# Sequential Area Interaction Sampling for Early Plant Disease Detection

François d'Alayer, Edith Gabriel, Samuel Soubeyrand

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François d'Alayer, Edith Gabriel, Samuel Soubeyrand. Sequential Area Interaction Sampling for Early Plant Disease Detection. 2025. hal-05367278

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Distributed under a Creative Commons Attribution 4.0 International License
Sequential Area Interaction Sampling for Early Plant Disease
Detection

François d'Alayer<sup>a,1</sup>, Edith Gabriel<sup>a</sup>, Samuel Soubeyrand<sup>a</sup>

<sup>a</sup>INRAE, BioSP, Avignon, 84914, France

**Abstract**

We develop general novel sequential point process models for plant disease surveillance, aiming to accelerate early detection of disease presence in a study domain. The framework adapts to the level of knowledge about the propagation parameters: with full information, it reduces to a sequential hard-core process, whereas under uncertainty it becomes a sequential area-interaction process. By incorporating past observations at each step, the proposed models optimize detection time in an adaptive setting. The dynamics of the proposed sequential point process models are analyzed and compared to alternative sampling strategies through a simulation study.

***Keywords:*** repulsive design, sequential sampling, plant disease surveillance, surveillance for early disease detection, area interaction point process

# 1. Introduction

Emerging infectious diseases of plants pose an escalating threat to ecosystems, food secu-
rity, and international trade, with their impacts often becoming irreversible once outbreaks are
firmly established. Because conventional measures such as quarantine and border inspections fre-
quently fail to prevent incursions, effective management increasingly hinges on early detection.
Surveillance systems must therefore be designed to identify invasions before pathogen preva-
lence exceeds critical thresholds, beyond which containment becomes unfeasible. Yet, most
methodological advances to date have relied on computationally intensive epidemic simulations
and optimisation frameworks to allocate surveillance resources [1, 2] or on risk-based surveil-
lance [3, 4, 5], as defined in [6]. Here, we introduce a novel early-detection method that departs
from this paradigm: it achieves reliable performance without recourse to heavy optimization
machinery and includes risk-based surveillance. By leveraging simple yet principled statistical
reasoning, our approach provides a transparent and tractable alternative, opening new method-
ological perspectives on how surveillance systems can be designed and assessed. A standard
formulation for early detection in a spatial domain $W \subset \mathbb{R}^2$ is: given a prescribed sample size
$n$, design a sampling scheme that maximizes the probability of detecting an infection (or equiv-
alently, minimizes the probability of a miss). Early detection can be alternatively formulated as
the minimization of a stopping time $n$, where $n$ is the index of the first sample in an ordered

Email address: francois.dalayer@inrae.fr (François d'Alayer)

1

Preprint submitted to Spatial Statistics

November 15, 2025
sequence that triggers detection. This formulation highlights the challenge in sequential surveillance for early detection: all outcomes up to $n-1$ are non-detections; hence, systems that exploit successive data must base decisions on where and when anterior samples were collected, since no positive labels are observed until the $n$-th sample. In practice, sampling is often risk-based (focusing tests in high-risk subregions), but our proposed sequential spatial strategies built upon risk-based sampling can substantially improve detection efficiency compared to classical sampling strategies.

A sequential spatial point process (SSPP) is a probability model for a finite ordered sequence of spatial points. Formally, one considers realizations $\vec{y}_n = (y_1, \dots, y_n) \in W^n$ (for a bounded region $W \subset \mathbb{R}^2$) whose elements are ranked by some criteria (e.g., arrival time or a mark-like size). The SSPP is defined by giving its joint density with respect to a Poisson reference measure. Equivalently, one can write the joint density with respect to a reference measure on $W$ as a product of conditional densities

$$p(\vec{y}_n) = p_1(y_1) \prod_{k=1}^{n-1} p_{k+1}(y_{k+1} | \vec{y}_k),$$

where $p_{k+1}(y_{k+1} | \vec{y}_k)$ is the density of the $(k+1)$th point given the history $\vec{y}_k$. This successive conditioning means each new point can depend on all past points (capturing spatial “memory” or self-interaction). van Lieshout [7] formalized SSPPs with random sequential-adsorption/inhibition models. More recently, Penttinen and Ylitalo [8] proposed an SSPP with self-interaction (motivated by eye-movement data) and Yazigi et al. [9] fitted an SSPP for forest trees by ordering trees by size. In the surveillance context, d’Alayer et al. [10] have applied marked SSPPs to epidemiological surveillance for an efficient estimation of the disease probability map.

In this article we present a sequential point-process approach that exploits negative test information to concentrate subsequent sampling in as-yet-unexcluded areas. Two related classes of sequential designs are introduced: (i) a sequential hard-core point process that enforces an exclusion radius around each negative sample, and (ii) a sequential area-interaction point process that allows overlap of exclusion discs when there is uncertainty about the spread speed (see [11] for the original area interaction point processes and [12] for hardcore point processes and point processes more generally). We provide some background in Section 2, explain the assumptions and the rationale behind the designs in Section 3 and show the results in Section 4.

