Research on Plant Disease and Pest Diagnosis Model Based on Generalized Stochastic Petri Net

Abstract

With the advancement of modern agricultural technology and the expansion of large-scale production, this article aims to solve the difficulties in plant disease and pest control through the application of artificial intelligence and automation technology, and provide accurate disease and pest warning mechanisms. This study first conducted a detailed identification and classification of plant disease and pest warning mechanisms, and established a dynamic model of disease and pests based on the environmental factors and symptoms of affected areas. On this basis, using the isomorphism relationship between generalized stochastic Petri nets and Markov chains, a plant disease and pest diagnosis model based on generalized stochastic Petri nets and an equivalent Markov chain model were constructed. The simulation results show that different combinations of infection rates have a significant impact on the probability of meeting treatment standards, with the combination of moderate and severe infection rates having the greatest impact on the probability of meeting treatment standards, while the impact of mild infection rates is relatively small. By comprehensively analyzing the interaction between mild, moderate, and severe infection rates, the critical zone surface under different disease and pest warning thresholds was obtained. Through actual data verification, the generalized stochastic Petri net model can effectively quantify the dynamic characteristics of disease and pest propagation. Combined with the equivalent analysis of Markov chains, it can provide key thresholds and decision support for disease and pest warning. This method provides a theoretical basis for automated monitoring and precise control of pests and diseases in large-scale agricultural planting, and it has high practical application value.

1. Introduction

The modernization of agriculture is characterized by achieving economies of scale through modern cultivation practices during production and management processes. It serves as an effective strategy to enhance management efficiency, reduce production costs, and ultimately improve agricultural economic benefits [1]. This strategy demands a higher yield efficiency and rational resource utilization, while also introducing a series of modernization requirements, such as resource management, pest and disease control, and environmental protection. To meet these demands, modern agriculture increasingly relies on high-tech solutions, particularly artificial intelligence and automation technologies. Plant pests and diseases refer to various afflictions and infestations that impact the health and growth of plants, including damage caused by fungi, bacteria, viruses, insects, and other pests. These diseases and pests not only deteriorate the appearance and utility value of plants but can also, in severe cases, lead to plant death, resulting in significant economic losses in the agricultural industry [2].

Traditional methods for identifying plant diseases and pests mainly rely on manual visual diagnosis, using expert knowledge along with textual and visual information from books or online resources. With the proliferation of machine vision devices equipped with cameras and internet capabilities, computer vision technology has provided new avenues for modern crop pest and disease monitoring. This study primarily involves two types of machine learning algorithms: unsupervised learning methods, such as k-means clustering, and supervised learning methods using support vector machines. In the context of plant pest and disease diagnosis, k-means clustering is mainly used for data-driven early anomaly detection and lesion classification. Meena Prakash et al. applied GLCM, KMC, and SVM algorithms to classify diseases in citrus leaf images, achieving a classification accuracy of 90% with their proposed strategy [3]. Similarly, Krithika and Veni used KMC and SVM classifiers to categorize cucumber leaf diseases [4]. SVM, on the other hand, is commonly used for multi-class classification tasks involving known disease types and for classifying small-sample, high-dimensional data. For instance, Motie et al. proposed a method based on drone aerial imagery and SVM classifiers. By analyzing near-infrared and visible light images, they effectively identified wheat plants affected by Sunn pest Eurygaster integriceps and mapped their distribution [5]. However, this technology requires significant manual intervention by experts or programmers, particularly during the feature extraction phase. It involves manually inputting extracted feature lists into traditional neural networks for data classification, which can be a labor-intensive process. The process is not only complex but also demands significant human and material resources. Despite the development of numerous disease diagnostic systems, their widespread application and adoption remain limited.

Due to the complexity and operational challenges associated with machine learning classification techniques, deep learning algorithms have become a major focus of current research both domestically and internationally. Hernández and Lópe propose a Bayesian deep learning approach for plant disease detection, emphasizing the significance of uncertainty quantification in deep learning models to enhance reliability and generalization in real-world agricultural applications [6]. Nagasubramanian et al. utilized MobileNetV2 to classify various plant stress phenotypes in soybean, demonstrating its efficient ability to detect and classify biotic and abiotic stresses while maintaining computational efficiency. This approach supports real-time, on-field applications for plant disease diagnosis [7]. In addition, various network architectures such as Alex Net [8],VGG-16 [9], GoogLeNet [10], ResNet [11], and Inception [11] have been proposed and applied in disease recognition. Although significant progress has been made in crop disease identification using deep learning methods, there are still limitations, including limited generalization ability, low accuracy in early and small lesion detection, and large computational requirements that make practical deployment difficult [12]. Furthermore, the significant symptom variability at different disease stages, symptom similarities among different diseases, and the coexistence of multiple diseases add complexity to the diagnostic process.

In the agricultural domain, the application of Petri nets has been gaining increasing attention, especially in agricultural planting and management systems. Yang et al. developed a Petri net model based on an Internet of Things system to achieve real-time monitoring of eggplant cultivation, validating the system’s effectiveness in precision agriculture [13]. Geng et al. utilized the stochastic Petri net to construct a predictive model for soil fertility trends, demonstrating its effectiveness using existing soil fertility data and highlighting the potential of Petri nets in analyzing complex agricultural systems and providing decision support [14]. Wang et al. applied a colored Petri net-based adaptive control approach in their study of an intelligent broad bean harvesting system, successfully achieving real-time optimization and decision support in the agricultural harvesting process [15]. These studies collectively demonstrate the potential of Petri nets in agricultural production, particularly in addressing the dynamics and complexities of agricultural production systems.

In light of these challenges, this paper proposes a plant pest and disease diagnostic model based on generalized stochastic Petri nets (GSPNs). The GSPN demonstrates significant advantages in handling system dynamics and complexity, effectively modeling and analyzing the progression of pests and diseases and their interactions with environmental factors. This method enables the development of dynamic and highly adaptive diagnostic systems. Moreover, GSPN models exhibit greater flexibility and efficiency in addressing diagnostic challenges involving multiple pests and diseases in complex backgrounds. Most existing Petri net research focuses on areas such as power grid failures [16], emergency response [17], logistics scheduling [18], and the spread and evolution of infectious diseases [19], often conducting sensitivity analysis on a single indicator. However, there has been limited application of Petri nets in the field of plant pest and disease early warning mechanisms. Therefore, based on the construction of a dynamic pest and disease model, this research establishes a generalized Petri net model and an equivalent Markov chain model for the propagation and evolution of plant pests and diseases, utilizing the isomorphic relationship between GSPN and Markov chains. Finally, the equilibrium states and variation patterns of the pest and disease propagation system are analyzed using Markov chains and related mathematical methods. This research direction is expected to provide more effective technical support for plant pest and disease management and is of great significance in promoting the modernization and scaling up of the agricultural industry.

2. Early Warning System for Plant Pests and Diseases

2.1. Influence of Environmental Factors

The occurrence of plant diseases is closely related to environmental factors such as water availability, nutrient levels, temperature, humidity, light exposure, and PH value. Proper control of these environmental conditions is key to preventing diseases and ensuring healthy plant growth. The classification and definition of environmental factors related to diseases are shown in

Table 1. Classification and definition of environmental factors for diseases.

C	Disease	C	Moisture	C	Fertilizer	C	Temperature	C	Humidity	C	Light	C	PH Value
D1	Downy Mildew	E1	Dry	A1	Excessive Nitrogen Fertilizer	TR1	15–20 °C	H1	High Humidity	L1	Low Light	PH1	Alkaline
D2	Powdery Mildew	E2	Insufficient Water	A2	Insufficient Fertilization	TR2	20–25 °C	H2	Moderate Humidity	L2	Moderate Light	PH2	Neutral
D3	Gray Mold	E3	Water Excess	A3	Proper Fertilizer Application	TR3	25–30 °C	H3	Low Humidity	L3	High Light	PH3	Acidic
…
Dm	Disease m	Ea	Moisture Condition a	Ab	Fertilization Status b	TRc	Temperature Condition c	Hd	Humidity Condition d	Le	Light Condition e	PHf	PH Value Condition f

. Among them, C represents the coding of various environmental factors.

The relationship between plant diseases and environmental factors can be described in the form of a function, where the disease ${\mathrm{D}}_{\mathrm{m}}$ is a function of various environmental factors. Taking the first pest and disease in Table 1 as an example, the relationship between downy mildew ${\mathrm{D}}_1$ and environmental factors can be expressed in the following form:

$${\mathrm{D}}_{11}=\mathrm{f}\left({\mathrm{E}}_3{, \mathrm{A}}_1{, \mathrm{TR}}_1{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_1\right)$$

(1)

Formula (1) indicates that downy mildew is more likely to occur in areas with standing water (${\mathrm{E}}_1$), especially when excessive nitrogen fertilizer (${\mathrm{A}}_1$) increases the number of susceptible young leaves. The disease develops rapidly within a temperature range of 15–20 °C ($\mathrm{T}{\mathrm{R}}_1$). High humidity (${\mathrm{H}}_1$) promotes the spread and development of downy mildew, while excessive shading and low light (${\mathrm{L}}_1$) also contribute to the expansion of the disease. Additionally, an alkaline soil environment ($\mathrm{P}{\mathrm{H}}_1$) further exacerbates the occurrence of downy mildew. These environmental factors interact to create high-risk conditions for the disease.

