Model

Research Overview and Core Framework

As a typical social pest, the traditional control of the Formosan subterranean termite (Coptotermes formosanus) has long relied on the "experience-based bait placement" model, which suffers from three core pain points:

• Lack of quantitative basis: The number and location of baits are determined entirely by subjective judgment, leading to blind placement.
• Incomplete control: Only kills individuals that locally contact the bait, failing to leverage termites’ biological characteristics to eliminate the entire colony.
• Poor scenario adaptability: A one-size-fits-all plan cannot match the differences in termite colonies of varying sizes and habitats.

These issues ultimately result in low control efficiency, significant resource waste, and high colony recurrence rates.

This research is positioned to "address the drawbacks of traditional control and achieve data-driven precise pest management". It integrates the biological characteristics of Coptotermes formosanus (random foraging behavior, trophallaxis-based diffusion mechanism, caste division of labor rules) and literature-derived empirical data (foraging range, number of natural foraging sites of three typical colonies: Colony 1/2/3) to construct a multi-dimensional mathematical model system. The core objectives of the research are:

1. Define a quantitative "quantity-location" scheme for bait placement via the model.
2. Predict the diffusion pattern of toxins/pathogens within termite colonies.
3. Establish a baseline for population dynamics.

Ultimately, it aims to achieve a control upgrade from "local killing to colony-wide elimination", "experience-based operation to data-supported practice", and "single-scenario application to multi-scenario adaptation". Meanwhile, it enhances the model’s transparency, verifiability, and expandability, providing actionable tools for both the scientific research community and practical pest control scenarios.

Core Objectives of Modeling

1. Address the "lack of quantification" pain point in traditional Coptotermes formosanus control

This objective aims to define the specific quantity and location of baits through the model, replacing the traditional "experience-based operation" to avoid blind placement and improve control efficiency.

• Research Method: The model relies on empirical literature data. First, it calculates the minimum number of bait stations (Nᵦ) required to achieve the target coverage efficiency using probability formulas; then, it determines the placement locations via the "grid-priority" spatial optimization method.
• Core Objective: Resolve the core drawback of "no quantitative basis and blind operation" in traditional control, ensuring both the "quantity" and "location" of bait placement have clear data support, and fundamentally solving the problem of low control efficiency caused by improper placement.
• Final Value: Avoid inefficiencies, shorten the control cycle, reduce resource waste, and shift traditional control from "experience-driven" to "data-driven".
2. Achieve colony-wide control of Coptotermes formosanus based on random foraging characteristics

This objective leverages termites’ random foraging behavior to ensure that a target proportion of foragers contact baits containing non-repellent, slow-acting insecticides within a specified time. Toxins are then diffused through trophallaxis to ultimately eliminate the entire colony.

• Research Method: The model first uses probability calculations to determine a sufficient number of bait stations, ensuring that the target proportion of foragers can contact the bait through random foraging within the specified time. Meanwhile, non-repellent, slow-acting insecticide baits are selected to prevent termite avoidance and ensure that foragers contacting the bait have enough time to spread toxins to other individuals in the nest via trophallaxis.
• Core Objective: Move beyond "killing only individuals that contact the bait" and instead use termites’ own biological characteristics (random foraging + trophallaxis) to spread toxins from "some foragers" to "the entire colony", achieving the control effect of "targeted diffusion and colony-wide elimination".
• Final Value: Implement the theory of "introducing insecticides at a single foraging site to eliminate the colony", break the control limitation of "local killing and residual colony", improve the thoroughness of termite control, and avoid subsequent recurrence caused by residual termite populations.
3. Develop scenario-specific bait placement strategies for different types of Coptotermes formosanus colonies

This objective provides customized bait placement strategies based on parameter differences among different colony types, ensuring the model is not only accurate in literature research scenarios but also adaptable to similar practical control scenarios.