## 2. Background

The problem of sampling for minimizing the detection time of a disease requires to formalize the notion of sampling from a point pattern $\xi$, which amounts to select individuals and assess their sanitary statuses. Such a sampling can be defined as a thinning of the pattern $\xi$, the points being thinned corresponding to sampled individuals (see [12] about the notion of thinning). One needs to account for multiple possibilities for this thinning as the selected points can be independent or dependent, and in the latter case sequential. We now formalize these notions.

### 2.1. Thinned point processes

Let $W \subset \mathbb{R}^2$ be the observation window, $N = N(W)$ the space of locally finite point configurations on $W$ and $\mathcal{F}$ the corresponding $\sigma$-field. Let $X$ be a point process on $W$ with law $P_X$.
### 2.1.1. Non-sequential thinning

A thinned process $\tilde{Y} \subseteq X$ is specified by a probability kernel

$$K: \mathcal{N} \times \mathcal{F} \to [0, 1],$$

such that for each $\xi \in \mathcal{N}$ the measure $K(\xi, \cdot)$ is supported on the set of sub-configurations of $\xi$: $K(\xi, \{\eta: \eta \subseteq \xi\}) = 1$. The marginal law of $\tilde{Y}$ is

$$P_{\tilde{Y}}(B) = \int_{\mathcal{N}} K(\xi, B) P_X(d\xi), \quad B \in \mathcal{F}.$$

*Independent thinning.* If there exists a measurable retention function $\pi: W \to [0, 1]$ and, conditional on $X = \xi$, independent Bernoulli marks with $\mathbb{P}(u \in \tilde{Y} | X = \xi) = \pi(u)$ for each $u \in \xi$, then the kernel factorizes on $\eta \subseteq \xi$ as

$$K(\xi, \{\eta\}) = \prod_{u \in \eta} \pi(u) \prod_{v \in \xi \setminus \eta} (1 - \pi(v)).$$

In particular, for any measurable $B \subset W$,

$$\mathbb{E}[|\tilde{Y} \cap B|] = \mathbb{E}[\sum_{x \in X} 1\{x \in B\} \pi(x)],$$

and if $X$ has intensity $\lambda_X(u)$ then $\tilde{Y}$ has intensity $\lambda_{\tilde{Y}}(u) = \pi(u) \lambda_X(u)$. Moreover, the $n$-th product density satisfies

$$\rho_{\tilde{Y}}^{(n)}(u_1, \dots, u_n) = \pi(u_1) \cdots \pi(u_n) \rho_{X}^{(n)}(u_1, \dots, u_n),$$

where $\rho_Z^{(n)}(u_1, \dots, u_n) du_1 \dots du_n$ is the probability that the point process $Z$ contains $n$ points in infinitesimal regions of volumes $du_i$ around $u_i$ for $1 \le i \le n$.

### 2.1.2. Sequential thinning

Conditional on a realization $X = \xi$, a sequential (ordered) selection of points

$$\vec{Y} = (y_1, y_2, \dots, y_{\tau}), \quad \tau \in \{1, 2, \dots, |\xi|\},$$

is specified by a family of measurable transition probabilities and stopping probabilities

$$p_i(u | \xi, y_1, \dots, y_{i-1}), \quad s_i(\xi, y_1, \dots, y_{i-1}) \in [0, 1], \quad s_{|\xi|+1}(\xi, y_1, \dots, y_{|\xi|}) = 1$$

for $i \ge 1$, satisfying for every $\xi$ and every previously chosen $y_1, \dots, y_{i-1} \in \xi$:

$$p_i(u | \xi, y_1, \dots, y_{i-1}) \ge 0, \quad u \in \xi, \qquad (1)$$

$$\sum_{u \in \xi} p_i(u | \cdot) = 1.$$

The function $u \to p_i(u | \xi, y_1, \dots, y_{i-1})$ obviously provides the conditional probability distribution of drawing $u$ given $\xi, y_1, \dots, y_{i-1}$, and the quantity $s_i(\xi, y_1, \dots, y_{i-1})$ provides the probability of stopping at time $i$ after drawing $y_{i-1}$. The probability of drawing the ordered tuple $(y_1, \dots, y_{\tau})$ conditional on stopping at time $\tau$ and $\xi$ is

$$K_{\text{ord}}(\xi; y_1, \dots, y_{\tau}) = \left( \prod_{i=1}^{\tau-1} p_i(y_i | \xi, y_1, \dots, y_{i-1}) \right) p_{\tau}(y_{\tau} | \xi, y_1, \dots, y_{\tau-1}).$$
For $i = 1, p_i(y_i | \xi, y_1, \dots, y_{i-1})(1 - s_i(\xi, y_1, \dots, y_i)) = p_1(y_1 | \xi)(1 - s_1(\xi))$.

The law of the ordered sequence $\tilde{Y}$ is given by integrating this kernel against $P_X$. The law of the

unordered retained set $\tilde{Y} = \{y_1, \dots, y_\tau\}$ is obtained by summing over permutations:

$$K(\xi, \{\eta\}) = \sum_{\pi \in \text{Perm}(\eta)} K_{\text{ord}}(\xi; \pi(\eta)), \quad \eta \subseteq \xi.$$

*Sequential hard-core example.* Given $\xi$, an exclusion radius $r > 0$ and already selected set

$S = \{u_1, \dots, u_{i-1}\}$, define the admissible set

$$A(\xi, S) = \{u \in \xi : \|u - v\| > r \text{ for all } v \in S\}.$$

A canonical sequential hard-core kernel chooses uniformly on $A(\xi, S)$ while nonempty and stops

when $A(\xi, S) = \emptyset$. Thus

$$p_i(u | \xi, u_1, \dots, u_{i-1}) = \frac{\mathbf{1}\{u \in A(\xi, S)\}}{|A(\xi, S)|},$$

$$s_i(\xi, u_1, \dots, u_{i-1}) = \mathbf{1}\{A(\xi, S) = \emptyset\},$$

where $|\cdot|$ is the counting measure.