Similarly, the disease dynamics model can be expressed as follows:

$${\mathrm{D}}_{21}=\mathrm{f}\left({\mathrm{E}}_1{, \mathrm{A}}_1{, \mathrm{TR}}_2{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_1\right)$$

(2)

$${\mathrm{D}}_{31}=\mathrm{f}\left({\mathrm{E}}_3{, \mathrm{A}}_1{, \mathrm{TR}}_1{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_3\right)$$

(3)

…

$${\mathrm{D}}_{\mathrm{m}1}=\mathrm{f}\left({\mathrm{E}}_{\mathrm{a}}{, \mathrm{A}}_{\mathrm{b}}{, \mathrm{TR}}_{\mathrm{c}}{, \mathrm{H}}_{\mathrm{d}}{, \mathrm{L}}_{\mathrm{e}}{, \mathrm{PH}}_{\mathrm{f}}\right)$$

(4)

In addition to plant diseases, the occurrence of pests is also greatly influenced by environmental factors.

Table 2. Classification and definition of environmental factors for pests.

C	Disease	C	Moisture	C	Fertilizer	C	Temperature	C	Humidity	C	Light	C	PH Value
D4	Aphids	E1	Dry	A1	Excessive Nitrogen Fertilizer	TR1	15–20 °C	H1	High Humidity	L1	Low Light	PH1	Alkaline
D5	Borers	E2	Insufficient Water	A2	Insufficient Fertilization	TR2	20–25 °C	H2	Moderate Humidity	L2	Moderate Light	PH2	Neutral
D6	Red Spider Mite	E3	Water Excess	A3	Proper Fertilizer Application	TR3	25–30 °C	H3	Low Humidity	L3	High Light	PH3	Acidic
…
Dn	Disease n	Ea	Moisture Condition a	Ab	Fertilization Status b	TRc	Temperature Condition c	Hd	Humidity Condition d	Le	Light Condition e	PHf	PH value Condition f

presents six environmental factors related to pest occurrence. By scientifically regulating these environmental parameters, the reproduction and spread of pests can be effectively suppressed, thereby ensuring the healthy growth of plants.

The pest dynamics model can be expressed sequentially as

$${\mathrm{D}}_{41}=\mathrm{f}\left({\mathrm{E}}_3{, \mathrm{A}}_1{, \mathrm{TR}}_2{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_3\right)$$

(5)

$${\mathrm{D}}_{51}=\mathrm{f}\left({\mathrm{E}}_2{, \mathrm{A}}_1{, \mathrm{TR}}_3{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_2\right)$$

(6)

$${\mathrm{D}}_{61}=\mathrm{f}\left({\mathrm{E}}_2{, \mathrm{A}}_1{, \mathrm{TR}}_1{, \mathrm{H}}_3{, \mathrm{L}}_1{, \mathrm{PH}}_2\right)$$

(7)

…

$${\mathrm{D}}_{\mathrm{n}1}=\mathrm{f}\left({\mathrm{E}}_{\mathrm{a}}{, \mathrm{A}}_{\mathrm{b}}{, \mathrm{TR}}_{\mathrm{c}}{, \mathrm{H}}_{\mathrm{d}}{, \mathrm{L}}_{\mathrm{e}}{, \mathrm{PH}}_{\mathrm{f}}\right)$$

(8)

In practical planting, a plant species is often simultaneously affected by multiple pests and diseases. Different pests and diseases tend to coexist under similar or complementary environmental conditions, resulting in cumulative effects on the plants. Therefore, comprehensively considering and regulating these environmental conditions is an important strategy to prevent the outbreak of multiple pests and diseases.

The following formula represents the influence of multiple pests and diseases on the plant ${\mathrm{P}}_{\mathrm{s}}$:

$${\mathrm{P}}_{\mathrm{s}}=\mathrm{f}\left({\mathrm{D}}_{11},{\mathrm{D}}_{21},{\mathrm{D}}_{31},\dots ,{\mathrm{D}}_{\mathrm{i}1}\right)$$

(9)

where each pest or disease ${\mathrm{D}}_{\mathrm{i}1}$ is a function of the environmental factors (Moisture ${\mathrm{E}}_{\mathrm{a}}$, Fertilizer ${\mathrm{A}}_{\mathrm{b}}$, Temperature $\mathrm{T}{\mathrm{R}}_{\mathrm{c}}$, Humidity ${\mathrm{H}}_{\mathrm{d}}$, Light ${\mathrm{L}}_{\mathrm{e}}$, and PH value $\mathrm{P}{\mathrm{H}}_{\mathrm{f}}$):

$${\mathrm{D}}_{\mathrm{i}1}=\mathrm{f}\left({\mathrm{E}}_{\mathrm{a}},{\mathrm{A}}_{\mathrm{b}},\mathrm{T}{\mathrm{R}}_{\mathrm{c}},{\mathrm{H}}_{\mathrm{d}},{\mathrm{L}}_{\mathrm{e}},\mathrm{P}{\mathrm{H}}_{\mathrm{f}}\right)$$

(10)

For example, flower ${\mathrm{P}}_1$ is more likely to develop downy mildew, powdery mildew, and aphids in a high-humidity environment. Formulas (11) and (12) represent the influence of multiple pests and diseases on ${\mathrm{P}}_1$:

$${\mathrm{P}}_1=\mathrm{f}\left({\mathrm{D}}_{11}, {\mathrm{D}}_{21}, {\mathrm{D}}_{41}\right)$$

(11)

$${\mathrm{P}}_1=\mathrm{f}\left({\mathrm{f}(\mathrm{E}}_3{, \mathrm{A}}_1{, \mathrm{TR}}_2{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_1),{\mathrm{f}(\mathrm{E}}_1{, \mathrm{A}}_1{, \mathrm{TR}}_2{, \mathrm{H}}_3{, \mathrm{L}}_1{, \mathrm{PH}}_1),{\mathrm{f}(\mathrm{E}}_3{, \mathrm{A}}_1{, \mathrm{TR}}_2{, \mathrm{H}}_1{, \mathrm{L}}_1{, \mathrm{PH}}_3)\right)$$

(12)

2.2. Pest and Disease Damage and Control

Different diseases and pests often affect specific parts of the plant, presenting distinct symptoms. By observing the health statuses of different plant parts such as roots, stems, leaves, flowers, fruits, young shoots, and buds, one can effectively diagnose and identify the type and severity of pests and diseases [20].

Table 3. Classification and definition of damage symptoms and control measures.

C	Root	C	Stem	C	Leaf	C	Flower	C	Fruit	C	Young Shoot	C	Bud	C	Control Measures
R1	No	S1	Disease Spots	Y1	Abscise	K1	Mold Layer	FR1	Poor	NS1	Death	B1	Mold Layer	C1	Environmental Control
R2	Indirect	S2	Powder Coating	Y2	Dried	K2	Disease Spots	FR2	Rot	NS2	Restricted Growth	B2	Spots Appear	C2	Nutritional Management
R3	Attached	S3	Pest	Y3	Disease Spots	K3	Death	FR3	Indirect	NS3	Disease Spots	B3	Obstruction	C3	Nutritional Management
	……
Rg	Root g	Sh	Stem h	Yj	Leaf j	Kk	Flower k	FRl	Fruit l	NSo	Young Shoot o	Bp	Bud p	Cq	Control Measures q

describes the impact of plant diseases and pests on the affected areas.

The specific impact of the plant disease and pest ${\mathrm{D}}_{\mathrm{i}}$ on different parts of the plant, such as ${\mathrm{R}}_{\mathrm{g}},{\mathrm{S}}_{\mathrm{h}},{\mathrm{Y}}_{\mathrm{j}},{\mathrm{K}}_{\mathrm{k}},\mathrm{F}{\mathrm{R}}_{\mathrm{l}},\mathrm{N}{\mathrm{S}}_{\mathrm{o}},{\mathrm{B}}_{\mathrm{p}}$, and the corresponding control measures ${\mathrm{C}}_{\mathrm{q}}$, can be represented by the following formulas:

$${\mathrm{D}}_{2\mathrm{i}}=\left({\mathrm{R}}_{\mathrm{g}},{\mathrm{S}}_{\mathrm{h}},{\mathrm{Y}}_{\mathrm{j}},{\mathrm{K}}_{\mathrm{k}},\mathrm{F}{\mathrm{R}}_{\mathrm{l}},\mathrm{N}{\mathrm{S}}_{\mathrm{o}},{\mathrm{B}}_{\mathrm{p}},{\mathrm{C}}_{\mathrm{q}}\right)$$

(13)

For example, downy mildew can cause varying degrees of damage to different parts of the rose plant, and corresponding control measures should be implemented for each affected part:

$${\mathrm{D}}_{21}=\left({\mathrm{R}}_2,{\mathrm{S}}_1,{\mathrm{Y}}_1,{\mathrm{Y}}_3,{\mathrm{K}}_1,\mathrm{F}{\mathrm{R}}_1,\mathrm{N}{\mathrm{S}}_1,\mathrm{N}{\mathrm{S}}_3,{\mathrm{B}}_1,{\mathrm{B}}_3,{\mathrm{C}}_1,{\mathrm{C}}_4\right)$$

(14)

Formula (14) represents the impacts of downy mildew infection on different parts of the rose plant. Specifically, the root experiences indirect effects (${\mathrm{R}}_2)$; the stem shows water-soaked lesions covered with a mold layer (${\mathrm{S}}_1$); the leaves develop irregularly shaped lesions (${\mathrm{Y}}_3$), and in severe cases, they fall off within 2–3 h (${\mathrm{Y}}_1$); the flowers become covered with mold, darken, and become deformed (${\mathrm{K}}_1$); the fruits show poor development ($\mathrm{F}{\mathrm{R}}_1$); young shoots exhibit sunken lesions, leading to wilting and death ($\mathrm{N}{\mathrm{S}}_1,\mathrm{N}{\mathrm{S}}_3$); and the buds are covered by mold, affecting their development and possibly leading to death (${\mathrm{B}}_1,{\mathrm{B}}_3$). To address these symptoms, the main control measures include improving ventilation (${\mathrm{C}}_1$), reducing leaf moisture, and using fungicides (${\mathrm{C}}_4$). These symptoms collectively form the typical characteristics of downy mildew, posing a serious threat to the health and growth of the rose plant, and provide corresponding scientific control methods.