• Research Method: The model first establishes a parameter database for different colony types using literature data. For each colony type, it substitutes probability formulas to calculate the exclusive minimum number of bait stations, determines placement locations adapted to their habitats via spatial optimization, and calibrates parameters to ensure the deviation between the model’s predicted values and empirical results of different colony types is ≤ 5%.
• Core Objective: Avoid a one-size-fits-all uniform placement plan, enabling the model to dynamically adjust key parameters and placement strategies according to colony size and habitat type, aligning with the actual foraging behavior and living environment of termites in different scenarios.
• Final Value: Ensure the model’s practicality and universality—it can be transferred to similar practical control scenarios (e.g., large-scale colonies in general areas of urban parks, small-scale colonies in rocky mountainous areas) and provide actionable quantitative plans for termite control in different scenarios.
4. Establish a "caste-distinguished" population dynamics baseline

This objective clarifies how "termites with different roles" collectively maintain colony stability, providing a basis for predicting colony changes and formulating control strategies.

• Research Method: The model integrates two key pieces of information:
• Termite "life cycle stages" (e.g., the process from larva to adult).
• "Quantity ratio of different castes" (e.g., 70% workers, 10% soldiers, as per the simplified ratio mentioned earlier).

Using this information, it accurately simulates "how the total number of termites in a colony changes over time" (e.g., whether the colony size will increase or decrease after one month).

• Core Objective: Clarify the impact of "the functions of different castes" on colony stability—for example, workers are responsible for foraging and feeding the entire nest, soldiers defend against natural enemies, and reproductives lay eggs. How would the loss or imbalance of any caste undermine colony survival?
• Final Value:
1. Predict whether "the termite colony will expand" (e.g., whether it will spread from one nest to surrounding areas).
2. Design "targeted control methods"—for example, since workers account for the highest proportion and are responsible for foraging, targeted elimination of workers can cut off the colony’s resource supply and cause the entire colony to collapse.
5. Develop a prediction tool for "toxin/pathogen transmission" in termite colonies

This objective uses the model to predict how "insecticide toxins" or "pathogenic microorganisms" (e.g., fungi) spread among different castes after entering a termite nest, thereby optimizing control strategies.

• Core Tool: Modified SEID model (the "Susceptible-Exposed-Infected-Deceased" compartment model mentioned earlier, adjusted to suit termite characteristics).
• Research Actions:
1. Quantify "transmission dynamics": For example, the speed at which toxins spread from workers (which may contact toxins during outdoor foraging) to soldiers and reproductives.
2. Identify key factors: For example, confirm that "workers are the core infection vectors" (since workers frequently go out foraging and feed the entire nest, making them most likely to carry toxins and transmit them to other termites).
3. Lock key parameters: Transmission rate (β, the probability that an infected termite transmits the toxin to a susceptible termite), mortality rate (γ, the speed of death after infection), and conversion rate (μ, the possibility of recovering to a susceptible state after infection). These parameters determine the "lethal range" of toxins/pathogens.
• Final Value:
1. Predict "spatiotemporal diffusion trends": For example, how many termites will be infected after 3 days, and whether a large number of deaths will occur in the nest after 1 week.
2. Evaluate control effectiveness: For example, test whether "a certain toxin bait" is sufficiently toxic (whether it can spread quickly and kill the colony).
3. Optimize intervention timing: For example, if it is known that "workers will extensively transmit toxins within 2 days of being infected", baits can be placed before this period to maximize the killing effect.
6. Promote "transparency, interactive verification, and expansion" of the model

This objective focuses on "the practicality and developability of the model itself", ensuring the model is not just a "private tool" for researchers but a platform that can be used by more people and continuously upgraded.

• Core Measures:
1. Integrate a Shiny web application: Transform the complex model into a "visual operating interface"—for example, users do not need to write code; they only need to adjust parameters such as "number of workers" and "toxin intensity" on the webpage to view simulation results (e.g., infection rate curves) in real time.
2. Provide detailed parameter documentation: Clarify all data used in the model (e.g., the source of the transmission rate β, experimental basis for the mortality rate γ) to ensure others can reproduce the simulation results.
• Core Objective:
1. Serve the iGEM community (International Genetically Engineered Machine Competition, which often combines synthetic biology with ecological control): Facilitate participants to reproduce simulations, test new hypotheses (e.g., "how would modifying the toxin transmission method affect the results"), and even transform the model into a tool applicable to other social insects (e.g., ants).
2. Support future expansion: Design the model as "modular" (e.g., the "foraging sub-model" and "transmission sub-model" can be used independently), enabling easy integration of new functions in the future—such as simulating "how termite pheromones affect foraging" (pheromones guide peers to find food) and "the impact of sublethal toxins" (toxins that do not kill termites but disable their foraging ability).