Hence for an ordered admissible sequence $(u_1, \dots, u_m)$,

$$K_{\text{ord}}^{\text{hc}}(\xi; u_1, \dots, u_m) = \prod_{i=1}^{m} \frac{\mathbf{1}\{u_i \in A(\xi, \{u_1, \dots, u_{i-1}\})\}}{|A(\xi, \{u_1, \dots, u_{i-1}\})|} \quad \mathbf{1}\{A(\xi, \{u_1, \dots, u_m\}) = \emptyset\},$$

and the unordered-law is obtained by summation over permutations of $\{u_1, \dots, u_m\}$.

In our setup, the process $X$ corresponds to the individuals locations and $Y \subseteq X$ corresponds to

the selected individuals to be tested.

In what follows, all sequential distributions are understood to be defined according to (1).

## 3. Models and assumptions

Let $W$ be a bounded observation window with finite area $|W|$ and $X$ be the point process of individuals locations in $W$. There is an unknown infected region $D_t \subset W$ evolving over time, and a diagnostic test, which returns positive when applied at location $x \in X$ if and only if the sampled individual at $x$ is within $D_t$.

We adopt the following assumptions, made explicit because they determine the exclusion rule:

1. **Infected region:** Given a prescribed sample size $n$, $n$ samples are taken in the time interval $[0, 1]$. The sampling times form a regular partition of the interval $[0, 1]$ so that the sampling time of the $i^{th}$ sample is $t_i = \frac{i-1}{n-1}$. Denoting by $D_t$ the infected region at time $t$, we suppose that $D_1 = W$ so that every individual is infected at final time.

It is helpful to represent the infected region $(D_t)_{t \in [0,1]}$ as a differentiable function $f: W \mapsto [0, 1]$ that gives the propagation pathway as follows:

$$\{x \in W : f(x) \ge 1 - t\} = D_t$$

This way, the set $\{f = 1\}$ in $W$ corresponds to the initial infected region, $\{f = 0\}$ corresponds to the set of the last infected individuals, $1 - f(x)$ corresponds to the time of infection of $x \in W$ and $\frac{1}{|\nabla f(x)|}$ is the propagation speed at point $x$, where $|\cdot|$ is the euclidean distance.
2. Bounds on propagation speed: We assume $\frac{1}{|\nabla f(x)|}$ to be uniformly bounded in $x$:

$$v_{\min} \le \frac{1}{|\nabla f(x)|} \le v_{\max}, \forall x \in W.$$

Then, if a sampled individual at $x$ tests negative at time $t$, the infection cannot be present within the disc $B(x, r)$, where $r$ is chosen from knowledge or estimates of $v$ and the relevant time interval: typically $r = v_{\min}t$.

3. Propagation and initial infection: we assume there is only one origin of the infection, that is $|\{z \in W : f(z) = 1\}| = 1$, the propagation is not necessarily isotropic and the prior spatial distribution of the initial infection is uniform on $W$ or informed by covariates.

Under these assumptions, each negative sample removes the disc $B(x_t, r_t)$ from the set of plausible locations for $D_t$, where $x_t$ is the individual sampled at time $t$ and $r_t = v_{\min}t$. The fundamental idea is to choose subsequent sampling locations so that their exclusion discs cover previously untested area as efficiently as possible.

We denote by $z \in W$ the origin of the disease, by $v_t \in [v_{\min}, v_{\max}]$ the mean spatial propagation speed at time $t$ and we consider a prior over these parameters, although we are only interested on the mean propagation speed parameter $\bar{v}_t = 1/t \int_0^t v_t dt$. We then consider the following uniform prior over the parameters $(z, \bar{v}_t)$:

$$p(z, \bar{v}_t) = p(z)p(\bar{v}_t) \propto \mathbf{1}\{z \in W\}\mathbf{1}\{\bar{v}_t \in (v_{\min}, v_{\max}]\}$$

Now given some sampling scheme $S = ((y_1, t_1), \dots, (y_i, t_i))$ consisting of non infected samples and their times of sampling, we update the information about $(z, \bar{v}_{t_i})$. Because the test results are deterministic, the likelihood of observing $S$ is:

$$L_S(z, \bar{v}_{t_i}) = \prod_{j=1}^{i} \mathbf{1}\{\bar{v}_{t_j} < \frac{\|y_j - z\|}{t_j}\}.$$

Thus, the posterior of $(z, \bar{v}_{t_i})$ satisfies

$$p(z, \bar{v}_{t_i} | S) \propto p(z) p(\bar{v}_{t_i}) L_S(z, \bar{v}_{t_i}).$$

We define the hitting probability of the infected set at time $t$ by the point $u \in W$ conditionally on the sampling set $S$ as the probability that the point $u$ lies in the infected set $D_t$:

$$ \begin{aligned} h_t(u|S) &= \mathbb{P}(u \in D_t|S) \\ &= \mathbb{P}(\bar{v}_t \ge \frac{\|u-z\|}{t}) \\ &= \int_W \int_{v_{\min}}^{v_{\max}} \mathbf{1}\{\bar{v}_t \ge \frac{\|u-z\|}{t}\} p(\bar{v}_t, z|S) d\bar{v}_t dz. \end{aligned} \quad (2) $$