2.3. Establishing a Plant Pest and Disease Early Warning System

To establish an effective plant pest and disease early warning system, the first step is to summarize the environmental conditions that promote pest and disease outbreaks, the primary plant parts affected, and the corresponding control technologies based on research from planting experts and the literature. By continuously monitoring these data, the system can proactively identify pests and diseases, preventing potential risks. Due to the diversity and rapid spread of pests and diseases, the damage to plants will continually change. Therefore, during the occurrence of pests and diseases, real-time monitoring of plant symptoms should be conducted based on the data collected. Additionally, feedback should be provided after treatment. Some pests and diseases may not be easily detected in the early stages; therefore, for newly emerging or changing pest and disease situations, timely adjustments and appropriate control measures should be implemented. In summary, a systematic plant pest and disease early warning mechanism consists of the following three stages: the Data Observation Stage, the Disease Judgment Stage, and the Control Execution Stage. The specific process is shown in Figure 1.

3. Generalized Random Petri Net Model for Plant Pest and Disease Spread Diagnosis

Based on the plant pest and disease early warning mechanism flowchart summarized above, it is evident that pest and disease management is a complex system involving multiple variables, stages, and high interactivity. In this context, the GSPN is used for its unique modeling capabilities to study the spread and diagnosis systems of plant pests and diseases. The Petri net structure can clearly depict and analyze the dynamic relationships between system states, where transitions represent changes in the system’s state, and places represent the various states the system may be in. This approach not only effectively simulates the transmission paths and potential development trends of pests and diseases but also provides a detailed reflection of the key control points and potential risks in the plant pest and disease management process. Moreover, this method supports quantitative analysis and optimization of complex systems, helping to propose more precise and practical control strategies. By introducing randomness and time parameters, the generalized stochastic Petri net can describe and analyze the randomness and uncertainty of plant pest and disease spread, providing scientific decision support for the sustainable development of the plant industry.

3.1. Generalized Stochastic Petri Net

Petri nets are a mathematical modeling tool initially proposed by Carl Adam Petri in the 1960s to describe and analyze the behavior of information transmission systems and other dynamic systems. The basic Petri net consists of places, transitions, and directed arcs. Places represent the states of the system, transitions represent the transitions between states, and directed arcs specify the relationships between places and transitions. In a Petri net, the state is changed by placing or removing tokens in the places, which allows the simulation of various dynamic behaviors of the system. The GSPN is an extension of the standard Petri net that introduces the concepts of time and randomness, enabling the model to more accurately reflect the uncertainty and time-dependence found in real-world systems [21]. In the process of plant pest and disease spread, time is a key parameter for disease development and control effectiveness. Therefore, this study introduces the generalized stochastic Petri net into the plant pest and disease spread diagnosis model.

• In GSPN, the system is described as a directed graph consisting of six basic elements, denoted as $\mathrm{P}\mathrm{N}=\left(\mathrm{P},\mathrm{T},\mathrm{F},\mathrm{W},\mathrm{M},\mathsf{\lambda }\right)$ [22]. These elements are specifically defined as follows:
• $\mathrm{P}=\left\{{\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_{\mathrm{n}}\right\}$ is the set of places, representing the various states or conditions in the system. In the plant pest and disease spread diagnosis model, places represent the key states and factors designed in the spread process, such as moisture factor ${\mathrm{E}}_{\mathrm{a}}$, fertilizer factor ${\mathrm{A}}_{\mathrm{b}}$, and so on. Here, $\mathrm{n}>0$ denotes the number of elements.
• $\mathrm{T}={\mathrm{T}}_{\mathrm{i}}\cup {\mathrm{T}}_{\mathrm{j}}\left({\mathrm{T}}_{\mathrm{i}}\cap {\mathrm{T}}_{\mathrm{j}}=\varnothing \right)$ is the set of transitions, triggered based on the conditions defined in ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}$. This set includes two types of transitions: Time transitions ${\mathrm{T}}_{\mathrm{i}}=\left\{\left.{\mathrm{t}}_1,{\mathrm{t}}_2,\dots ,{\mathrm{t}}_{\mathrm{k}}\right\}\right.$,which simulate activities with delays, such as the disease development stages. Instantaneous transitions ${\mathrm{T}}_{\mathrm{j}}=\left\{{\mathrm{t}}_{\mathrm{k}+1},{\mathrm{t}}_{\mathrm{k}+2},\dots ,{\mathrm{t}}_{\mathrm{m}}\right\}$, $\mathrm{m}>0$, used to represent immediate events, such as the rapid diagnosis of a disease.
• $\mathrm{F}\subseteq {\mathrm{a}}^{+}\cup {\mathrm{a}}^{-}$ is the set of directed arcs, connecting places and transitions, determining the flow of tokens. ${\mathrm{a}}^{-}\subseteq \mathrm{P}\times \mathrm{T}$ represents the input arcs of transitions in the Petri net; ${\mathrm{a}}^{+}\subseteq \mathrm{T}\times \mathrm{P}$ represents the output arcs of transitions in the Petri net. The association matrix in the Petri net is defined as $\mathrm{a}={\mathrm{a}}^{+}-{\mathrm{a}}^{-}$. Set $\mathrm{F}$ has direction, representing the path through which pests and diseases transition from one state to another.
  $\mathrm{W}:\mathrm{F}\to {\mathrm{N}}^{+}$ is the arc weight function vector, where ${\mathrm{N}}^{+}=\left\{1,2,\dots ,\mathrm{m}\right\}$, $\mathrm{m}>0$ represents the capacity of each system transition.
• $\mathrm{M}:\mathrm{P}\to \mathrm{N}$ is the state-marking vector of the Petri net, which represents the possible states of the pest and disease spread system during its dynamic operation by defining the number of tokens in each place. ${\mathrm{M}}_0$ represents the initial marking of the system. If there is no function defined on an arc, the default weight is 1.
• $\mathsf{\lambda }=\left\{{\mathsf{\lambda }}_1,{\mathsf{\lambda }}_2,\dots ,{\mathsf{\lambda }}_{\mathrm{m}}\right\}$ represents the average triggering frequency associated with time transitions [23]. Each ${\mathsf{\lambda }}_{\mathrm{i}}$ is an exponential distribution parameter used to describe the average time interval between the triggers of the corresponding time transition. This allows the model to probabilistically express the occurrence of transitions, with the $\mathsf{\lambda }$ value for instantaneous transitions defaulting to zero, reflecting their characteristic of occurring immediately with no delay.

The GSPN has an isomorphic relationship with continuous-time homogeneous Markov chains, which allows for the establishment of a corresponding Markov chain model by analyzing the reachability set of the GSPN. This isomorphism means that once the Markov chain reaches its equilibrium distribution, the long-term steady-state probabilities of the system can be calculated. By constructing and analyzing such a Markov chain, it becomes possible to effectively analyze the starting points of the plant pest and disease spread diagnostic system.

3.2. Pest and Disease Spread Evolution Modeling Methods

Based on the structural description of the plant pest and disease spread evolution process mentioned earlier, the pest and disease spread evolution model using the GSPN is established. The specific steps are as follows:

Step 1: Construct the GSPN model. Build the GSPN model based on the characteristics and requirements of pest and disease spread. This includes defining the places in the system (representing the states of pest and disease), transitions (representing the changes between states), and their relationships (directed arcs). Additionally, associate time delays with the corresponding transitions.

Step 2: Generate the reachability graph and construct the isomorphic Markov chain. The distribution and movement of tokens represent changes in the system’s state. Based on the calculation results of ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}$ (whether the conditions are met), tokens can move from one place to another, simulating the results of the ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}$ conditions through transitions. This generates the reachability graph $\mathrm{R}\left(\mathrm{m}\right)$, where each state or marking is denoted as ${\mathrm{M}}_1,{\mathrm{M}}_2,\dots ,{\mathrm{M}}_{\mathrm{n}}$ (n being the total number of states), representing a specific configuration in the Petri net. Then, assign the corresponding firing rate to each arc and construct the isomorphic Markov chain [24], followed by an analysis of the model’s validity.

Step 3: Analyze the Markov chain. Convert the obtained reachability graph into a Markov model to calculate the steady-state probabilities. The steady-state probability $\mathrm{P}\left[{\mathrm{M}}_{\mathrm{i}}\right]$ can be solved using the relevant theorems of the stationary distribution of the Markov chain and the Chapman–Kolmogorov equation:

$$\mathrm{P}=\left(\mathrm{P}\left[{\mathrm{M}}_0\right],\mathrm{P}\left[{\mathrm{M}}_1\right],\dots ,\mathrm{P}\left[{\mathrm{M}}_{\mathrm{n}-1}\right]\right)$$

(15)

$$\left\{\begin{array}{c}\mathrm{P}\mathrm{Q}=0\\ {\displaystyle \sum _{\mathrm{i}=0}^{\mathrm{n}-1}\mathrm{P}\left[{\mathrm{M}}_{\mathrm{i}}\right]}=1\end{array}\right.$$

(16)

where $\mathrm{n}$ is the number of states in the model, $\mathrm{i}=\left\{0,1,\dots ,\mathrm{n}-1\right\}$, $\mathrm{P}$ is the state probability vector, and $\mathrm{Q}=\left[{\mathrm{q}}_{\mathrm{i}\mathrm{j}}\right]$, where $\mathrm{i}=\left\{0,1,\dots ,\mathrm{n}-1\right\}$ and $\mathrm{j}=\left\{0,1,\dots ,\mathrm{n}-1\right\}$. For the elements off the main diagonal ${\mathrm{q}}_{\mathrm{i}\mathrm{j}}$, if there is an arc from state ${\mathrm{M}}_{\mathrm{i}}$ to state ${\mathrm{M}}_{\mathrm{j}}$, the value is the firing rate ${\mathsf{\lambda }}_{\mathrm{k}}$ associated with the exponential distribution of the transition ${\mathrm{T}}_{\mathrm{k}}$ from ${\mathrm{M}}_{\mathrm{i}}$ to ${\mathrm{M}}_{\mathrm{i}}$; if no arc connects the two states, this element is 0. The elements on the main diagonal ${\mathrm{q}}_{\mathrm{i}\mathrm{j}}$ follow the equation $-\left({{\displaystyle \sum }}_{\mathrm{i}\ne \mathrm{j}}{\mathrm{q}}_{\mathrm{i}\mathrm{j}}\right)$, ensuring that the sum of each row equals zero [19].