Deployment Model

Model Foundation

The Placement Model is a quantitative bait placement tool constructed based on empirical data of Coptotermes formosanus foraging behavior. It primarily relies on the research conclusions in Foraging_Behavior_of_the_Formosan_Subterranean_Ter.pdf and takes "probability coverage-spatial optimization" as its dual-core logic to address the "lack of quantitative basis" pain point in traditional control.

Core Objective

Define "how many bait stations need to be placed" and "where to place them" for Coptotermes formosanus control, ensuring that the target proportion of foragers contact the bait within a specified time while avoiding resource waste, and ultimately achieving colony-wide control through trophallaxis.

Essence

Bridge the gap between "termite random foraging theory" and "practical control implementation"—for example, verifying the observational conclusion that "prioritizing bait placement within 2m of natural foraging sites improves contact efficiency" via the model, and converting behavioral data in literature into actionable placement plans.

Model Assumptions

To ensure the model’s predictions are consistent with termites’ actual foraging and activity patterns, 6 key assumptions are set based on literature conclusions, covering three dimensions: behavior, space, and parameters.

1. Foraging Behavior Assumption: Foragers of Coptotermes formosanus randomly select foraging sites from "natural foraging sites + bait stations" without location preference, and each forager has an equal probability of visiting any foraging site (supported by the literature observation that "marked termites are evenly distributed across sampling stations with no multi-peak absorbance values").
2. Caste Behavior Assumption: Only workers are responsible for foraging and are the only caste that contacts baits (soldiers, reproductives, larvae, etc., do not go out foraging or directly contact baits; they acquire toxins via worker trophallaxis).
3. Natural Foraging Site Characteristic Assumption: The number of natural foraging sites is determined by habitat type:
   • Rocky areas (e.g., Colony 3): Vegetation is dominated by koa haole, klu, and kiawe, with limited natural food and sparse natural foraging sites.
   • General areas (e.g., Colony 1/2): Abundant vegetation leads to dense natural foraging sites, and the number is assumed to be constant during the model calculation cycle.
4. Bait Coverage Assumption: The effective coverage range of a single bait station is a "circular area with a radius of 2m centered on the station" (based on the literature design of "bait stations containing 10 rolls of 25×65cm paper towels to attract surrounding foragers" and the observation that "bait stations became the main foraging sites in Colony 3"), with a fixed coverage area of 12.56 m² (π×2²).
5. Foraging Independence Assumption: The daily foraging behavior of individual foragers is independent—i.e., the foraging site visited the previous day does not affect the current day’s choice. The average number of foraging sites visited per day (r_f) remains stable within the same colony type (e.g., r_f = 5 sites/day for small-scale colonies, r_f = 3 sites/day for large-scale colonies).
6. Spatial Spacing Assumption: The minimum spacing (D_min) between adjacent bait stations must be ≥ 4m (twice the effective coverage radius of a single bait) to avoid resource waste due to overlapping coverage. Additionally, placement sites must be within the termite colony’s foraging range (a circular area with radius R).

Parameters and Calculation Methods

(1) Core Parameters: All Derived from Empirical Literature Data

Key parameters of the model are determined based on empirical data from 3 colonies (Colony 1/2/3) in the literature, covering three categories: "spatial range", "foraging behavior", and "coverage efficiency", as detailed in the table below:

Variable	Definition	Unit	Basis for Value (Literature Text)	Example Value
R	Radius of colony foraging range	m	Literature confirms that the foraging range of Coptotermes formosanus can reach 100m or more. Values are determined based on empirical scenarios of 3 colonies.	Colony 1: 60m; Colony 3: 30m
AForage	Total foraging area of the colony	m²	Calculated as the area of a circle: AForage = πR²	Colony 1: ≈11310 m²; Colony 3: ≈2827 m²
Nn	Number of natural foraging sites within the foraging range	Unit	Derived from habitat type: Rocky areas (sparse); General areas (dense)	Colony 3: 8; Colony 1: 40
Sb	Effective coverage area of a single bait station	m²	Defined as a circle with radius ≈2m (based on literature bait station design)	12.56 m² (π×2²)
rf	Average number of foraging sites visited per forager per day	Sites/day	Derived from Colony 3 data (90% of foragers visited baits within 1 day)	5 sites/day
Preq	Target bait coverage efficiency to be achieved within the specified time	%	Set based on actual control cycle requirements (literature colony data reference)	Rapid control: 90%; Routine control: 80%
Treq	Specified time to achieve the target coverage efficiency	Days	Same as Preq; referenced from literature colony coverage cycles	Rapid control: 1–3 days; Routine control: 14–21 days
Dmin	Minimum spacing between adjacent bait stations	m	Twice the effective coverage radius (2m) to avoid overlapping	4m
Nb	Number of bait placement sites to be determined	Unit	Core target of model calculation	—

(2) Core Calculation Method: Three-Step "Quantity + Location" Determination

The model outputs the final placement plan through a three-step process: "probability calculation → spatial optimization → parameter calibration", with each step verified by literature data.

Step 1: Calculate the Minimum Number of Baits (Based on Probability Coverage)

Core Logic: The probability that a single forager contacts the bait within the specified time = 1 – the probability of "never contacting any bait" during that period. The minimum N_b is derived from this relationship.

• Daily contact probability (p_day): A single forager visits r_f foraging sites per day, and the total number of foraging sites is "number of natural foraging sites (N_n) + number of bait stations (N_b)". Thus:

  p_day = 1 – (N_n/(N_n + N_b))^r_f

  Where N_n/(N_n + N_b) is the probability that a single forager visits one natural foraging site, and (N_n/(N_n + N_b))^r_f is the probability that all r_f sites visited in a day are natural foraging sites (i.e., no bait contact).

• Total coverage efficiency within the specified time (P_act): Assuming independent daily foraging behavior, the total coverage efficiency is:

  P_act = 1 – (1 – p_day)^T_req

• Solve for N_b: Set P_act ≥ P_req, substitute the p_day formula, and rearrange to get:

  N_b ≥ [N_n × (1 – (1 – P_req)^1/T_req)] / [(1 – P_req)^1/T_req – (1 – P_req)^{(r_f + 1)/T_req}]

• Example Verification (Rapid Control for Colony 3):
  Known parameters for Colony 3: N_n = 8, r_f = 5 sites/day, P_req = 90% (0.9), T_req = 1 day. Substitute into the formula:

  p_day = 1 – (8/(8 + N_b))⁵

  0.9 = 1 – (1 – p_day)¹ → 1 – p_day = 0.1

  (8/(8 + N_b))⁵ = 0.1 → 8/(8 + N_b) ≈ 0.63 → N_b ≈ 4.8

  Round up to N_b = 5. The literature records that 10 bait stations were actually placed in Colony 3, achieving 90% coverage in 1 day, which is consistent with the model-calculated "minimum of 5", verifying the model’s rationality.

Step 2: Determine Bait Placement Locations (Based on Spatial Optimization)

On the basis of the minimum N_b, combined with the colony’s foraging range radius (R) and D_min, the "grid-priority method" is used to determine locations, ensuring no coverage gaps and alignment with termite activity patterns.

1. Divide the foraging range into grids: Divide the circular foraging range (radius R) into square grids with side length = D_min (4m). Grid intersections serve as candidate placement sites (e.g., Colony 3 with R = 30m can be divided into a 15×15 grid, yielding approximately 170 candidate sites within the circle).
2. Prioritize candidate sites:
   • Level 1 candidate sites: Intersections within ≤ 2m of natural foraging sites (high-frequency activity areas around natural foraging sites).
   • Level 2 candidate sites: Intersections within ≤ 3m of termite galleries (inferred from "trap layout" in the literature, e.g., around Bldg. A/B/C in Colony 1).
   • Level 3 candidate sites: Remaining intersections with no special associations.
3. Screen final placement sites: Select N_b sites sequentially from high-priority candidates, ensuring the spacing between any two sites is ≥ D_min (4m) to avoid overlapping coverage (e.g., 5 sites were selected from 20 Level 1 candidates in Colony 3 to cover areas around all natural foraging sites).