The marginal posterior propagation speed at time $t$ conditional on the sampling set $S$ satisfies

$$p(\bar{v}_t|S) = \int_W p(\bar{v}_t, z|S) dz. \quad (3)$$
### 3.1. Sequential hard-core sampling

#### 3.1.1. Definition

A sequential hard-core sampling scheme of exclusion radii ($r_{in}$)$_{1<n,1\le i\le n} \ge 0$ is defined as follows. Let $\xi$ be a point pattern realization of X and $y_1$ be the first sample location chosen according to some initialization rule (e.g., risk-based or random). For $k \ge 1$, having obtained negative test results at locations $y_1, \dots, y_k$, choose the next sampling location $y_{k+1}$ uniformly at random on the remaining feasible set

or, more generally, according to a design density supported on $F_{k+1}$. The procedure halts either when a positive test is observed or after $n$ tests. If, at a given time step $l$, the feasible set is empty, then no sample is collected at that step, and sampling resumes at the subsequent time step. This corresponds to sampling with the sequential density given by

$$p_{k+1}(u | \xi, y_1, \dots, y_k) = \frac{\mathbf{1}\{u \in F_{k+1}\}}{|F_{k+1}|} \quad (4)$$

When furthermore choosing the design density on $F_{k+1}$ as the hitting probability of the infected set, we denote it by $p_{k+1}^h$:

$$p_{k+1}^h(u | \xi, y_1, \dots, y_k) \propto \mathbf{1}\{u \in F_{k+1}\}h_{k+1}(u|S_k) \quad (5)$$
where $S_k = (y_1, t_1), \dots, (y_k, t_k)).

#### 3.1.2. Rationale

The exclusion radii $r_{i,k}$ encodes how much spatial area is excluded by a single negative observation. By forbidding subsequent samples to lie near previously-tested points, the design maximizes the marginal new area excluded by each additional negative test. Intuitively, we seek to cover W with $n$ discs $B(Y_i, r_{i,n})$ so as to minimize the remaining uncovered individuals

$$U_n = |\xi \cap W \setminus \bigcup_{i=1}^{n} B(Y_i, r_{i,n})|,$$

If the infection location were uniform on W, the worst-case probability of missing the infection after $n$ negative tests is $U_n/|W|$. Thus minimizing $U_n$ is equivalent to minimizing the miss probability when the prior infection location is uniform over W.

This maximum coverage objective function satisfies the adaptive submodularity and monotonicity properties.

**Adaptive Submodularity.** In the study of stochastic optimization problems, the notion of *adaptive submodularity* is the extension of the classical submodularity to sequential decision-making under uncertainty. Formally, a set function is said to be adaptive submodular if, conditioned on any partial realization of the random outcomes, the expected marginal gain of selecting an additional element does not increase as more information is revealed. Intuitively, this property ensures a “diminishing returns” behavior in adaptive settings, as in the classical submodular structure. A fundamental consequence of adaptive submodularity is that the simple *adaptive greedy*
policy, which iteratively selects the element with the largest expected marginal contribution given the current knowledge, achieves provably near-optimal performance [13]. Indeed, under adaptive submodularity and adaptive monotonicity, the greedy policy guarantees a constant-factor approximation to the optimal adaptive strategy: the performance of the greedy policy is at least as good as $1 - 1/e$ times the performance of the optimal strategy.

In our context, this implies that, when transitioning from step $i$ to step $i + 1$, selecting the location that maximizes the marginal coverage starting from $U_i$ as in the sequential hard-core sampling guarantees a coverage that is at least a factor of $1 - 1/e$ of the optimal coverage.

Importantly, our sequential sampling method never relies on the infection status of untested individuals; the algorithm updates its feasible region solely from the negative tests already collected, meaning that the full underlying infection pattern remains unknown and does not need to be inferred.

### 3.1.3. Radii specification

Let's suppose the disease propagation speed of the infected region depends on time, as denoted by $v_t$ for the speed at time $t$. Then, when sampling a negative point $x$ at time $t$, we know the disease was not present in a radius $r_t = \int_0^t v_t dt$. At time $t' > t$, this radius shrinks because of the possible disease spread, giving the radius $r_{t,t'} = (\int_0^t v_u du - \int_t^{t'} v_u du)_+$, where $x_+ = \max(x, 0)$. For sample indices $i, k \in (1, \dots, n)$, we denote by $r_{i,k}$ and $r_i = r_{i,i}$ the following data dependent radii:

$$r_{i,k} = r_{t_i, t_k} = (t_i \bar{v}_{t_i}^{S(k-1)} - (t_k - t_i) \bar{v}_{t_k}^{S(k-1)})_+ \quad (6)$$ $$r_i = r_{t_i} = t_i \bar{v}_{t_i}^{S(i-1)}$$

where $S_k = ((Y_1, t_1), \dots, (Y_k, t_k))$ and $\bar{v}_t^S$ is the marginal posterior mean speed at time $t$ from the samples $S$:

$$\bar{v}_t^S = \int_{v_{\min}}^{v_{\max}} v p(v|S) dv, \quad (7)$$

with $p(v|S)$ the marginal posterior propagation speed expressed in Equation (3). This radius $r_{i,k}$ corresponds to the disease-free radius around a point sampled at the $i^{th}$ step, measured at the time of the $k^{th}$ sample. The radius $r_i$ denotes the disease-free radius associated with the point sampled at the $i^{th}$ step, as measured at the moment of its sampling.