Step 4: Performance evaluation and optimization. After calculating the steady-state probabilities, evaluate and optimize the performance of the pest and disease spread system constructed by the generalized stochastic Petri net model. The busy probability of places can be calculated using Formula (17). The idle probability of places can be calculated using Formula (18). Additionally, the utilization rate of system transitions can be calculated using Formula (19) [25].

$$\begin{array}{c}\mathrm{P}\left[\mathrm{M}\left({\mathrm{p}}_{\mathrm{i}}\right)=1\right]={\displaystyle \sum _{\mathrm{j}}\mathrm{P}\left[{\mathrm{M}}_{\mathrm{j}}\right]},{\mathrm{M}}_{\mathrm{j}}\in [{\mathrm{M}}_0>{\mathrm{M}}_{\mathrm{j}}(\mathrm{p})=\mathrm{i}\end{array}$$

(17)

$$\mathrm{P}\left[\mathrm{M}\left({\mathrm{p}}_{\mathrm{i}}\right)=0\right]=1-\mathrm{P}\left[\mathrm{M}\left({\mathrm{p}}_{\mathrm{i}}\right)=1\right]$$

(18)

$$\begin{array}{c}\mathrm{U}\left(\mathrm{t}\right)=\end{array}{\displaystyle \sum _{\mathrm{M}\in \mathrm{E}}\mathrm{P}\left[\mathrm{M}\right]}$$

(19)

3.3. The GSPN Model for Plant Pest and Disease Spread Evolution

Based on the plant pest and disease early warning system diagram shown in Figure 1, define the places, transitions, and directed arcs. Following Step 1 as mentioned earlier, construct the GSPN model for plant pest and disease spread evolution, as shown in Figure 2.

As shown in the figure above, the model contains 12 places and 22 transitions. The specific meanings of each place and transition are provided in

Table 4. The meanings of places and transitions in the GSPN model for plant pest and disease spread evolution.

Places	Meaning of Places	Transitions	Meaning of Transitions
P1	Water and Fertilizer Data	t1	Information Filtering
P2	Temperature Data	t2	Root Symptom Observation
P3	Humidity Data	t3	Stem Symptom Observation
P4	Light Data	t4	Leaf Symptom Observation
P5	pH Value	t5	Flower Symptom Observation
P6	Data Integration	t6	Fruit Symptom Observation
P7	Infection Severity Assessment	t7	Young Shoot Symptom Observation
P8	Invasion Severity Assessment	t8	Bud Symptom Observation
P9	Control Method Decision	t9	Mild Infection
P10	Implementation of Control Measures	t10	Moderate Infection
P11	Data Monitoring	t11	Severe Infection
P12	End of Treatment	t12	Reaching Treatment Standards
		t13	Environmental Control
		t14	Nutritional Management
		t15	Physical Methods
		t16	Chemical Methods
		t17	Biological Control
		t18	Field Cleanliness and Standard Cultivation
		t19	Recovery of Health
		t20	Plant Death
		t21	Treatment Effectiveness Assessment
		t22	Data Feedback

.

According to Step 2, the plant pest and disease spread evolution process is modeled and run using PIPEv5.2.0 to determine the reachability marking set of the Petri net. The initial marking of the GSPN is assumed to be ${\mathrm{M}}_1=\left(1,1,1,1,1,0,0,0,0,0,0\right)$, indicating that there is one token in places ${\mathrm{P}}_1,{\mathrm{P}}_2,{\mathrm{P}}_3,{\mathrm{P}}_4 \mathrm{and} {\mathrm{P}}_5$. By applying different transitions, the corresponding reachability set can be obtained.

$$\begin{gathered} {\mathrm{M}}_1=\left(1,1,1,1,1,0,0,0,0,0,0,0\right) \\ {\mathrm{M}}_2=\left(0,0,0,0,0,1,0,0,0,0,0,0\right) \\ {\mathrm{M}}_3=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_4=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_5=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_6=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_7=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_8=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_9=\left(0,0,0,0,0,0,1,0,0,0,0,0\right) \\ {\mathrm{M}}_{10}=\left(0,0,0,0,0,0,0,1,0,0,0,0\right) \\ {\mathrm{M}}_{11}=\left(0,0,0,0,0,0,0,1,0,0,0,0\right) \\ {\mathrm{M}}_{12}=\left(0,0,0,0,0,0,0,1,0,0,0,0\right) \\ {\mathrm{M}}_{13}=\left(0,0,0,0,0,0,0,0,1,0,0,0\right) \\ {\mathrm{M}}_{14}=\left(0,0,0,0,0,0,0,0,0,1,0,0\right) \\ {\mathrm{M}}_{15}=\left(0,0,0,0,0,0,0,0,0,1,0,0\right) \\ {\mathrm{M}}_{16}=\left(0,0,0,0,0,0,0,0,0,1,0,0\right) \\ {\mathrm{M}}_{17}=\left(0,0,0,0,0,0,0,0,0,1,0,0\right) \\ {\mathrm{M}}_{18}=\left(0,0,0,0,0,0,0,0,0,1,0,0\right) \\ {\mathrm{M}}_{19}=\left(0,0,0,0,0,0,0,0,0,1,0,0\right) \\ {\mathrm{M}}_{20}=\left(0,0,0,0,0,0,0,0,0,0,1,0\right) \\ {\mathrm{M}}_{21}=\left(0,0,0,0,0,0,0,0,1,1,1,0\right) \\ {\mathrm{M}}_{22}=\left(0,0,0,0,0,0,0,0,0,0,0,1\right) \end{gathered}$$

By constructing the reachability set of 22 states (${\mathrm{M}}_1,{\mathrm{M}}_2,{\mathrm{M}}_3,\dots ,{\mathrm{M}}_{22}$), the equivalent isomorphic Markov chain of the GSPN model for plant pest and disease spread evolution is formed, as shown in Figure 3. The directed arcs in the figure represent the state transitions within the plant pest and disease spread evolution system.

Based on the construction of the Markov chain, and following Step 3 in this chapter, the steady-state probability equations for the GSPN model of plant pest and disease spread evolution are constructed using Formulas (15) and (16) to analyze the system’s operational efficiency [26]. Let $\mathrm{P}\left({\mathrm{M}}_{\mathrm{i}}\right)$, $\left( \mathrm{i}=1,2,\dots ,22\right)$ be the probability that the system is in state ${\mathrm{M}}_{\mathrm{i}}$ in the steady state. ${\mathsf{\lambda }}_{\mathrm{j}}$ represents the transition rate ($\mathrm{j}>0$). From this, the relationship between the state probabilities can be derived using Equation (20).