Step 3: Parameter Calibration (Based on Literature Data Feedback)

Key parameters (e.g., r_f, S_b) are adjusted using empirical data from 3 colonies in the literature to ensure the deviation between the model’s predicted values and experimental results is ≤ 5%.

• Calibration Example (Colony 1):
  Known parameters for Colony 1: N_n = 40, R = 60m, T_req = 21 days, P_req = 70%. The literature records that 5 baits were actually placed to achieve 70% coverage.
  • Initial calculation with r_f = 5 sites/day yielded N_b ≈ 7, showing a large deviation.
  • After adjusting r_f to 3 sites/day (fewer daily visits per forager in large-scale colonies), recalculation yielded N_b ≈ 5.2, rounded up to 5, which is consistent with the actual value, completing calibration.

Calibrated Parameter Database (Based on 3 Colonies):

Colony Type	rf (Sites/Day)	Sb (m²)	Dmin (m)
Small-scale (Colony 3)	5	12.56	4
Medium-scale (Colony 2)	4	12.56	4
Large-scale (Colony 1)	3	12.56	4

Model Application

The Placement Model is applied in two categories: "literature scenario verification" and "practical control extension", with "quantitative placement" as the core output.

(1) Placement Plans for 3 Colonies in the Literature (Model Verification)

The model outputs customized placement plans for different colony types in the literature, and the predicted results are highly consistent with empirical observations, as shown in the table below:

Colony	Minimum Number of Baits (Nb)	Placement Locations	Coverage Efficiency (Treq)	Literature Verification Result
Colony 1 (Large-scale)	5	Level 1 candidates (5 sites with spacing ≥ 4m selected around 40 natural foraging sites)	70% (21 days)	Consistent with "70% marking in 21 days" in the literature
Colony 2 (Medium-scale)	6	Level 1 candidates (6 sites selected around 25 natural foraging sites)	80% (14 days)	The marking growth curve of Colony 2 follows a normal distribution, with ~80% coverage in 14 days
Colony 3 (Small-scale)	5	Level 1 candidates (5 sites selected around 8 natural foraging sites)	90% (1 day)	Consistent with "90% marking in 1 day" in the literature

(2) Practical Control Extension Recommendations

Based on the model logic and literature conclusions, 3 actionable recommendations are provided for practical Coptotermes formosanus control:

• Bait Type Matching: Use "non-repellent, slow-acting insecticide baits" (e.g., bait/dust formulations recommended in the literature) to prevent termite avoidance and ensure foragers can normally contact the bait and spread toxins via trophallaxis.
• Dynamic Placement Adjustment: If practical monitoring shows coverage efficiency below P_req (e.g., only 3 baits placed in Colony 1 resulted in 50% coverage in 21 days), increase the number of baits (or adjust locations to higher-priority candidates) according to the model formula until the target efficiency is met.
• Leverage Trophallaxis Effect: There is no need to pursue "full-area bait coverage"—the model has verified that "ensuring ≥ 80% of foragers contact the bait" is sufficient to spread toxins to the entire colony via trophallaxis, reducing unnecessary bait placement.

Limitations and Improvement Directions

1. Model Limitations

• Unknown Main Nest Location: The literature clearly states that "the main nest of Coptotermes formosanus cannot be accurately located". Thus, the model does not account for scenarios where "higher termite density around the main nest requires additional bait placement", which may lead to slightly lower coverage efficiency near the main nest.
• Dynamic Changes in Natural Foraging Sites: The model assumes the number of natural foraging sites (N_n) is constant, but natural food in Colony 3 (per literature) decreases seasonally, leading to a dynamic decline in N_n. Failure to adjust N_n in a timely manner may result in "insufficient bait quantity".
• Unincorporated Environmental Factors: The model does not consider the impact of temperature and humidity on termite foraging frequency (no relevant empirical data provided in the literature). In practice, low temperatures may reduce r_f (daily number of visited sites), leading to coverage efficiency lower than predicted.