## 3.2. Sequential area interaction sampling

### 3.2.1. Motivation

In practice, precise knowledge of propagation speed(s) $v$ and relevant time-window is rarely available. Using a hard exclusion may be overconfident: an actual infection could lie within the forbidden radius if $v$ is overestimated, or temporal dynamics are more complex. To accommodate uncertainty we introduce a softer rule that permits some overlap of exclusion discs but penalizes overlap so that new samples still favor new area.

This is a sequential *area interaction* point process: the probability density for the next point depends on the areas already covered by previous discs.
### 3.2.2. Construction

Let $\phi \ge 0$ be an interaction parameter controlling the penalty for overlap. After negative tests at $y_1, \dots, y_k$, define a sequential sampling density on $W$ of the form

$$p_{k+1}(u | \xi, y_1, \dots, y_k) \propto \phi^{-C_{k+1}(u)}, \qquad (8)$$

$$C_{k+1}(x) = \frac{|B(u, r_{k+1}) \setminus \bigcup_{i=1}^{k-1} B(y_i, r_{i,k+1})|}{\pi r_{k+1}^2} - 1,$$

where the radii $(r_{i,k})$ and $(r_i)$ are chosen as the data dependent radii in (3.1.3). $C_k(u)$ measures the overlap of a ball centered at $u$ with balls centered on previous samples. The density (8) corresponds to the Papangelou conditional intensity of the classical area interaction point process with non stationary radii (see [11] for a detailed account of the original area-interaction point process with a fixed interaction radius $r > 0$).

For $\phi = 0$, we retrieve the sequential hard-core process.

For $0 < \phi < 1$, the corresponding process tends to be repulsive, meaning that we penalize the discs overlap.

For $\phi = 1$, the density is uniform over $W$.

For $\phi > 1$, the process tends to be clustered.

As for the sequential hard-core sampling scheme, we also extend the area-interaction sampling strategy by introducing a non-stationary base intensity:

$$p_{k+1}(u | \xi, y_1, \dots, y_k) \propto h_{t_{k+1}}(u | S_k) \phi^{-C_{k+1}(u)}, \qquad (9)$$

which assigns weights to regions proportionally to their posterior probability of being reached by the disease front. The rationale underlying the two terms is as follows:

* If the disease were to be detected at the next step, the optimal choice would be to sample location $u$ with probability $h_{t_{k+1}}(u | S_k)$, i.e. the probability of belonging to the infected set.

* Conversely, if the disease were not detected, the most effective strategy would be to maximize the contraction of the domain of the hitting probability, thereby favouring repulsive sampling.

Relying solely on the first term would amount to a greedy strategy, whereas combining both terms promotes a balanced approach that ensures long-term efficiency.

*Remark.* Building on the preceding rationale, the sequential densities may be generalized as

$$p_{k+1}(u | \xi, y_1, \dots, y_k) \propto h_{t_{k+1}}(u | S_k)^{a_{k+1}} (\phi^{-C_{k+1}(u)})^{b_{k+1}},$$

where the sequences $(a_i)_{1 \le i \le n}$, $(b_i)_{1 \le i \le n}$ take values in $[0, 1]$. $(a_i)_{1 \le i \le n}$ may, for instance, be chosen as non-decreasing. The underlying idea is that, at the final time step $n$, one aims to maximize the probability of detecting the disease, which corresponds to setting $a_n = 1$. Conversely, during the earlier stages, when the estimated hitting probability is less reliable, smaller values of $a_i$ may be preferable.
### 3.2.3. Rationale

Assume that the initial infected set $D$ is uniformly distributed on $W$. If after $k$ samples the union of exclusion discs has area $E_k$, then the probability that the $(k+1)^{th}$ sample misses the infection is bounded by

$$ \mathbb{P}(\text{miss at } k+1) \le 1 - \frac{\mathbb{E}[E_k]}{|W|}. $$

Therefore designs that maximize the expected covered area $\mathbb{E}[E_n]$ minimize the miss probability. The hard-core rule is efficient because it maximizes the marginal increase in $E_k$ at each step. The area-interaction rule trades some marginal increase for robustness when $v$ is uncertain.

### 3.3. Algorithmic implementation

A practical implementation with a budget of $n$ tests proceeds as follows.

1. Choose an initial sample $Y_1$ (e.g., random, risk-based).

2. For $k = 1, \dots, n$:

(a) Test at $Y_k$. If the result is positive, stop (detection).

(b) Otherwise update the feasible region: $F_{k+1} = W \setminus \cup_{i=1}^k B(Y_i, r_{k+1} + r_{i,k+1})$.

(c) For hard-core: sample $Y_{k+1}$ uniformly on $F_{k+1}$ with distribution (4), or from the distribution (5).

