Kinetic growth percolation: Epidemic processes with immunization

Using an agent-based model to evaluate the effect of producer specialization on the epidemiological resilience of livestock production networks

An agent-based computer model that builds representative regional U.S. hog production networks was developed and employed to assess the potential impact of the ongoing trend towards increased producer specialization upon network-level resilience to catastrophic disease outbreaks. Empirical analyses suggest that the spatial distribution and connectivity patterns of contact networks often predict epidemic spreading dynamics. Our model heuristically generates realistic systems composed of hog producer, feed mill, and slaughter plant agents. Network edges are added during each run as agents exchange livestock and feed. The heuristics governing agents’ contact patterns account for factors including their industry roles, physical proximities, and the age of their livestock. In each run, an infection is introduced, and may spread according to probabilities associated with the various modes of contact. For each of three treatments—defined by one-phase, two-phase, and three-phase production systems—a parameter variation experiment examines the impact of the spatial density of producer agents in the system upon the length and size of disease outbreaks. Resulting data show phase transitions whereby, above some density threshold, systemic outbreaks become possible, echoing findings from percolation theory. Data analysis reveals that multi-phase production systems are vulnerable to catastrophic outbreaks at lower spatial densities, have more abrupt percolation transitions, and are characterized by less-predictable outbreak scales and durations. Key differences in network-level metrics shed light on these results, suggesting that the absence of potentially-bridging producer–producer edges may be largely responsible for the superior disease resilience of single-phase “farrow to finish” production systems.

How does the resistance threshold in spatially explicit epidemic dynamics depend on the basic reproductive ratio and spatial correlation of crop genotypes?

We examined the fraction of resistant cultivars necessary to prevent a global pathogen outbreak (the resistance threshold) using a spatially explicit epidemiological model (SIR model) in a finite, two-dimensional, lattice-structured host population. Infectious diseases in our model could be transmitted to susceptible nearest-neighbour sites, and the infected site either recovered or died after an exponentially distributed infectious period. Threshold behaviour of this spatially explicit SIR model cannot be reduced to that of bond percolation, as was previously noted in the literature, unless extreme assumptions (synchronized infection events with a fixed lag) are imposed on infection process. The resistance threshold is significantly lower than that of conventional mean-field epidemic models, and is even lower if the spatial configuration of resistant and susceptible crops are negatively correlated. Finite size scaling applied to the resistance threshold for a finite basic reproductive ratio ρ of pathogen reveals that its difference from static percolation threshold (0.41) is inversely proportional to ρ. Our formula for the basic reproductive ratio dependency of the resistance threshold produced an estimate for the critical basic reproductive ratio (4.7) in a universally susceptible population, which is much larger than the corresponding critical value (1) in the mean-field model and nearly three times larger than the critical growth rate of a basic contact process (SIS model). Pair approximation reveals that the resistance threshold for preventing a global epidemic is factor 1/(1-η) greater with spatially correlated planting than with random planting, where η is initial correlation in host genotypes between nearest-neighbour sites. Thus the eradication is harder with a positive spatial correlation (η>0) in mixed susceptible/resistant plantings, and is easier with a negative correlation (η<0). The effect of finite field size (L), which corresponded to the mean distance between sources of infections, is given by the increased resistance threshold (by the amount L⁻⁰·⁷⁵) from its infinite size limit. Implications of these results on effective planting strategies in multi-line control plans are discussed.

Pathogen invasion and host extinction in lattice structured populations

We examined the propagation of an infectious disease and the eventual extinction of the host population in a lattice-structured population. Both the host colonization and pathogen transmission processes are assumed to be restricted to act between the nearest neighbor sites. The model is analyzed by an improved version of pair approximation (IPA). Pair approximation is a technique to trace the dynamics of the number of nearest neighbor pairs having particular states, and IPA takes account of the clustering property of lattice models more precisely. The results are checked by computer simulations. The analysis shows: (i) in a one-dimensional lattice population, a pathogen cannot invade a host population no matter how large is the transmission rate; (ii) in a two-dimensional lattice population, pathogens will drive the host to extinction if the transmission rate is larger than a threshold. These results indicate that spatially structured population models may give qualitatively different results from conventional population models, such as Lotka-Volterra ones, without spatial structure.