When and why direct transmission models can be used for environmentally persistent pathogens

Variants of the susceptible-infected-removed (SIR) model of Kermack & McKendrick (1927) enjoy wide application in epidemiology, offering simple yet powerful inferential and predictive tools in the study of diverse infectious diseases across human, animal and plant populations. Direct transmission models (DTM) are a subset of these that treat the processes of disease transmission as comprising a series of discrete instantaneous events. Infections transmitted indirectly by persistent environmental pathogens, however, are examples where a DTM description might fail and are perhaps better described by models that comprise explicit environmental transmission routes, so-called environmental transmission models (ETM). In this paper we discuss the stochastic susceptible-exposed-infected-removed (SEIR) DTM and susceptible-exposed-infected-removed-pathogen (SEIR-P) ETM and we show that the former is the timescale separation limit of the latter, with ETM host-disease dynamics increasingly resembling those of a DTM when the pathogen’s characteristic timescale is shortened, relative to that of the host population. Using graphical posterior predictive checks (GPPC), we investigate the validity of the SEIR model when fitted to simulated SEIR-P host infection and removal times. Such analyses demonstrate how, in many cases, the SEIR model is robust to departure from direct transmission. Finally, we present a case study of white spot disease (WSD) in penaeid shrimp with rates of environmental transmission and pathogen decay (SEIR-P model parameters) estimated using published results of experiments. Using SEIR and SEIR-P simulations of a hypothetical WSD outbreak management scenario, we demonstrate how relative shortening of the pathogen timescale comes about in practice. With atttempts to remove diseased shrimp from the population every 24h, we see SEIR and SEIR-P model outputs closely conincide. However, when removals are 6-hourly, the two models’ mean outputs diverge, with distinct predictions of outbreak size and duration.

Since the full posterior distribution of parameters and exposure times, p( β, δ, γ, t E | t I , t R ) is intractable, we use MCMC methods to sample dependent sequences β i , δ i , γ i , t E i as follows: We first set initial values for the parameters and exposure times, β 0 , δ 0 , γ 0 , t E 0 and then, for some T ≥ 1.0 (the temperature -see below), iterating through the following steps for i = 1, . . . , n: 1. Update δ via Metropolis-Hastings, i.e., propose δ ∼ N (δ i−1 , σ 2 δ ) and with probability a 1 set δ i = δ , otherwise δ i = δ i−1 , where f T is the marginal conditional density for δ, t E at temperature T (see below). The parameter σ δ is tuned during an number of iterations in order to get an acceptance rate of between 20% and 40%.
2. Choose an exposure time to update (with index j) uniformly at random. Propose t E , where Update β by the Gibbs sampler, i.e. sample from the full conditional distribution β i ∼ Γ(m, A+ω β ).
The above follows a fully-centred parameterisation, as discussed by Neal and Roberts for the SIR model [1]. Note that there is no need to sample γ as part of the above routine, since having assumed an exponentially-distributed prior p(γ) = ω γ e −ωγ γ , its posterior density is and therefore Due to the high dimensionality of the sample space and the likelihood function perhaps having local maxima, the sampling chains can sometimes be slow to converge to stationarity and even become stuck at certain parameter values, with an acceptance ratio going towards zero. Metropolis coupled MCMC, or is the strategy adopted here to alleviate poor mixing and is summarised as follows: several of the above chains are run with several closely spaced temperatures 1.0 = T 1 < T 2 < · · · < T r . The first chain, with temperature 1.0, is termed the cold chain and is the only chain from which we obtain samples.
The other chains are known as the heated chains. After performing a fixed number of iterations for each chain in parallel, two are selected uniformly at random, with temperatures T and T and current states X = β , δ , t E and X = β , δ , t E . The states are then exchanged with probability a 3 , where Although the samples from the heated chains are ultimately discarded, the method has the advantage that is easily parallelised on a multi-core machine. For this work, this was achieved using Python's multiprocessing module. Six chains were run in parallel with the temperatures T = 1.00, 1.02, 1.04, 1.06, 1.08, 1.10 and exchanges of state were attempted every 400 iterations.
We can therefore marginalise the posterior density at temperature T , obtaining f T which is, for

B Force of infection in Reed-Frost epidemic model
and the probability that there is transmission from at least one infected individual to this specified susceptible is therefore and the expected number of new infections occuring is The force of infection is therefore C Estimation of α, and ρ for SEIR-P model of WSD in shrimp C.1 Estimation of pathogen decay rate, ρ By a challenge experiment in which P. monodon were immersed in sterile seawater that had been spiked with WSSV a variable number of days prior to immersion, Kumar et.al. were able to estimate how long a known quantity of WSSV remains viable in seawater under laboratory conditions. Ten experimental and one control bucket were filled with 10l of sterile seawater. To the experimental buckets pure WSSV was added to a final concentration of 1000 virion ml −1 , meaning that around 10 7 particles were present in each bucket. On days 0 up to 18, 10 juvenile P. monodon were added to one of the unoccupied buckets and all shrimp were monitored at 8h intervals for mortality or signs of WSSV infection. Dead shrimp were removed from the buckets and proportions of living shrimp were recorded daily for each bucket.
The authors found that under the conditions of the experiment seawater-borne WSSV remains infective for up to 12 days.
The plots given in [5, Figure 2] for buckets 0 to 8 indicate similar rates of mortality across these buckets, suggesting that the WSSV lost little of its infectivity during the first eight days in seawater.
Total mortality occurred at around the four day mark following introduction to the 0 to 8-day buckets.
A reduction in mortality rates is then noticeable following immersion in the 10 day and 12 day buckets, where 100% mortality was observed each at the 7 day mark (i.e 7 days after immersing the shrimp). No mortalities occurred following immersion in the 14, 16 and 18 day buckets, at least during the experimental period, suggesting that the amount of viable WSSV had decayed significantly by 12-14 days in seawater.
Since at 8 days there was still a sufficient quantity of WSSV to inoculate all 10 of the shrimp, we expect that the mean infectious lifetime a WSSV particle in seawater to be no less than 8 days, giving pathogen decay rate, ρ, no greater than 0.005 h −1 .

C.2 Estimation of environmental transmission rate, α
The second column of Table A contains the time in hours, t 100 , from immersion to 100% mortality of shrimp in each of the buckets, labelled by the number of days after the introduction of WSSV to the bucket that the shrimp were immersed. A lower, order of magnitude, estimate for α (indirect rate of WSD transmission) can be obtained from the same data by assuming that mortality comes immediately upon infection (thus underestimating the infectivity) and assuming that the pathogen density remained at its initial level of 1000 virion ml −1 , at least for buckets 0 to 8, where the rates of mortality were similar.
The transmission rate α should be no more than 10/(10 3 × t 100 ), since 10 exposures occurred in t 100 hours, and the rate of new exposures is αP = 10 3 α. These values are in Column 3 of Table A Fig 2]). Buckets are labelled by time in days between addition of WSSV to bucket and immersion of shrimp. t 100 -the time in hours to 100% mortality.
Upper and lower estimates of α are given to 1 decimal place and calculations are described in main body of text. independently. Trace plots show all iterations, including those later discarded for burn in, in order to demonstrate convergence of two independent chains to stationarity.