$$\left\{\begin{array}{l}7{\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)={\lambda }_{22}\mathrm{P}\left({\mathrm{M}}_{22}\right)\\ 3{\lambda }_2\mathrm{P}\left({\mathrm{M}}_2\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ 3{\lambda }_3\mathrm{P}\left({\mathrm{M}}_3\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ 3{\lambda }_4\mathrm{P}\left({\mathrm{M}}_4\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ 3{\lambda }_5\mathrm{P}\left({\mathrm{M}}_5\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ 3{\lambda }_6\mathrm{P}\left({\mathrm{M}}_6\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ 3{\lambda }_7\mathrm{P}\left({\mathrm{M}}_7\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ 3{\lambda }_8\mathrm{P}\left({\mathrm{M}}_8\right)={\lambda }_1\mathrm{P}\left({\mathrm{M}}_1\right)\\ {\lambda }_9\mathrm{P}\left({\mathrm{M}}_9\right)={\lambda }_2\mathrm{P}\left({\mathrm{M}}_2\right)+{\lambda }_3\mathrm{P}\left({\mathrm{M}}_3\right)+{\lambda }_4\mathrm{P}\left({\mathrm{M}}_4\right)+{\lambda }_5\mathrm{P}\left({\mathrm{M}}_5\right)+{\lambda }_6\mathrm{P}\left({\mathrm{M}}_6\right)+{\lambda }_7\mathrm{P}\left({\mathrm{M}}_7\right)+{\lambda }_8\mathrm{P}\left({\mathrm{M}}_8\right)\\ {\lambda }_{10}\mathrm{P}\left({\mathrm{M}}_{10}\right)={\lambda }_2\mathrm{P}\left({\mathrm{M}}_2\right)+{\lambda }_3\mathrm{P}\left({\mathrm{M}}_3\right)+{\lambda }_4\mathrm{P}\left({\mathrm{M}}_4\right)+{\lambda }_5\mathrm{P}\left({\mathrm{M}}_5\right)+{\lambda }_6\mathrm{P}\left({\mathrm{M}}_6\right)+{\lambda }_7\mathrm{P}\left({\mathrm{M}}_7\right)+{\lambda }_8\mathrm{P}\left({\mathrm{M}}_8\right)\\ {\lambda }_{11}\mathrm{P}\left({\mathrm{M}}_{11}\right)={\lambda }_2\mathrm{P}\left({\mathrm{M}}_2\right)+{\lambda }_3\mathrm{P}\left({\mathrm{M}}_3\right)+{\lambda }_4\mathrm{P}\left({\mathrm{M}}_4\right)+{\lambda }_5\mathrm{P}\left({\mathrm{M}}_5\right)+{\lambda }_6\mathrm{P}\left({\mathrm{M}}_6\right)+{\lambda }_7\mathrm{P}\left({\mathrm{M}}_7\right)+{\lambda }_8\mathrm{P}\left({\mathrm{M}}_8\right)\\ 7{\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)={\lambda }_9\mathrm{P}\left({\mathrm{M}}_9\right)+{\lambda }_{10}\mathrm{P}\left({\mathrm{M}}_{10}\right)+{\lambda }_{11}\mathrm{P}\left({\mathrm{M}}_{11}\right)\\ 3{\lambda }_{13}\mathrm{P}\left({\mathrm{M}}_{13}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)\\ 3{\lambda }_{14}\mathrm{P}\left({\mathrm{M}}_{14}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)\\ 3{\lambda }_{15}\mathrm{P}\left({\mathrm{M}}_{15}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)\\ 3{\lambda }_{16}\mathrm{P}\left({\mathrm{M}}_{16}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)\\ 3{\lambda }_{17}\mathrm{P}\left({\mathrm{M}}_{17}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)\\ 3{\lambda }_{18}\mathrm{P}\left({\mathrm{M}}_{18}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)\\ {\lambda }_{19}\mathrm{P}\left({\mathrm{M}}_{19}\right)={\lambda }_{13}\mathrm{P}\left({\mathrm{M}}_{13}\right)+{\lambda }_{14}\mathrm{P}\left({\mathrm{M}}_{14}\right)+{\lambda }_{15}\mathrm{P}\left({\mathrm{M}}_{15}\right)+{\lambda }_{16}\mathrm{P}\left({\mathrm{M}}_{16}\right)+{\lambda }_{17}\mathrm{P}\left({\mathrm{M}}_{17}\right)+{\lambda }_{18}\mathrm{P}\left({\mathrm{M}}_{18}\right)\\ {\lambda }_{20}\mathrm{P}\left({\mathrm{M}}_{20}\right)={\lambda }_{13}\mathrm{P}\left({\mathrm{M}}_{13}\right)+{\lambda }_{14}\mathrm{P}\left({\mathrm{M}}_{14}\right)+{\lambda }_{15}\mathrm{P}\left({\mathrm{M}}_{15}\right)+{\lambda }_{16}\mathrm{P}\left({\mathrm{M}}_{16}\right)+{\lambda }_{17}\mathrm{P}\left({\mathrm{M}}_{17}\right)+{\lambda }_{18}\mathrm{P}\left({\mathrm{M}}_{18}\right)\\ {\lambda }_{21}\mathrm{P}\left({\mathrm{M}}_{21}\right)={\lambda }_{12}\mathrm{P}\left({\mathrm{M}}_{12}\right)+{\lambda }_{13}\mathrm{P}\left({\mathrm{M}}_{13}\right)+{\lambda }_{14}\mathrm{P}\left({\mathrm{M}}_{14}\right)+{\lambda }_{15}\mathrm{P}\left({\mathrm{M}}_{15}\right)+{\lambda }_{16}\mathrm{P}\left({\mathrm{M}}_{16}\right)+{\lambda }_{17}\mathrm{P}\left({\mathrm{M}}_{17}\right)+{\lambda }_{18}\mathrm{P}\left({\mathrm{M}}_{18}\right)+{\lambda }_{19}\mathrm{P}\left({\mathrm{M}}_{19}\right)+{\lambda }_{20}\mathrm{P}\left({\mathrm{M}}_{20}\right)\\ {\lambda }_{22}\mathrm{P}\left({\mathrm{M}}_{22}\right)={\lambda }_{21}\mathrm{P}\left({\mathrm{M}}_{21}\right)\\ {\displaystyle \sum }_{\mathrm{i}=1}^{22}\mathrm{P}\left({\mathrm{M}}_{\mathrm{i}}\right)=1\end{array}\right.$$

(20)

3.4. Stimulation Experiment of the Model

To verify the rationality and feasibility of the GSPN model for plant pest and disease spread diagnosis, a simulation experiment was conducted using the PIPE [27]. First, the model was drawn in the PIPE software, and initial markings were placed in the initial places. Then, by running the State Space Analysis function, a detailed analysis of the model’s state space was performed. The simulation results are shown in Figure 4. The analysis results indicate that the optimized Petri net model maintains a bounded state during its operation, without deadlocks, and exhibits good safety. These results confirm that the model design is reasonable and feasible.

4. Simulation Analysis

4.1. Performance Analysis of Pest and Disease Spread Evolution Process Based on the GSPN Model

By solving the above linear system of equations, the steady-state probabilities of the pest and disease spread evolution events in different states can be obtained. From the steady-state probabilities, the key areas for improving the system’s operational efficiency can be identified, which is of significant practical importance for enhancing the efficiency of emergency decision-making in pest and disease outbreaks. First, solve the place busy probability, place idle probability, and the transition utilization rate to analyze the model’s performance and identify the main factors and key nodes in the pest and disease spread process [28]. Subsequently, the average implementation rates of these key nodes are adjusted, and MATLAB is used to generate a trend graph of the steady-state probability changes, thus providing robust data support and a theoretical foundation for pest and disease control.

Based on the evolution process of plant pest and disease spread, this study divides the entire process into three stages: the data observation stage, the disease assessment stage, and the treatment execution stage. The relevant λ values are reasonably set according to the characteristics and practical conditions of each stage. The basis for these settings is illustrated in

Table 5. Basis for setting $\mathsf{\lambda }$ value.

Data Stage	λ Value	Specific Value	Source
Data Observation Stage	λ1	18	Grünig et al., Experimental Data
	λ2	10	Expert Recommendations
	λ3	12	Expert Recommendations
	λ4	8	Expert Recommendations
	λ5	7	Expert Recommendations
	λ6	6	Expert Recommendations
	λ7	9	Expert Recommendations
	λ8	8	Expert Recommendations
Disease Assessment Stage	λ9	18	Appeltans et al.
	λ10	8	Appeltans et al.
	λ11	3	Appeltans et al.
	λ12	8	Expert Recommendations, Experimental Data
Treatment Execution Stage	λ13	7	Alimzhanova et al.
	λ14	6	Alimzhanova et al.
	λ15	5	Alimzhanova et al.
	λ16	4	Alimzhanova et al.
	λ17	3	Mubeen et al.
	λ18	2	Mubeen et al.
	λ19	3	Expert Recommendations
	λ20	14	Expert Recommendations
	λ21	9	Expert Recommendations, Experimental Data
	λ22	20	Expert Recommendations, Experimental Data

.

In the data observation stage, following the study by Grünig et al. [29], the data collection and processing rates in similar agricultural pest and disease monitoring projects were optimized based on actual sampling and processing times. Drawing upon the practical experience and data processing methodologies outlined in the literature, we set ${\mathsf{\lambda }}_1=18$. In consultation with agricultural experts, the setting of λ values for observing symptoms in various plant parts was determined by considering the difficulty and time required for observation. Experts indicated that symptom observation for roots and stems is conducted at a moderate pace, with ${\mathsf{\lambda }}_2=10$ and ${\mathsf{\lambda }}_3=12$, respectively. In contrast, the observation of symptoms in leaves, flowers, fruits, young shoots, and buds necessitates a more meticulous inspection, leading to the following $\mathsf{\lambda }$ values: ${\mathsf{\lambda }}_4=8, {\mathsf{\lambda }}_5=7, {\mathsf{\lambda }}_6=6, {\mathsf{\lambda }}_7=9$, and ${\mathsf{\lambda }}_8=8$.

In the disease assessment stage, referring to the work of Appeltans et al., which classifies the severity of leek white tip disease into early, moderate, and severe stages, we set the corresponding $\mathsf{\lambda }$ values for disease symptoms in accordance with the gradual acceleration characteristics of disease spread discussed in the literature [30]. In the early stage of the disease, the symptoms in affected plants are not immediately apparent; however, as the disease progresses, the plants gradually exhibit disease susceptibility characteristics, ultimately leading to rapid wilting and death within a short period. Based on this, we set the infection rates for mild, moderate, and severe stages as ${\mathsf{\lambda }}_9=18, {\mathsf{\lambda }}_{10}=8$, and ${\mathsf{\lambda }}_{11}=3$, respectively. The determination of whether the treatment standard has been met requires a comprehensive evaluation, for which ${\mathsf{\lambda }}_{12}=8$.

