related to the nature of spatial distribution of immobile susceptible units, which is explicitly demonstrated in our analysis.

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.

Spatial heterogeneity of daphniid parasitism within lakes

Spatially explicit models show that local interactions of hosts and parasites can strongly influence invasion and persistence of parasites and can create lasting spatial patchiness of parasite distributions. These predictions have been supported by experiments conducted in two-dimensional landscapes. Yet, three-dimensional systems, such as lakes, ponds, and oceans, have received comparatively little attention from epidemiologists. Freshwater zooplankton hosts often aggregate horizontally and vertically in lakes, potentially leading to local host-parasite interactions in one-, two-, or three-dimensions. To evaluate the potential spatial component of daphniid parasitism driven by these local interactions (patchiness), we surveyed vertical and horizontal heterogeneity of pelagic Daphnia infected with multiple microparasites in several north temperate lakes. These surveys uncovered little evidence for persistent vertical patchiness of parasitism, since the prevalence of two parasites showed little consistent trend with depth in four lakes (but more heterogeneity during day than at night). On a horizontal scale of tens of meters, we found little systematic evidence of strong aggregation and spatial patterning of daphniid hosts and parasites. Yet, we observed broad-scale, basin-wide patterns of parasite prevalence. These patterns suggest that nearshore offshore gradients, rather than local-scale interactions, could play a role in governing epidemiology of this open water host-parasite system.