(c') For area-interaction: sample $Y_{k+1}$ from one of the densities (8, 9).

3. If $n$ tests are exhausted and are all negatives, declare no detection.

## 4. Simulation study

To assess the efficiency of our sampling strategy, we consider the following experimental setting. The spatial window is $W = [0, 10]^2$, and a total budget of $n = 100$ samples is available for disease detection. For each simulation, the underlying population $\xi$ consists of 1000 individuals independently and uniformly distributed over $W$. The disease origin $z$ is drawn uniformly from $W$, and the disease propagation is given by

$$ f(x) = \left( 1 - \frac{1}{v} \|x - z\| \right)_+, $$

where the propagation speed $v$ is uniformly drawn in the interval $[2, 8]$ for each simulation.

### 4.1. Illustration

To provide some intuition regarding the behaviour of the sequential area interaction model with sampling density (9), Figures 1 and 2 illustrate a realization of the sampling procedure with $\phi = 0.1$. In the early stages of disease propagation, the influence of a previously sampled point is limited, as the potential range of propagation remains small. Consequently, the resulting sampling density closely resembles the estimated hitting probability derived from earlier samples. As time progresses, however, the informational value of a negative sample increases, and its corresponding interaction extends over a broader spatial range. This effect gradually “sparsifies” the sampling space: locations in close proximity to a negative sample get a reduced sampling density. As a result, the overall sampling distribution becomes more contracted compared to the raw hitting probability. In this sense, the interaction term can be interpreted as a form of regularization imposed on the base sampling intensity.
<figure><img src="image_3.png" /><figcaption>Figure 1: Temporal evolution of the sequential area interaction model’s components with density (9). The first column’s map displays the estimated hitting probability (2). The second column’s map displays the interaction component (8) associated with the sampling points. The final column displays the resulting sampling density (9), where warmer (orange) tones indicate higher density values. Contour lines are the level sets of the corresponding functions. Rows correspond to the sequence of samples up to 25, 55 and 65.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2: One realization of the model. The first column displays the propagation function *f*, where darker purple shades indicate earlier times of infection. The second column displays the hitting probability computed from all locations except the detection point. The third column displays the interaction component, also excluding the detection point. The final column shows the complete sampling density. Detection is at sample 68.</figcaption></figure>

## 4.2. Benchmark

Because the propagation speed is unknown, we benchmark our sequential area interaction approach against three competing sampling strategies over 1000 realizations of disease propagation:

* *Uniform:* *n* samples are drawn independently and uniformly from ξ.

* *Cluster:* 10 cluster centers are uniformly chosen among ξ. Around each center, 10 points are sampled according to a Gaussian kernel with bandwidth 0.5. The clusters are then assigned a random order, and their points are sampled sequentially.

* *Repulsive:* *n* individuals are selected using the Spatially Correlated Poisson Sampling (SCPS) method [14], as implemented in the `BalancedSampling` package [15]. This method yields spatially balanced samples. The selected individuals are then ordered to maximize the cumulative inter-sample distances $\sum_{i=1}^{n-1} \|y_{i+1} - y_i\|$, which amounts to solving a variant of the Traveling Salesman Problem, implemented using the `TSP R` package [16]. This yields overall regular samples in space and time.

* *SAIS (Sequential Area Interaction Sampling)*: our Bayesian Area Interaction approach with $\phi = 0.1$, in which, at each step, the sampling radius is determined by the posterior mean of the marginal posterior propagation speed $p(\bar{v}_t | S)$. We also experimented with alternative specifications of our model, such as using different quantiles of the distribution $p(\bar{v}_t | S)$, varying the interaction parameter with $\phi \in \{0.5, 1\}$ (where $\phi = 1$ corresponds to the absence of interaction), as well as a Thompson Sampling version in which a realization from $p(\bar{v}_t | S)$ was drawn at each step. All these variants outperformed the three baseline models; however, the choice $\phi = 0.1$ combined with the posterior mean of $p(\bar{v}_t | S)$ for the radii specification yielded the best performance in terms of average detection time of the disease.

Figure 3 displays the distribution of detection times across the 1000 simulations. As also illustrated in Figure 4, the miss probability is null for our method. Somewhat unexpectedly, the regular method does not outperform the uniform one, while the cluster method is clearly dominated by the three other approaches. This indicates that additional structure must be incorporated into a regular sampling pattern in order to enhance early detection, and that naïve inhibitory strategies are ill-suited to this problem. The underlying reason is that samples progressively lose
informational value over time, as the advancing disease front reduces their discriminatory power. Regular strategies, however, tend to assign equal weights to all samples, treating them as mutually repulsive irrespective of their sampling time. Finally, our method achieves a 7.7% reduction in mean detection time relative to the uniform method. Of course, the magnitude of the improvement depends on the amount of prior information available regarding the disease propagation speed and its initial location.

Figure 5 presents the pairwise comparison of the four competing methods. Conditionally on the detection time of a reference method, we assess the relative performance of the others. The results indicate that the Uniform and SCPS methods are not significantly different, while all approaches outperform the Cluster method. Furthermore, our method consistently outperforms all alternatives. The plot also reveals that when the mean detection times of two methods are low, the differences between them are not statistically meaningful, suggesting that early detections are largely attributable to chance. Conversely, when the mean detection time increases, our method shows a clear advantage, underscoring the benefit of leveraging information from past samples.

