Analytic approximation of spatial epidemic models of foot and mouth disease

Equilibrium properties of the spatial SIS model as a point pattern dynamics - How is infection distributed over space?

We revisit the classical epidemiological SIS model as a stochastic point pattern dynamics with special focus on its spatial distribution at equilibrium. In this model, each point on a continuous space is either susceptible S or infectious I, and infection occurs with an infection kernel as a function of distance from I to S. This stochastic process has been mathematically described by the hierarchical dynamics of the probabilities that a point, a pair made by two points, and a triplet made by three points, etc., is in a specific configuration of status. Using a simple closure thereby triplet probabilities that appear in the dynamics are approximated, we show that the average singlet probabilities and the pair probabilities that describe spatial distributions of Ss and Is at equilibrium can be explicitly derived using the infection kernel; Is are spatially clustered in the same order of the infection kernel. The results highlight the advantage of point pattern approach to model spatial population dynamics in general ecology where local interactions among individuals likely depend on distance between them.

Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions

Pair approximation models have been used to study the spread of infectious diseases in spatially distributed host populations, and to explore disease control strategies such as vaccination and case isolation. Here we introduce a pair approximation model of individual uptake of non-pharmaceutical interventions (NPIs) for an acute self-limiting infection, where susceptible individuals can learn the NPIs either from other susceptible individuals who are already practicing NPIs ("social learning"), or their uptake of NPIs can be stimulated by being neighbours of an infectious person ("exposure learning"). NPIs include individual measures such as hand-washing and respiratory etiquette. Individuals can also drop the habit of using NPIs at a certain rate. We derive a spatially defined expression of the basic reproduction number R0 and we also numerically simulate the model equations. We find that exposure learning is generally more efficient than social learning, since exposure learning generates NPI uptake in the individuals at immediate risk of infection. However, if social learning is pre-emptive, beginning a sufficient amount of time before the epidemic, then it can be more effective than exposure learning. Interestingly, varying the initial number of individuals practicing NPIs does not significantly impact the epidemic final size. Also, if initial source infections are surrounded by protective individuals, there are parameter regimes where increasing the initial number of source infections actually decreases the infection peak (instead of increasing it) and makes it occur sooner. The peak prevalence increases with the rate at which individuals drop the habit of using NPIs, but the response of peak prevalence to changes in the forgetting rate are qualitatively different for the two forms of learning. The pair approximation methodology developed here illustrates how analytical approaches for studying interactions between social processes and disease dynamics in a spatially structured population should be further pursued.

Impacts of constrained culling and vaccination on control of foot and mouth disease in near-endemic settings: a pair approximation model

Many countries have eliminated foot and mouth disease (FMD), but outbreaks remain common in other countries. Rapid development of international trade in animals and animal products has increased the risk of disease introduction to FMD-free countries. Most mathematical models of FMD are tailored to settings that are normally disease-free, and few models have explored the impact of constrained control measures in a 'near-endemic' spatially distributed host population subject to frequent FMD re-introductions from nearby endemic wild populations, as characterizes many low-income, resource-limited countries. Here we construct a pair approximation model of FMD and investigate the impact of constraints on total vaccine supply for prophylactic and ring vaccination, and constraints on culling rates and cumulative culls. We incorporate natural immunity waning and vaccine waning, which are important factors for near-endemic populations. We find that, when vaccine supply is sufficiently limited, the optimal approach for minimizing cumulative infections combines rapid deployment of ring vaccination during outbreaks with a contrasting approach of careful rationing of prophylactic vaccination over the year, such that supplies last as long as possible (and with the bulk of vaccines dedicated toward prophylactic vaccination). Thus, for optimal long-term control of the disease by vaccination in near-endemic settings when vaccine supply is limited, it is best to spread out prophylactic vaccination as much as possible. Regardless of culling constraints, the optimal culling strategy is rapid identification of infected premises and their immediate contacts at the initial stages of an outbreak, and rapid culling of infected premises and farms deemed to be at high risk of infection (as opposed to culling only the infected farms). Optimal culling strategies are similar when social impact is the outcome of interest. We conclude that more FMD transmission models should be developed that are specific to the challenges of FMD control in near-endemic, low-income countries.

Dynamics and control of foot-and-mouth disease in endemic countries: a pair approximation model

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.

Analytical methods for quantifying environmental connectivity for the control and surveillance of infectious disease spread

The sustained transmission and spread of environmentally mediated infectious diseases is governed in part by the dispersal of parasites, disease vectors and intermediate hosts between sites of transmission. Functional geospatial models can be used to quantify and predict the degree to which environmental features facilitate or limit connectivity between target populations, yet typical models are limited in their geographical and analytical approach, providing simplistic, global measures of connectivity and lacking methods to assess the epidemiological implications of fine-scale heterogeneous landscapes. Here, functional spatial models are applied to problems of surveillance and control of the parasitic blood fluke Schistosoma japonicum and its intermediate snail host Oncomelania haupensis in western China. We advance functional connectivity methods by providing an analytical framework to (i) identify nodes of transmission where the degree of connectedness to other villages, and thus the potential for disease spread, is higher than is estimated using Euclidean distance alone and (ii) (re)organize transmission sites into disease surveillance units based on second-order relationships among nodes using non-Euclidean distance measures, termed effective geographical distance (EGD). Functional environmental models are parametrized using ecological information on the target organisms, and pair-wise distributions of inter-node EGD are estimated. A Monte Carlo rank product analysis is presented to identify nearby nodes under alternative distance models. Nodes are then iteratively embedded into EGD space and clustered using a k-means algorithm to group villages into ecologically meaningful surveillance groups. A consensus clustering approach is taken to derive the most stable cluster structure. The results indicate that novel relationships between nodes are revealed when non-Euclidean, ecologically determined distance measures are used to quantify connectivity in heterogeneous landscapes. These connections are not evident when analysing nodes in Euclidean space, and thus surveillance and control activities planned using Euclidean distance measures may be suboptimal. The methods developed here provide a quantitative framework for assessing the effectiveness of ecologically grounded surveillance systems and of control and prevention strategies for environmentally mediated diseases.