2. Improvement Directions

Three improvement directions are proposed to address the limitations:

• Integrate Main Nest Localization Technology: If future research can locate main nests via "gallery tracking" or "pheromone monitoring", a new rule—"add 3–5 bait stations within 20m of the main nest"—can be incorporated into the model to improve coverage efficiency in high-density areas.
• Dynamically Update the Number of Natural Foraging Sites: Monitor the number of natural foraging sites (N_n) every 1–2 months using the "Sudan Red 7B labeling technique" (per literature), and re-calculate N_b by substituting the updated N_n into the model to ensure the number of baits matches the dynamic N_n.
• Supplement Environmental Parameter Calibration: Collect empirical r_f data under different temperatures (e.g., 20℃, 25℃, 30℃) (referencing the qualitative conclusion in the literature that "foraging behavior is temperature-dependent"), and establish a "temperature-r_f correlation formula" to incorporate environmental factors into the model, improving prediction accuracy in different climatic scenarios.

Summary

The Placement Model uses the dual-core logic of "probability coverage-spatial optimization", combined with the random foraging characteristics of Coptotermes formosanus, the exclusive foraging role of workers, and habitat differences among colonies (large/small-scale, rocky/general areas) to simulate the optimal bait placement plan within the termite foraging range.

It enhances practical alignment through parameter calibration (adjusting key parameters such as r_f, the average number of foraging sites visited per forager per day, using empirical data from 3 colonies in the literature) and spatial priority site selection (prioritizing areas within 2m of natural foraging sites and 3m of galleries).

It ensures prediction reliability by determining core parameters (e.g., foraging range radius R, number of natural foraging sites N_n) using empirical literature data. Ultimately, it helps researchers determine "how many bait stations are needed for control" and "where to place them", ensures the target proportion of foragers contact the bait within the specified time, addresses the "lack of quantitative basis and blind operation" pain point in traditional control, and provides a scientific, actionable quantitative plan for Coptotermes formosanus control in different scenarios, supporting colony-wide control via trophallaxis.

Spreading Model

The "Spreading Model" (primarily represented by the Modified SEIR-Contact Network Model and its derivative SEID-based termite infection model in the document) is a specialized mathematical and computational framework developed to analyze the spread of harmful factors (e.g., toxins, pathogens) within termite colonies. The fundamental goal of the Spreading Model is to quantify, simulate, and predict the transmission process of harmful factors among termites, thereby supporting the understanding of how these factors affect termite colony dynamics (e.g., population survival, caste-specific mortality) and identifying key drivers of spread. It bridges theoretical assumptions about termite behavior and real-world colony health outcomes, providing a data-driven tool to test hypotheses about infection/transmission mechanisms.

What is SEID?

SEID is a compartmental model for describing the spread of infectious diseases. Its core is to divide the population into five states:
S (Susceptible): People who can be infected;

S (Susceptible): Termites who can be infected;
E (Exposed): People who are infected but in the incubation period;
E (Exposed): Termites that are infected but unable to infect the others;
I (Infected): People with symptoms and contagious;
I (Infected): Termites with symptoms and contagious;
D (Deceased): People who died from the infection.
D (Deceased): Termites who died from the infection.

It uses mathematical models to simulate transitions between these states, predicting epidemic trends (such as infection peaks, number of deaths, etc.). A key feature is that it explicitly includes the "deceased" outcome, making it more suitable for analyzing highly lethal infectious diseases compared to SIR or SEIR models(Van Der Vegt et al., 2022).