<figure><img src="image_5.png" /><figcaption>Figure 3: Comparison of the detection time distribution of different instances of our model with the uniform sampling method. The sample progression is the fraction of sampling budget at the time of detection. The points represent the mean detection time for each strategy.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4: Tail probability of the fraction of sampling budget at the time of detection for each sampling model.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Cross comparison for the different sampling strategies. Each column corresponds to a base strategy with detection times ($T_i$)$_i$. For every other strategy with detection times ($T'_i$)$_i$, we plot the mean ($T_i + T'_i$)/2 in the x-axis and $T_i - T'_i$ in the y-axis.</figcaption></figure>

These simulation outcomes are intuitively explained by the efficient incremental coverage property of inhibitory designs: negative tests “carve out” confirmed-safe discs, and forcing subsequent points outside those discs prevents redundant testing of already-excluded areas.

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## 5. Discussion

We discuss how the model defined in Section 3 accounts for some apparently more general settings.

**Multiple disease origins.** In theory, this case can be addressed with appropriate choices of pri-ors. However, in practice, it becomes challenging since Bayesian inference of the locations is currently performed on a grid. Monte Carlo integration could provide a solution, as it is in-dependent of the dimensionality. Nevertheless, this scenario is not particularly relevant to our objective, as we are only concerned with detecting a single infected individual, regardless of whether it belongs to one disease cluster or another. Moreover, under the situation where only negative test results are observed, an identifiability issue arises: the number of disease origins and the propagation speed cannot be separated, since an increase in the number of origins combined with a decrease in propagation speed would yield similar observational patterns.

**Knowledge about the starting time of disease propagation.** Assuming prior knowledge of the exact time at which the disease propagation begins may appear to be a restrictive hypothesis. However, this uncertainty can be incorporated in a natural way by broadening the prior interval $[v_{min}, v_{max}]$ for the propagation speed. Indeed, if the disease had started before time 0, this would be equivalent to a disease with a higher propagation speed during the initial time periods, starting at time 0. Conversely, if the disease were to start after time 0, this would be equivalent to a slower propagation process beginning at time 0. For this model, the likelihood can be written as

$$L_S(z, v, t_0) = \prod_{j=1}^{i} \left( \mathbf{1}\{t_j < t_0\} + \mathbf{1}\{t_j \ge t_0\} \mathbf{1}\{v < \frac{\|y_j - z\|}{t_j}\} \right),$$
where $t_0 \in \mathbb{R}$ denotes the starting time of propagation. Now, consider $t'_0 = 0$. Then, there exists $v' \ge 0$, such that, taking the expectation over the sample locations under independent uniform draws, we obtain

$$\mathbb{E}[L_S(z, v, t_0)] = \mathbb{E}[L_S(z, v', t'_0)],$$
which implies that an appropriate transformation of the prior interval $[v_{min}, v_{max}]$ is sufficient to account for different possible starting times of propagation.

**Partial disease propagation.** In our framework, at each time step, all individuals within a certain distance of the disease cluster are assumed to be infected. A more realistic situation would be that only a proportion $q \in (0, 1]$ of these individuals are infected, with the complementary proportion $1-q$ remaining uninfected. Moreover, $q$ could depend on the distance from the cluster. This extension can be naturally incorporated into the Bayesian analysis by introducing $q$ as an additional parameter, or more generally by using a parameter that governs the spatial propagation of the disease, such as the bandwidth of a kernel.
This situation is equivalent to redefining the propagation speed as $v' = v \sqrt{q}$ and $q' = 1$. In cases where $q$ is uncertain, this corresponds to adopting a lower prior bound for the minimal propagation speed $v_{min}$.

Indeed, for this model, the likelihood can be written as

$$L_S(z, v, q) = \prod_{j=1}^{i} \left( \mathbf{1}\{v < \frac{\|y_j - z\|}{t_j}\} + (1-q)\mathbf{1}\{v \ge \frac{\|y_j - z\|}{t_j}\} \right).$$
Now, consider $v' = v\sqrt{q}$ and $q' = 1$. Then, taking the expectation over the sample locations under independent uniform draws, we obtain

$$ \mathbb{E}[L_S(z, v, q)] = \mathbb{E}[L_S(z, v', q')], $$

which implies that, in expectation, the decision rule based on the collected data remains unchanged. We argue that, in this case of a “weak” likelihood principle, one should prefer the simpler model over the more complex one, since the additional complexity does not yield meaningful improvements in inference.

*Imperfect test method.* In practice, diagnostic tests may not always be perfectly reliable. For instance, a test might correctly identify the health status of an individual with only 80% accuracy. The approach described for partial disease propagation can also be applied here, by introducing a parameter $q$ that accounts for the test sensitivity or adapting the prior bounds for the propagation speed. If multiple diagnostic tests are available, each associated with specific costs and sensitivities, [17] provides a general rule of thumb for selecting the most appropriate diagnostic test under such circumstances.

## 6. Conclusion

Sequential Area Interaction Sampling that exploits negative-test information via exclusion discs provides a conceptually simple and practically effective strategy for early detection of plant disease. The hard-core design maximizes the newly covered area at each step and thereby yields lower miss probabilities, while the area-interaction variant offers robustness to uncertainty in the disease propagation speed. The proposed approach is flexible, as it can incorporate risk covariates, and it remains straightforward to implement in practice.