In the treatment execution stage, environmental control and nutrient management, as foundational measures, generally require a longer period to gradually exhibit their effects. Accordingly, we set ${\mathsf{\lambda }}_{13}=7$ and ${\mathsf{\lambda }}_{14}=6$. Alimzhanova et al. noted that physical and chemical methods (such as the use of chemical pesticides and fungicides) are commonly employed treatment approaches, which can yield significant effects in a short period, leading to faster disease control [31]. Therefore, we set ${\mathsf{\lambda }}_{15}=5$ and ${\mathsf{\lambda }}_{16}=4$. Mubeen et al. highlighted that biological control and field sanitation practices, as long-term measures, rely on the natural regulation and resistance enhancement of plants, and their effects appear gradually and more slowly [32]. Based on this, we set ${\mathsf{\lambda }}_{17}=3$ and ${\mathsf{\lambda }}_{18}=2$. Through the corresponding treatment measures, plants gradually recover their health, with ${\mathsf{\lambda }}_{19}=3$. The plant death rate is relatively slow, with ${\mathsf{\lambda }}_{20}=14$. Finally, the determination of the treatment effect is made with ${\mathsf{\lambda }}_{21}=9$. To ensure timely updates and processing of all data, the data feedback rate is rapid, and thus, ${\mathsf{\lambda }}_{22}=20$.

Using Formulas (15), (16), and (20) from Section 3.2, the steady-state probabilities are calculated and shown in

Table 6. Steady-state probabilities.

Mark	Steady-State Probability	Mark	Steady-State Probability
P(M1)	0.013443	P(M12)	0.030248
P(M2)	0.008066	P(M13)	0.011523
P(M3)	0.006722	P(M14)	0.013443
P(M4)	0.010083	P(M15)	0.016132
P(M5)	0.011523	P(M16)	0.020165
P(M6)	0.013443	P(M17)	0.026887
P(M7)	0.008962	P(M18)	0.04033
P(M8)	0.010083	P(M19)	0.161321
P(M9)	0.031368	P(M20)	0.034569
P(M10)	0.070578	P(M21)	0.188208
P(M11)	0.188208	P(M22)	0.084694

.

4.1.1. Busy Probability of a Place

In the GSPN system, when a place contains resources, it indicates that the corresponding place is busy during the pest and disease spread evolution process. First, the token distribution in each place under each marking state is presented to identify which parts of the system are active or utilized at each point. The analysis results are shown in

Table 7. Distribution of tokens in the library under different identification states.

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12
M1	1	1	1	1	1	0	0	0	0	0	0	0
M2	0	0	0	0	0	1	0	0	0	0	0	0
M3	0	0	0	0	0	0	1	0	0	0	0	0
M4	0	0	0	0	0	0	1	0	0	0	0	0
M5	0	0	0	0	0	0	1	0	0	0	0	0
M6	0	0	0	0	0	0	1	0	0	0	0	0
M7	0	0	0	0	0	0	1	0	0	0	0	0
M8	0	0	0	0	0	0	1	0	0	0	0	0
M9	0	0	0	0	0	0	1	0	0	0	0	0
M10	0	0	0	0	0	0	0	1	0	0	0	0
M11	0	0	0	0	0	0	0	1	0	0	0	0
M12	0	0	0	0	0	0	0	1	0	0	0	0
M13	0	0	0	0	0	0	0	0	1	0	0	0
M14	0	0	0	0	0	0	0	0	0	1	0	0
M15	0	0	0	0	0	0	0	0	0	1	0	0
M16	0	0	0	0	0	0	0	0	0	1	0	0
M17	0	0	0	0	0	0	0	0	0	1	0	0
M18	0	0	0	0	0	0	0	0	0	1	0	0
M19	0	0	0	0	0	0	0	0	0	1	0	0
M20	0	0	0	0	0	0	0	0	0	0	1	0
M21	0	0	0	0	0	0	0	0	1	1	1	0
M22	0	0	0	0	0	0	0	0	0	0	0	1

, which details the number of tokens present in each place under each reachable marking.

Next, in conjunction with the place busy probability from Formula (17), the busy probability for each place is calculated as shown in

Table 8. Busy probability for each place.

Place	Busy Probability	Place	Busy Probability
M1	0.013443	M7	0.092184
M2	0.013443	M8	0.289034
M3	0.013443	M9	0.199731
M4	0.013443	M10	0.466486
M5	0.013443	M11	0.222777
M6	0.008066	M12	0.084694

.

In the GSPN model, the relatively busy places are ${\mathrm{P}}_8$ and ${\mathrm{P}}_{10}$, with busy probabilities reaching 29% and 47%, respectively. This indicates that invasion severity assessment and implementation of control measures are the two most active stages in the entire pest and disease spread model. These stages are prone to information accumulation and thus represent critical nodes for model optimization. The invasion severity assessment stage (${\mathrm{P}}_8$) occupies a pivotal position in the decision-making chain. Any delay or error at this stage may have significant cascading effects on the overall system. To enhance the accuracy of this assessment, it is often necessary to conduct multi-level analyses and verifications, including preliminary diagnosis, refined analysis, and final confirmation. The high busy rate at the implementation of control measure stage (${\mathrm{P}}_{10}$) suggests that this process involves the coordination of multiple resources, such as pesticides, labor, equipment, and funding. The scheduling and allocation of these resources are characterized by high levels of dynamism and uncertainty. Furthermore, the pest and disease control process requires real-time monitoring of treatment efficacy and timely adjustment of strategies in response to continuously evolving pest and disease symptoms.

4.1.2. Place Idle Probability

The place idle probability represents the likelihood that a specific place is in a non-busy state at a given moment. In other words, it indicates the probability that there are no tokens in that place under steady-state conditions. In accordance with Equation (18), the idle probabilities for each place were calculated, as shown in

Table 9. Idle probabilities for each place.

Place	Idle Probability	Place	Idle Probability
M1	0.986557	M7	0.907816
M2	0.986557	M8	0.710966
M3	0.986557	M9	0.800269
M4	0.986557	M10	0.533514
M5	0.986557	M11	0.777223
M6	0.991934	M12	0.915306

.

In the GSPN model, the places ${\mathrm{P}}_1$ to ${\mathrm{P}}_6$ corresponding to the data observation stage exhibit relatively high idle probabilities, indicating that information accumulation is less likely to occur during the early phase of the plant pest and disease early warning system. Therefore, these stages can be considered key targets for optimization. By enhancing the efficiency of information collection and processing at these points, the overall early warning capability of the system can be improved, thereby playing a crucial role in the effective prevention of pest and disease outbreaks.

4.1.3. Transition Utilization Rate

The transition utilization rate reflects the activity level and resource usage efficiency of each transition node within a specific time frame. In accordance with Equation (19), the utilization rates for each system transition are calculated as shown in

Table 10. Transition utilization rate.

Transition	Utilization Rate	Transition	Utilization Rate
t1	0.013443	t12	0.289034
t2	0.008066	t13	0.011523
t3	0.008066	t14	0.011523
t4	0.008066	t15	0.011523
t5	0.008066	t16	0.011523
t6	0.008066	t17	0.011523
t7	0.008066	t18	0.011523
t8	0.008066	t19	0.278278
t9	0.092184	t20	0.278278
t10	0.092184	t21	0.188208
t11	0.092184	t22	0.084694

.

Compared to other transitions, the utilization rates of ${\mathrm{t}}_{19},{\mathrm{t}}_{20}$, and ${\mathrm{t}}_{21}$ are relatively high. The frequent triggering of recovery of health (${\mathrm{t}}_{19}$) and plant death (${\mathrm{t}}_{20}$) transitions indicates a strong system demand for evaluating treatment outcomes. This reflects the necessity of continuously monitoring and assessing the health status of plants to ensure the effectiveness of control measures and to promptly adjust strategies to prevent further spread of pests and diseases. Among all, the treatment effectiveness assessment (${\mathrm{t}}_{21}$) transition has the highest utilization rate. This is because, following the implementation of control measures, the system must assess their effectiveness to verify that the pest or disease is effectively managed. The high utilization rate of this transition node is critical for ensuring that pest and disease control measures are appropriate, strategies are adjusted in a timely manner, and overall system performance is optimized.

4.2. Stimulation Analysis of the Pest and Disease Diagnosis Model

Simulation analysis was performed on the plant pest and disease diagnosis model to verify the effectiveness of the proposed method and to uncover key patterns in the evolution of pest and disease spread. By comprehensively analyzing place busy probabilities, place idle probabilities, and transition utilization rates, several critical nodes within the spread evolution process were identified: namely, information filtering (${\mathrm{t}}_1$), mild infection (${\mathrm{t}}_9$), moderate infection (${\mathrm{t}}_{10}$), severe infection (${\mathrm{t}}_{11}$), reaching treatment standards (${\mathrm{t}}_{12}$), and treatment effectiveness assessment (${\mathrm{t}}_{21}$). By altering the average firing rates of these transitions, the analysis explored their impact on the overall pest and disease propagation process and investigated the underlying mechanisms of each node within the diagnostic workflow. This approach contributes to optimizing the efficiency and effectiveness of the early warning system, enhancing the timeliness and accuracy of pest and disease control measures, and ultimately safeguarding the healthy development of plant cultivation.

4.2.1. Data Observation Stage

By varying ${\mathsf{\lambda }}_1$ from 1 to 20 while keeping all other $\mathsf{\lambda }$ value constants, the relationship between ${\mathsf{\lambda }}_1$ and the steady-state probabilities of each system state is uncovered, as illustrated in Figure 5. As ${\mathsf{\lambda }}_1$ increases—indicating a faster data-processing rate in the data observation stage—the probability of the system remaining in the information filtering state $\mathrm{P}\left({\mathrm{M}}_1\right)$ significantly decreases, while the probability of reaching the treatment effectiveness assessment state $\mathrm{P}\left({\mathrm{M}}_{21}\right)$ increases most rapidly. This suggests that enhancing the data-filtering efficiency during the early stages of pest and disease spread can substantially improve the system’s overall processing performance, thereby accelerating progression toward the final stage of treatment assessment. The observed trend reflects the presence of an effective feedback mechanism: speeding up the information filtering and integration phase directly drives higher efficiency in subsequent evaluation, decision-making, and control processes. Such changes highlight the critical role of the information-processing speed in pest and disease diagnosis. In practical applications, it is essential to optimize the information-processing pipeline to improve the system’s overall response speed and effectiveness.

