Supplementary Materials

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A number of rows is also drawn from a Poisson distribution P(. √ n), from ... qq qq q q qq qq. q q q qq qqq qq q q q qq q qq q qq qqqq qqq q qqq qq q. q q q q q q.
Supplementary Materials A

The estimation model as an online app

For practical use, we produced an approximation of the upper bound of the confidence interval of the disease incidence. This requires few simple algebraic operations that can be solved “on the back of an envelop” if 5

needed. However, to make our estimation more widely accessible, we deployed it online as a simple browser app that can be found on shinyapps.io. The app derives the probability distribution of the disease incidence given tweakable inputs regarding the epidemic (growth rate r and asymptomatic period σ), the required level of confidence in the estimation, the

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rates of imperfect detection, and, of course, a sampling series. The sampling series is a 3-column table in which the user can fill (or upload) the dates of the sampling events, their size N and their outcome M . Detailed information can be found on the “Help” panel of the app. Because online access to the app might be limited, one can also download it from GitLab and use it locally

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(distributed under GNU GPL).

B

Specific approximations for disease incidence estimation

As discussed in the introduction, early detection of epidemics and the establishment of disease absence have received attention in epidemiological literature. We show here that these are two special cases of our model. 20

Early detection: Parnell et al. (2012) discuss the case where all monitoring rounds return zero infected hosts and only the last monitoring round returns one or more infected hosts. They ask the question “what the disease incidence can be when the first symptomatic host is detected?”. Referring to equation (6), we thus are calculating Pearly (M |q) =

Nk X

P (0, 0, ..., 0, Mk |q),

(B.1)

Mk =1

which, considering (1 − q)N ≈ e−qN , can be shown to be Pearly (M |q) = 1 − e

−ZK NK q



exp −

K−1 X

! Z k Nk q .

(B.2)

k=1 25

Substituting (1−e−ZK NK q ) ≈ ZK NK q and using Baye’s equation we find as approximation for the probability distribution of the incidence at first detection: P (q|M ) = A2 q e−Aq .

(B.3)

This is a slightly different result from that of Parnell et al. (2012) which is due to a different way of approximation. Parnell et al. (2012) calculate the expected incidence at first detection when monitoring interval and number of hosts sampled are constant. From this probability density we find that the expected value of q is E(q) =

r∆ 1 1 − e−r∆ ≈ . −r∆K N 1−e N 1

(B.4)

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With the last approximation step for small values of ∆ and large K, and this is the expression Parnell et al. (2012) derived. Monitoring for disease absence: Several authors have asked the question “What can the disease incidence be when no disease is detected?”. Bourhis et al. (2018) included epidemic growth into the existing binomial sampling methods, similar to our approach here. When no disease is detected in all K monitoring rounds we

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find, using equations (6) and (4): P (q|notf ound) ≈ Ae−qA , where A=

K X

exp −r σ +

k=1

K X

(B.5)

!! ∆i

Nk .

(B.6)

i=k

The upper bound of the X% confidence interval is then given by q˜X =

C 40

−ln(1 − X/100) . A

(B.7)

A neutral model of host distribution in field-like patterns

For the purpose of testing our incidence estimation model against spatially explicit and stochastic simulations of epidemics, we develop a landscape model. This landscape model is in fact a point pattern process marking the locations of hosts in a 2D-space. The hosts are aggregated in in field-like structures mimicking orchards, where trees are set in rows inside rectangular shapes. The point pattern process is a clustered process and more precisely a case of Neyman-Scott process (Illian

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et al., 2008). First, a parent Poisson process locates field locations (seeds) across the spatial extent. A lower scale daughter process then disposes the hosts in regular structures on seed locations. The process is defined as F(λ, µ) where λ is the intensity of the process, i.e. number of hosts by surface unit, and µ is the mean population size of the fields. The seeds are disposed according to the Poisson process

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P(λ/µ) where λ/µ is the intensity of the parent process. For each seeds, a population size, n, is drawn from a √ Poisson distribution P(µ). A number of rows is also drawn from a Poisson distribution P( n), from which the complementary dimension is derived to result in a grid accommodating the population size of the field, with respect to specified spacings between hosts. Finally, the fields are given an orientation from uniform draws of angles U(0, 2π). Figure 1 below illustrates the resulting host distributions from varying values of field population size µ and

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spatial extent. The point pattern process code (in R) can be found on GitLab and used according to the GNU GPL.

2

µ=1 1.00

µ = 20

µ = 60







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µ = 180

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Nb. of hosts = 1929

Nb. of hosts = 2001

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Nb. of hosts = 342

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Nb. of hosts = 435

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Nb. of hosts = 509

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Side = 10000 m

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Nb. of hosts = 506 ●

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0.25

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Nb. of hosts = 177

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Nb. of hosts = 89

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Side = 5000 m

Normalised distance from origin in North direction

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1.000.00

Nb. of hosts = 1924 0.25

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1.00

Normalised distance from origin in East direction

Figure 1: Examples of host distribution generating for varying aggregation µ and varying spatial extent (referred as ”Side” in row headers). The landscape dimensions are scaled to 1 for illustrative reasons. The bottom row, Side = 10000m, illustrates the spatial extent used in the spatial simulations detailed in this paper.

References Bourhis, Y., Gottwald, T. R., Lopez-Ruiz, F. J., Patarapuwadol, S., and van den Bosch, F. (2018). Sampling for disease absence—deriving informed monitoring from epidemic traits. Journal of Theoretical Biology. 60

Illian, D. J., Penttinen, P. A., Stoyan, D. H., and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley & Sons. Google-Books-ID: U6BER2stYsC. Parnell, S., Gottwald, T., Gilks, W., and van den Bosch, F. (2012). Estimating the incidence of an epidemic when it is first discovered and the design of early detection monitoring. Journal of Theoretical Biology, 305:30–36.

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