Simulation results demonstrate that naive repulsive designs do not improve upon uniform designs, since they fail to account for the time-dependent information carried by individual samples. By contrast, our methodology, which integrates the information from previously sampled points into a repulsive design, outperforms uniform independent sampling, naive repulsive sampling, and clustered sampling strategies.

Future work will aim to formalize theoretical coverage guarantees under specific stochastic models for $D_t$, for instance by modelling $f$ as a Gaussian process. Another important direction concerns the development of sequential stopping rules, namely determining the number of negative tests required to confidently conclude the absence of disease with high probability. This line of research would generalize the framework of [18], which prescribes the sample size prior to a surveillance campaign. Furthermore, temporal aspects of surveillance design will be investigated in order to extend beyond the regular temporal sampling schedules considered in this study.

## Acknowledgements

This work was supported by the BEYOND project (grant ANR-20-PCPA-0002) and the HORIZON BeXyl Project (grant 101060593).
References

[1] A. J. Mastin, T. R. Gottwald, F. van Den Bosch, N. J. Cunniffe, S. Parnell, Optimising risk-based surveillance for early detection of invasive plant pathogens, PLoS biology 18 (2020).

[2] D. Yemshanov, R. G. Haight, F. H. Koch, R. C. Venette, T. Swystun, R. E. Fournier, M. Marcotte, Y. Chen, J. J. Turgeon, Optimizing surveillance strategies for early detection of invasive alien species, Ecological Economics 162 (2019) 87–99.

[3] S. Parnell, T. Gottwald, T. Riley, F. Van Den Bosch, A generic risk-based surveying method for invading plant pathogens, Ecological Applications 24 (2014) 779–790.

[4] V. Martin, X. Zhou, E. Marshall, B. Jia, G. Fusheng, M. A. FrancoDixon, N. DeHaan, D. U. Pfeiffer, R. J. S. Magalhães, M. Gilbert, Risk-based surveillance for avian influenza control along poultry market chains in south China: the value of social network analysis, Preventive veterinary medicine 102 (2011) 196–205.

[5] D. Martinetti, S. Soubeyrand, Identifying lookouts for epidemio-surveillance: application to the emergence of *Xylella fastidiosa* in France, Phytopathology 109 (2019) 265–276.

[6] C. E. Morris, G. Géniaux, C. Nédellec, N. Sauvion, S. Soubeyrand, One Health concepts and challenges for surveillance, forecasting, and mitigation of plant disease beyond the traditional scope of crop production, Plant Pathology 71 (2022) 86–97.

[7] M. N. M. van Lieshout, Campbell and moment measures for finite sequential spatial processes, in: Proceedings
Prague Stochastics 2006 (Prague, Czech Republic, August 21-25, 2006), Matfyzpress, 2006, pp. 215–224.

[8] A. Penttinen, A.-K. Ylitalo, Deducing self-interaction in eye movement data using sequential spatial point processes, Spatial Statistics 17 (2016) 1–21.

[9] A. Yazigi, A. Penttinen, A.-K. Ylitalo, M. Maltamo, P. Packalen, L. Mehtätalo, Modeling forest tree data using sequential spatial point processes, Journal of Agricultural, Biological and Environmental Statistics 27 (2022) 88–108.

[10] F. d'Alayer, E. Gabriel, S. Soubeyrand, A marked sequential point process for disease surveillance: Modeling and optimization, Spatial Statistics 68 (2025). URL: https://www.sciencedirect.com/science/article/pii/S2211675325000351. doi:https://doi.org/10.1016/j.spasta.2025.100913.

[11] A. J. Baddeley, M. Van Lieshout, Area-interaction point processes, Annals of the Institute of Statistical Mathematics 47 (1995) 601–619.

[12] S. N. Chiu, D. Stoyan, W. S. Kendall, J. Mecke, Stochastic geometry and its applications, John Wiley & Sons, 2013.

[13] D. Golovin, A. Krause, Adaptive submodularity: Theory and applications in active learning and stochastic optimization, Journal of Artificial Intelligence Research 42 (2011) 427–486.

[14] A. Grafström, Spatially correlated poisson sampling, Journal of Statistical Planning and Inference 142 (2012) 139–147. URL: https://www.sciencedirect.com/science/article/pii/S0378375811002734. doi:https://doi.org/10.1016/j.jspi.2011.07.003.

[15] A. Grafström, J. Lisic, W. Prentius, M. A. Grafström, Package ‘balancedsampling’, Open Journal of Statistics 3 (2018) 1.

[16] M. Hahsler, K. Hornik, Tsp—infrastructure for the traveling salesperson problem, Journal of Statistical Software 23 (2008) 1–21.

[17] A. J. Mastin, F. van den Bosch, F. van den Berg, S. R. Parnell, Quantifying the hidden costs of imperfect detection for early detection surveillance, Philosophical Transactions of the Royal Society B 374 (2019).

[18] European Food Safety Authority (EFSA), A framework to substantiate absence of disease: the risk based estimate of system sensitivity tool (RiBESS) using data collated according to the EFSA Standard Sample Description - An example on *Echinococcus multilocularis*, EFSA Supporting Publications 9 (2012).