Table 1.
Summary of symbols and their units used throughout this paper.
Table 2.
Summary of abbreviations employed in the text.
Fig 1.
Susceptible-exposed-infectious-removed-pathogen (SEIR-P) environmental transmission model diagram illustrating the four host compartments ((S)usceptible, (E)xposed, (I)nfectious and (R)emoved, or recovered) and single pathogen compartment (P) of the model.
The solid arrows indicate the movement of hosts between host compartments and the loss of viable pathogen from the system. The dotted arrows indicate how the host and pathogen parts of the model influence each other. The parameters α and β are the environmental and direct transmission rates while ϵ and ρ are the rates of pathogen emission and decay. The time that hosts spend in the E and I compartments is determined by the chosen exposed and infectious lifetime distributions, e.g., gamma distributions with shape parameters νδ and νγ and rate parameters λδ and λγ, as shown here. When exponential distributions are assumed, then the rate parameters are denoted respectively by δ and γ (see Table 1 for a summary of all symbols used throughout the paper). When α = ϵ = ρ = 0 we obtain the SEIR direct transmission model as a submodel.
Fig 2.
Host S, I and R outputs from a single realisation of SIR-P model with environmental transmission only (i.e., β = 0 host−1 h−1) among a host population of size N = 2000.
The rates of environmental transmission and pathogen emission are α = 1.325 × 10−10 virion−1 h−1 and ϵ = 3462.5 virion host−1 h−1 and the rates of pathogen decay, ρ, and host recovery, γ, are both 0.029 h−1 (equivalent to a half-life of 24 h). The inset in the central panel is the key feature of interest and shows how the number of infectious hosts goes to zero roughly between the times t = 2075 h and 2120 h. Under the SIR model, as soon as the number of infectious hosts reaches zero no further secondary infections are possible. However, within the SIR-P model with comparable pathogen and host infectious lifetimes, outbreaks can appear to die off but later continue, due to the force of infection from long-living pathogens.
Fig 3.
Susceptible, infectious and removed host sub-population sizes of SIR-P (blue) process averaged over 5000 simulations for fixed environmental transmission rate α = 5.95 × 10−7 virion−1 h−1 and host mortality rate γ = 5.95 × 10−3 h−1.
The rates of pathogen emission, ϵ, and pathogen removal, ρ, are increased while keeping their ratio fixed, . Top row: ϵ = 2.98 × 10−2 host−1 h−1, ρ = 5.95 × 10−4 h−1, middle row: ϵ = 2.98 host−1 h−1, ρ = 5.95 × 10−2 h−1, bottom row: ϵ = 2.98 × 102 host−1 h−1, ρ = 5.95 h−1. For comparison, the same sub-population sizes for the DTA SIR process with fixed direct transmission rate
and γ′ = 5.95 × 10−3 h−1 are plotted in (red). Median population sizes indicated by bold lines, dashed lines indicate 5th and 95th percentiles. The top row of panels show two processes that are visibly distinct in their outputs, but with a hundred-fold increase in the pathogen decay rate, a closer alignment between the two sets of trajectories can be seen in the middle row. In the last case (bottom row) there is no difference on the scale of the plots between the SIR-P and SIR model outputs.
Table 3.
Scenarios for simulation study.
Fig 4.
Density estimates of SEIR parameter posterior distributions and R0 for long-lived pathogen (A), intermediate pathogen (B), short-lived pathogen (C) and direct-transmission only (D) data sets.
The red dot in the leftmost panels indicates , where
and
. The red vertical lines in the central and rightmost panels indicate
and
, respectively. The marginal posterior distributions γ and (β, δ) are conditionally independent and so are plotted separately.
Fig 5.
Observed outbreak size trajectories, It, over course of a single simulated outbreak (solid red line): long-lived (A), intermediate (B) and short-lived pathogen (C) and direct transmission only (D).