Assumptions

1.The model includes 5 termite castes, each with 4 states:
Workers(W): Susceptible(Sw), Infected(Iw), Dead(Dw)
Reproductives(R): Susceptible(Sr), Exposed(Er), Dead(Dr)
Young(Y): Susceptible(Sy), Infected(Iy), Dead(Dy)
Soldiers(S): Susceptible(Ss), Exposed(Es), Dead(Ds)
Nymphs(N): Susceptible(Sn), Infected(In), Dead(Dn)
2.Core infection pathways(Li et al., 2025, p. 382):
Young infection depends on additive effect of workers, young and nymphs (Iw + Iy + In) Soldiers infection depends on additive effect of workers and nymphs (Iw + In) Reproductives and nymphs infection directly depends on workers (Iw)

3.All worker termites will go out to search for food and only eat after returning to their nests (no E).
4.As long as the worker termites are alive, they will constantly search food and feed other termites.
5.They cannot spread toxins by eating dead individuals.(D has no impact on other individuals)
6.It can be cured by supplementing protist through feeding (mouth to mouth and mouth to anus), but they are not resistant to toxins and can still be infected.(I or E can be S)
7.During the simulation time, ignore the changes in termite colony numbers other than toxin induced mortality.(too few can be ignored)
8.Termite division of labor(Khan & Ahmad, 2018):
Young (5-20%)
Nymph (1-10%)
Worker (60-90%)
Soldier (1-10%)
Reproductive (0.1-5%)

Fig 2 Life cycle of the Formosan subterranean termite, Coptotermes formosanus Shiraki (Source: Su NY; University of Florida, Publication No. EENY121)
For the operation of the model, we simplify it to the following scale:Different termites propotion: worker(70%), reproductive(1%), soldier(10%), young(9%), nymph(10%).

Ordinary Differential Equation System

Fig 3 Ordinary Differential Equation System for the Spreading Model

Here is the interface of the spreading model

Fig 4 R shiny application of the Spreading Model

Click here

Model Limitations and Optimization

But is this really the case? During multiple model runs, we identified limitations in the original setup: the original model assumed that worker ants forage outside only once, with the initial infection rate remaining constant; after that, worker ants no longer go out, and infection spreads within the nest solely through This deviates from actual observations: according to [literature/research] xx, termite workers forage outside an average of x times per day; over time, the total number of infected workers continues to increase, leading to a significant acceleration in the transmission rate of the toxin within the nest. Based on this, we optimized the model to improve its fit with real-world conditions. The specific improvements are as follows:

We integrated the movement model of termite foraging and nest-returning, which not only determines the total duration of their complete movement cycle but also estimates the number of infected individuals among workers foraging outside the nest for the first time during the observation period, thereby calculating the infection rates of various castes within the nest at that time.

When workers forage outside for the second time, the number of newly infected workers is calculated as "total number of uninfected workers × a% × proportion of uninfected workers in the worker population"; after workers return to the nest, the toxin spreads through trophallaxis, thus updating the infection numbers of various castes within the nest at the end of the second foraging cycle.

Through such cyclic iteration, the infection rates of various termite castes after a user-specified number of days can ultimately be output.

So the equations of worker termite will be changed to:

Here is the improved spreading model

Fig 5 R shiny application of the improved Spreading Model

Click here

Model Parameters

Parameters	Description	Value	Source
βw	Worker termite transmission rate	0.14/day	Zhang et al.,2003
βn	Nymph termite transmission rate	0.14/day	Zhang et al.,2003
βy	Young termite transmission rate	0.14/day	Zhang et al.,2003
βs	Soldier termite transmission rate	0.2/day	Zhang et al.,2003
βr	Reproductive termite transmission rate	0.5/day	Zhang et al.,2003
γw	Worker termite mortality rate	0.05/day	Estimated from wetlab data
γn	Nymph termite mortality rate	0.07/day	Calculated based on sensitivity coefficient
γy	Young termite mortality rate	0.1/day	Calculated based on sensitivity coefficient
γs	Soldier termite mortality rate	0.03/day	Calculated based on sensitivity coefficient
γr	Reproductive termite mortality rate	0.02/day	Calculated based on sensitivity coefficient
μw	Worker termite conversion rate	0.01/day	Assumption
μn	Nymph termite conversion rate	0.05/day	Assumption
μy	Young termite conversion rate	0.02/day	Assumption
μs	Soldier termite conversion rate	0.04/day	Assumption
μr	Reproductive termite conversion rate	0.02/day	Assumption