4.2.2. Disease Judgment Stage

A systematic simulation analysis was conducted on the disease judgment phase in the plant pest and disease spread evolution process. First, the relationships between the mild infection rate, moderate infection rate, and severe infection rate with the probability of meeting treatment criteria were explored. Then, the interactions between the mild infection rate, moderate infection rate, and severe infection rate were analyzed. Based on this, the simulation derived the critical surfaces for three different pest and disease early warning thresholds. These findings provide scientific evidence for developing effective pest and disease control strategies.

While keeping all other $\mathsf{\lambda }$ values constant except for ${\mathsf{\lambda }}_9$ and ${\mathsf{\lambda }}_{10}$, the relationship between the mild infection rate, moderate infection rate, and the probability of meeting treatment criteria is derived from Formula (20):

$$\begin{array}{c}\mathrm{P}\left({\mathrm{M}}_{12}\right)={\displaystyle \frac{945{\lambda }_9{\lambda }_{10} }{17640{\lambda }_9+17640{\lambda }_{10} +28057{\lambda }_9{\lambda }_{10}}}\end{array}$$

(21)

The plant pest and disease early warning threshold is set at 0.005. The contour lines in Figure 6 show the regions where the probability of meeting treatment criteria is the same under different infection rate combinations. For example, when ${\mathsf{\lambda }}_9=1.8$ and ${\mathsf{\lambda }}_{10}=2.4$, $\mathrm{P}\left({\mathrm{M}}_{12}\right)=0.0209$. This indicates that under this infection rate combination, the probability of meeting treatment criteria is 2.09%. The system needs to activate the early warning mechanism and promptly implement corresponding control measures. This combination is particularly important in the prevention and early intervention stages. By enhancing monitoring, integrated control, and preventive measures, plant health and efficient production can be ensured.

Under the condition that all other $\mathsf{\lambda }$ values remain constant except for ${\mathsf{\lambda }}_{10}$ and ${\mathsf{\lambda }}_{11}$, the relationship between the moderate infection rate, severe infection rate, and the probability of meeting treatment criteria is derived from Formula (20):

$$\mathrm{P}\left({\mathrm{M}}_{12}\right)={\displaystyle \frac{105{\mathsf{\lambda }}_{10} {\mathsf{\lambda }}_{11} }{1960{\mathsf{\lambda }}_{10}+1960{\mathsf{\lambda }}_{11} +2573{\mathsf{\lambda }}_{10}{\mathsf{\lambda }}_{11}}}$$

(22)

The changes in the contour lines in Figure 7 show the regions where the probability of meeting the treatment criteria is the same under different infection rate combinations. For example, when ${\mathsf{\lambda }}_{10}=1.8$ and ${\mathsf{\lambda }}_{11}=2.4$, $\mathrm{P}\left({\mathrm{M}}_{12}\right)=0.0234$. This indicates that under this infection rate combination, the probability of meeting the treatment criteria is 2.34%. The system needs to activate the early warning mechanism and promptly implement corresponding control measures. Compared to the ${\mathsf{\lambda }}_9$, ${\mathsf{\lambda }}_{10}$ combination, the combination of ${\mathsf{\lambda }}_{10}$ and ${\mathsf{\lambda }}_{11}$ has a greater influence on the probability of meeting the treatment criteria, with more regions reaching higher probabilities. This combination requires special attention in pest and disease control measures. For areas with high rates of moderate and severe infection, the monitoring frequency should be increased. Combined physical, chemical, and biological control methods should be employed to comprehensively manage moderate and severe infections.

Under the condition that all other $\mathsf{\lambda }$ values remain constant except for ${\mathsf{\lambda }}_{11}$ and ${\mathsf{\lambda }}_9$, the relationship between the severe infection rate, mild infection rate, and the probability of meeting treatment criteria is derived from Formula (20):

$$\begin{array}{c}\mathrm{P}\left({\mathrm{M}}_{12}\right)={\displaystyle \frac{945{\mathsf{\lambda }}_9 {\mathsf{\lambda }}_{11} }{17640{\mathsf{\lambda }}_9+17640{\mathsf{\lambda }}_{11} +24382{\mathsf{\lambda }}_9{\mathsf{\lambda }}_{11}}}\end{array}$$

(23)

The changes in the contour lines in Figure 8 show the regions where the probability of meeting the treatment criteria is the same under different infection rate combinations. For example, when ${\mathsf{\lambda }}_{11}=1.8$ and ${\mathsf{\lambda }}_9=2.4$, $\mathrm{P}\left({\mathrm{M}}_{12}\right)=0.0231$. This indicates that under this infection rate combination, the probability of meeting the treatment criteria is 2.31%. The system needs to activate the early warning mechanism and promptly implement corresponding control measures. Compared to the ${\mathsf{\lambda }}_9$, ${\mathsf{\lambda }}_{10}$ combination and the ${\mathsf{\lambda }}_{10}$, ${\mathsf{\lambda }}_{11}$ combination, the ${\mathsf{\lambda }}_{11}$, ${\mathsf{\lambda }}_9$ rate combination has an intermediate impact on the probability of meeting the treatment criteria. This indicates that, although the individual effect of mild infection is relatively small, its combined impact with severe infection remains significant. Therefore, in the pest and disease control process, it is essential to consider the combined effects of different infection stages and develop multi-level control strategies to ensure timely and effective management of disease progression.

Through the comparative analysis of the different infection rate combinations, the key impacts of different infection stages on pest and disease control can be identified. To further understand the complex spread mechanisms of plant pests and diseases, and to develop more effective control strategies in practical applications, the interactions between the mild infection rate, moderate infection rate, and severe infection rate will continue to be explored in order to establish graded early warning and control strategies.

Three different early warning activation thresholds are set to measure the severity of pest and disease occurrence and trigger the corresponding control measures. The specific definitions are as follows: When the probability of pest and disease occurrence reaches 0.005, the system activates the blue alert. This is applicable when pests or diseases have just started to appear or have a low level of spread, and the main goal is early detection and prevention to avoid further spread of the condition. When the probability of pest and disease occurrence reaches 0.01, the system activates the yellow alert. This applies when the pest and disease spread is more significant but not yet severe, requiring more active control measures to manage the disease progression. When the probability of pest and disease occurrence reaches 0.015, the system activates the red alert. This applies when the pest and disease spread is rapid and severe, requiring immediate and strong control measures to prevent widespread economic losses and plant death.

As shown in Figure 9, when ${\mathsf{\lambda }}_9=1.15,{\mathsf{\lambda }}_{10}=0.13,{\mathsf{\lambda }}_{11}=0.95$, the system reaches the blue alert critical surface, indicating the need for attention and early control measures. When ${\mathsf{\lambda }}_9=1.17,{\mathsf{\lambda }}_{10}=0.37,{\mathsf{\lambda }}_{11}=1.81$, the system reaches the yellow alert critical surface, suggesting the need for more active control measures to prevent further deterioration of the condition. When ${\mathsf{\lambda }}_9=1.59,{\mathsf{\lambda }}_{10}=0.88,{\mathsf{\lambda }}_{11}=1.84$, the system reaches the red alert critical surface, requiring immediate and strong control measures to prevent widespread economic losses.

4.2.3. Treatment Execution Stage

In the simulation, keeping all other $\mathsf{\lambda }$ values constant, ${\mathsf{\lambda }}_{21}$ was varied from 1 to 20. The relationship between ${\mathsf{\lambda }}_{21}$ and the steady-state probabilities of each system state is shown in Figure 10. As ${\mathsf{\lambda }}_{21}$ increases, which indicates that the speed of the treatment effectiveness assessment process is accelerated, the probability of meeting the treatment effectiveness assessment, $\mathrm{P}\left({\mathrm{M}}_{21}\right)$, significantly decreases. This results in an overall increase in the probabilities of other states, with the probabilities of severe infection $\mathrm{P}\left({\mathrm{M}}_{11}\right)$ and recovery of health $\mathrm{P}\left({\mathrm{M}}_{19}\right)$ increasing the most. This behavior occurs because the system has more opportunities to try different treatment methods until a viable and effective solution is found, thus improving the treatment speed. Additionally, the changes in treatment effectiveness assessment also affect data monitoring and subsequent control measure implementation through feedback. Since the system needs to continuously adjust control measures based on treatment effectiveness, this change leads to iterative and repeated adjustments in the treatment process. This, in turn, enhances the overall system’s efficiency and resource allocation. Therefore, to improve treatment efficiency and effectiveness, the system needs to automatically adjust its treatment strategies based on real-time plant growth data and pest/disease conditions, ensuring a responsive and optimized treatment process.

5. Case Study

5.1. The GSPN Model for the Spread and Evolution of Grape Downy Mildew

Grape downy mildew is a common plant disease caused by Plasmopara viticola, widely distributed in grape-growing areas, and especially severe under humid and warm climatic conditions. The disease is transmitted through airborne spores that infect grape plants, often initially attacking the leaves and forming typical white mold layers on the leaf surfaces. If not promptly controlled, downy mildew can cause significant losses to both the yield and quality of grapes [33]. As the spread of downy mildew is influenced by environmental factors such as temperature and humidity, establishing an effective early warning system is crucial for the prevention and control of this disease.