These are compared with the trajectories obtained from SEIR model with MCMC-sampled parameters values, with small outbreaks (≤ 50) discarded. The time axis was discretised (400 points) and 5th, 25th, 50th (median), 75th and 95th percentiles of the SEIR-predicted outbreak size were estimated at each discrete time point. The solid blue line indicates the median outbreak size while the dashed blue lines are the other percentiles. In the short-lived and direct transmission only cases, the shape of the predicted outbreak size trajectories (as indicated by the blue lines) mirrors that of the observed outbreak size, with the solid red and blue lines aligning at the initial exponential growth phase, as well as at the end of the outbreak when the number of infective hosts dies out. This is not the case for the long-lived pathogen case, for which the model predicts earlier onset of growth of the outbreak, peaking somewhat earlier than was observed.
Fig 6.
Graphical comparisons of final outbreak size (total hosts infected during outbreak) vs. outbreak duration (latest removal time minus time of first onset of infectivity) with their posterior predictive distributions.
The red dot indicates observed value of statistics from one outbreak: long-lived (A), intermediate (B) and short-lived pathogen (C) and direct transmission only (D). The shading and contours were obtained from a kernel density estimate after simulating 15000 SEIR outbreak trajectories with parameter values taken from the MCMC samples obtained while fitting the SEIR model, with small outbreaks (≤ 50) discarded. In the case of long-lived pathogen, the fitted model tends to predict shorter duration outbreaks but otherwise agrees with the data in terms of final outbreak size. This is indicated by the red dot aligning horizontally with the darkest part of the density estimate but being shifted vertically. Better agreement between the data and fitted model is evident in the short-lived and intermediate pathogen and DT-only cases.
Fig 7.
Graphical comparisons of size of outbreak peak, i.e., the size of It at its largest, and time of outbreak peak, as defined in main body of text with their posterior predictive distributions.
The red dot indicates observed value of statistics from one outbreak: long-lived (A), intermediate (B) and short-lived pathogen (C) and direct transmission only (D). The shading and contours were obtained from a kernel density estimate after simulating 15000 SEIR outbreak trajectories with parameter values taken from the MCMC samples obtained while fitting the SEIR model, with small outbreaks (≤ 50) discarded. For long-lived pathogen, the fitted SEIR model predicts that outbreaks peak, on average, at the size observed in the data. However, the model predicts outbreaks that peak earlier. This is evident in panel (A), where the predicted outbreak size trajectories clearly peak earlier than the observed outbreak size trajectory indicated by the solid red line. Better agreement between data and model predictions are visible in panels (C) and (D).
Fig 8.
Summary of SEIR-P model of WSD among penaeids.
Parameter values are listed in Table 4. The arrow from I to R, labelled Γ(νγ, λγ), represents removal of dead hosts after a gamma-distributed time to full natural decay. The curved arrow from I to R represents removal at one of the x-hourly removal attempts, with probability πI, similarly for the curved arrow from E to I. The direct transmission approximation (DTA) of the SEIR-P model is obtained by replacing αStPt + βStIt above the leftmost arrow with and setting ϵ = ρ = 0.
Table 4.
Parameter estimates and sources for SEIR-P model of WSSV in penaeids.
Fig 9.
Estimated density plots of final outbreak size (left panels) and outbreak duration (right) for the SEIR-P (blue) and DTA (red) models of WSD under (A) 24-hourly removals, where both quantities are distributed very similarly under the two models, and (B) 6-hourly removals.
Increasing the removal frequency tends to reduce the size and duration of outbreaks, although some larger outbreaks still occur. The benefit of increasing the removal frequency, in terms of reduction in mean final outbreak size, is underestimated slightly by the DTA and the reduction in outbreak duration is over-estimated.
Fig 10.
Simulations of the SEIR-P (blue) and DTA (red) models of WSD in penaeid shrimp with removals of exposed (E) and dead (I) hosts at 24-hourly intervals, with probabilities of success 0.05 and 0.95, respectively.
Single outbreak trajectories (top row) and averages over 30 000 independent simulations with small outbreaks (fewer than 10 infections) excluded (bottom row). The zig-zag pattern in the 3rd panel on the bottom row is due to the periodic removals. The averaged model outputs show a high degree of similarity between SEIR-P and DTA, meaning that at these timescales the environmental transmission of WSD can be well approximated with direct transmission among the hosts.
Fig 11.
Simulations of the SEIR-P (blue) and DTA (red), as in Fig 10, with removals at 6-hourly intervals.
Although the outbreaks of single SEIR-P and DTA trajectories appear similar, a small but definite divergence between the two models appears when studying their averaged outputs.