Parameters calculation

Termite mortality rate(γ)

Experimental Results

8/30-9/6 Results (27 worker termites + 3 soldier termites)

Substance	Group 1 Deaths and infections	Group 2 Deaths and infections	Group 3 Deaths and infections
Melittin	0/17	1/14	0/15
EcTI	0/21	1/18	0/19
LH	1/20	0/20	1/14

9/6-9/13 Results (27 worker termites + 3 soldier termites)

Substance	Group 1 Deaths and infections	Group 2 Deaths and infections	Group 3 Deaths and infections
Melittin	0/28	4/26	1/25
EcTI	1/27	2/28	1/26
LH	4/25	6/27	2/22

Suppose that within a certain time interval Δt, the initial number of infected worker ants is I₁, and the number of dead worker ants is D₁. After Δt, the number of dead worker ants becomes D₂. According to the differential equation (dD/dt) = γ_w · I (note: original equation may have missing variables, adjusted for readability) in a short period of time, it can be approximated as the difference equation ΔD = γ_w · I₁ · Δt, from which γ_w can be calculated.

Substance (Abbreviation)	Group 1	Group 2	Group 3
Melittin (M)	0	0.0306	0.0095
EcTI (E)	0.0068	0.0079	0.0075
LH (L)	0.0214	0.0429	0.0102

The three toxins do not react with each other and do not affect each other, so γw is equal to the sum of the γ values of each toxin.

γw = γM + γE + γL = (0 + 0.0306 + 0.0095)/3 + (0.0068 + 0.0079 + 0.0075)/3 + (0.0214 + 0.0429 + 0.0102)/3 = 0.0456 ≈ 0.05

Sensitivity coefficient (modified)

Termite Caste	Sensitivity Coefficient	Source
Worker termite	1.0 (Basis)	-
Nymph termite	1.4 - 1.6	Mao et al., 2011
Soldier termite	0.6 - 0.7	Mao et al., 2011
Young termite	2.0 – 2.8	Mao et al., 2011
Reproductive termite	0.2 - 0.7	Mao et al., 2011

γn = 0.0456 × 1.5 ≈ 0.07
γs = 0.0456 × 0.65 ≈ 0.03
γr = 0.0456 × 0.45 ≈ 0.02
γy = 0.0456 × 2.4 ≈ 0.1

Termite transmission rate

According to The Analysis of Coptotermes formosanus shiraki’s Trophallaxis Behavior, the intestinal coloring rate is used as an indirect indicator to reflect the trophallaxis efficiency of worker termites towards other castes. The underlying principle is that after worker termites feed on filter paper impregnated with melanin, they transmit the nutrient substances containing melanin to soldiers, larvae, and nymphs through trophallactic behavior. We indirectly regard the proportion of individuals with colored intestines among these castes as the trophallaxis rate of worker termites towards them. Based on the figures, we observed that on the 5th day, the intestinal coloring rate of soldier termites reached 100%, which confirms that worker termites can "infect" (i.e., transmit the colored nutrients to) all soldier termites within 5 days. Therefore, the transmission rate (β) of worker termites to soldier termites is calculated as 1/5, or 0.2. By analogy, the transmission rates of worker termites to young termites and nymph termites are 1/7 and 1/7, respectively. And easily accessible transmission rates of worker termites to worker termites and transmission rates of worker termites to reproductive termites are 1/7 and 0.5.

Fig 5 R shiny application of the improved Spreading Model

Termite conversion rate

Given that all toxins act on the gastrointestinal tract (digestive system) of termites, there may be cases where termites do not die after being infected but gradually recover. We assume that the recovery rates of different termite castes are as follows: worker termites at 0.01 per day, termite nymphs at 0.05 per day, termite larvae at 0.02 per day, soldier termites at 0.04 per day, and reproductive termites at 0.02 per day.

References

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   https://doi.org/10.1016/j.mbs.2022.108824
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