In the early warning system for grape downy mildew, the leaves are the most critical observation area, while the monitoring of young shoots also holds certain significance. In comparison, the observation of roots, stems, flowers, fruits, and buds has a lesser role in disease assessment. To better align the GSPN model with the actual prevention and control needs of grape downy mildew, the model of plant pest and disease spread evolution is simplified in this case study. Based on practical considerations, we set ${\mathsf{\lambda }}_2={\mathsf{\lambda }}_3={\mathsf{\lambda }}_5={\mathsf{\lambda }}_6={\mathsf{\lambda }}_7={\mathsf{\lambda }}_8=0$, thereby creating a more simplified GSPN model focused specifically on the leaf disease analysis of grape downy mildew.

5.2. Verification of the Grape Downy Mildew Pest and Disease Early Warning System

Based on the GSPN model for the spread and evolution of grape downy mildew, this study evaluated the severity of the disease and implemented corresponding control strategies through simulation analysis of different disease occurrence probability combinations. To validate the effectiveness of this model, data collection was conducted from April to May 2025 at a grape planting base in Kunming, Yunnan Province, with the experimental period lasting two months. During the experiment, the health status of grape leaves was closely monitored, and high-resolution imaging equipment was used for periodic image data collection of these areas. Researchers collected images of selected grape plants at a fixed time each day (from 9:00 am to 10:00 am) to ensure stable environmental lighting and avoid interference with image quality. At least three images were taken of each grape plant to comprehensively document growth changes and disease development. The captured images were annotated by professionals, particularly marking the infected areas for downy mildew lesions. The annotation and segmentation of the leaves and lesions were performed using Labelme, which allows for precise labeling of the leaf and lesion areas and generates corresponding JSON format files. These files were ultimately used to train and validate the segmentation model’s performance.

The indicators for classifying disease severity are typically set based on the extent of the disease’s impact on crops. These indicators help farmers, horticulturists, and plant protection experts assess the severity of the disease and take appropriate control measures accordingly. For downy mildew, the area of the lesions is an intuitive indicator for evaluating disease severity. By measuring the proportion of the lesion area relative to the leaf area, one can roughly determine the stage of development and severity of the disease. The calculation involves determining the ratio of the number of pixels in the lesion area to the number of pixels in the leaf area, which gives the percentage of the leaf area affected by the lesions. The formula for this calculation is shown in Formula (24):

$$\mathrm{P}={\displaystyle \frac{{\mathrm{A}}_{\mathrm{d}}}{{\mathrm{A}}_1}}\times 100\%$$

(24)

In Formula (24), $\mathrm{P}$ represents the percentage of the downy mildew lesion area relative to the entire leaf area; ${\mathrm{A}}_{\mathrm{d}}$ is the area of the lesion region; and ${\mathrm{A}}_1$ is the area of the leaf region. Since the dataset used in this study consists of single-leaf images, to better assess the severity of the disease, the severity of grape downy mildew was classified by calculating the ratio of the pixel area of the downy mildew lesions on a single leaf to the pixel area of the entire grape leaf.

In the disease identification process, a two-step segmentation method is used to process grape leaf images. The first step involves using a leaf segmentation model to extract the grape leaf, separating it from the complex background and obtaining an image of the leaf with a simple background. In the second step, based on the leaf image with a simple background, a lesion segmentation model is used to extract the lesion area of downy mildew. After obtaining the lesion area, the pixel area of the lesion is calculated and compared with the pixel area of the leaf. This ratio is used to calculate the proportion of the lesion area relative to the total leaf area, and the severity of the disease is then determined.

According to the Chinese national standard GB/T17980.122-2004 [34], the classification of grape downy mildew is based on the proportion of the lesion area relative to the leaf area. In this study, the classification is specifically divided into mild, moderate, and severe infections. The classification standards are as follows: mild infection is defined as when the lesion occupies less than 25% of the leaf area; moderate infection is when the lesion occupies between 25% and 75% of the leaf area; severe infection is when the lesion occupies more than 75% of the leaf area. The specific grading standards are shown in

Table 11. Classification standard of downy mildew.

Severity Level	Description
Mild Infection	Lesions occupy less than 25% of the leaf area
Moderate Infection	Lesions occupy 25% to 75% of the leaf area
Severe Infection	Lesions occupy more than 75% of the leaf area

.

Based on the GSPN model for the spread and evolution of grape downy mildew, the analysis of different combinations of disease occurrence probabilities enables the assessment of disease severity and the implementation of corresponding control strategies. Through comparative analysis of various infection stages, this study illustrates the spread of downy mildew in grape-growing areas of Kunming and demonstrates the effectiveness of the early warning mechanism. The specific descriptions of grape downy mildew severity are detailed in

Table 12. Description of grape downy mildew severity.

Date	P(M12)	Detailed Disease Description	Warning Level
10 April	0.002	Small, translucent spots appear on the front side of the leaves; under high humidity, mold is visible; lesion edges are unclear.	Blue Warning
11 April	0.004	Lesions begin to turn yellow and gradually take on a circular or irregular shape; a small amount of white mold appears on the leaf underside.	Blue Warning
18 April	0.007	Lesions expand; under high humidity, mold increases significantly; infected leaves begin to wither, affecting grape photosynthesis.	Blue Warning
22 April	0.012	Lesions merge; mold spreads further with increased humidity; fruits begin to show signs of mold infection, affecting fruit development.	Yellow Warning
26 April	0.014	Lesions have extensively expanded; mold covers most leaves; some fruits show severe rot, impacting yield.	Yellow Warning
30 April	0.018	Lesions nearly cover all leaves; fruits rot; mold almost fully covers surfaces, causing severe damage to grapevine health.	Red Warning
3 May	0.017	Lesions decrease; some leaves recover, but mold persists; fruit conditions improve somewhat but full recovery has not yet occurred.	Yellow Warning
10 May	0.013	Most mold disappears; lesion area gradually reduces; grapevines begin to recover and resume growth; further management is still required.	Yellow Warning

.

According to the early warning activation rules, the early warning probability for grape downy mildew is determined to be $\mathrm{P}\left({\mathrm{M}}_{12}\right)=0.018>0.015$, thereby triggering a red warning. Experimental data further show that on 30 April 2025, lesions almost completely covered all leaves, fruits were rotting, and mold had nearly entirely spread across the plant surfaces, resulting in severe damage to grapevine health—consistent with the findings of this study. The disease progression trend of grape downy mildew is illustrated in Figure 11.

6. Conclusions

In the context of modern agricultural demands generated by large-scale planting, the spread and control of plant pests and diseases have become important research topics in agricultural production and plant protection, which are critical for improving production efficiency and ensuring crop health. However, existing studies often lack an in-depth analysis of the dynamic evolution of pest and disease spread and complex system modeling, especially the spatial and temporal dynamics of pest and disease spread, which have not been systematically quantitatively analyzed.

This paper first identifies and classifies plant pest and disease early warning mechanisms. Based on environmental factors and symptomatology of affected plant parts, a dynamic model of pest and disease spread is established, and a systematic description of the plant pest and disease early warning mechanism system is provided. Secondly, a generalized GSPN model is constructed to analyze the spread process and influencing factors of pests and diseases. By utilizing the isomorphic relationship between Petri nets and Markov chains, the reachability graph is generated, and a Markov chain model is constructed. The model’s validity and feasibility are verified using PIPE.

On this basis, three performance indicators—place busy probability, place idle probability, and transition utilization rate—are calculated to identify the key factors and critical nodes in the pest and disease spread process, namely, information filtering (${\mathrm{t}}_1$), mild infection (${\mathrm{t}}_9$), moderate infection (${\mathrm{t}}_{10}$), severe infection (${\mathrm{t}}_{11}$), reaching treatment standards (${\mathrm{t}}_{12}$), and treatment effectiveness assessment (${\mathrm{t}}_{21}$). By changing the values of these key factors, the impact of these factors on the overall pest and disease evolution process is analyzed. The simulation results show that different infection rate combinations significantly affect the probability of meeting treatment criteria. Among them, the combination of a moderate infection rate and severe infection rate has the most significant impact on the probability of meeting treatment criteria, while the effect of the mild infection rate is relatively smaller. By considering the interactions between mild, moderate, and severe infection rates, critical surfaces for different pest and disease early warning thresholds (0.005, 0.01, 0.015) are derived. The plant pest and disease early warning critical surfaces illustrate the probability distribution of reaching the respective early warning thresholds under different infection rate combinations, providing an intuitive basis for early warning decisions and treatment measures.

In practice, the plant pest and disease diagnosis model based on the GSPN developed in this paper is designed to provide effective decision support for agricultural production and plant protection. This model can be used to monitor and alert for pest and disease situations in plant cultivation by simulating the probability of meeting treatment criteria under different infection rate combinations, helping farmers and agricultural management departments take appropriate control measures in a timely manner. Based on the characteristics of the spread and evolution of grape downy mildew, this study established a simplified GSPN model focusing on the analysis of leaf lesions. Experimental data were collected at a grape cultivation base in Kunming, Yunnan Province, China. The experimental results demonstrate that the predictions made by the GSPN model are consistent with the actual data collected. During the early warning and control of grape downy mildew, the model accurately identified the disease transmission process and severity, successfully predicting the occurrence of the disease. This experimental validation further confirms the practical applicability and value of the model in real-world agricultural environments.

However, there are still some limitations in this study. Firstly, in the process of agricultural pest and disease spread, multiple environmental factors, different crop types, and complex interactions are often involved. This means the application of Petri nets in these areas potentially faces challenges such as high model complexity and difficulties in directly applying simulation results to practice. In addition, with large and rapidly changing datasets, ensuring the real-time performance and accuracy of the model is also an important direction for future research. Second, although this paper considers the interactions between mild, moderate, and severe infection rates, other factors, such as climatic conditions, soil moisture, and nutritional status, which can also influence pest and disease spread, have not been thoroughly addressed. These factors will be the focus of more in-depth research in future experiments.