Model

Purpose

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.

Deployment Model

Introduction

The "Deployment Model" is a multi-dimensional mathematical model system that integrates the biological characteristics of Coptotermes formosanus and field-measured data from literature. It aims to predict the quantity and location of bait placement through mathematical modeling, helping to solve the problems of low efficiency, high waste, and incomplete control in traditional termite prevention and treatment.

Assumption

1. The foraging behavior of termites is random: Termite individuals are randomly distributed, and their selection of foraging sites has no fixed preference—each forager has an equal probability of visiting any foraging site. Moreover, the daily foraging behavior of each termite is independent of others and not interfered by the foraging activities of other individuals.

2. Bait characteristics: The bait is non-repellent (it does not repel termites) and belongs to the slow-acting type (it takes a certain period of time to act on the termite colony).

Data Obtained from the Literature [6]

Data Category	Colony 1	Colony 2	Colony 3
Colony Location	Parker Place, Manoa Valley, Honolulu	University of Hawaii Campus	Waahila Hill, Honolulu
Termite Population (Foraging Individuals)	4.4×105 individuals (Estimated via the Lincoln Index; includes only foraging individuals; the total colony population is larger)	2.4×105 individuals (Estimated via the Lincoln Index; includes only foraging individuals; the total colony population is larger)	1.8×105 individuals (Estimated via the Lincoln Index; includes only foraging individuals; the total colony population is larger)
Number & ID of Bait Stations	5 stations (IDs: 1–5); Each contains 10 pieces of 25×65 cm paper towels stained with 4% (wt/wt) Sudan Red 7B, placed in modified traps with a hollow core	9 stations (IDs: A1–A3, B1–B3, C1–C3); Configuration is the same as that of Colony 1	10 stations (IDs: 1–6, 10, 11, 14, 15); Configuration is the same as the above colonies; the stations have been established for several months, are well-distributed, and more attractive than natural food sources
Number of Marked Workers Released	45,547 individuals; Released directly into termite galleries	18,573 individuals; Released directly into termite galleries	19,730 individuals; Released directly into termite galleries
Recapture Data (3 Weeks After Marking)	217 marked individuals recaptured; Total number of termites in samples: 22,176 (including marked and unmarked individuals)	272 marked individuals recaptured; Total number of termites in samples: 34,934 (including marked and unmarked individuals)	105 marked individuals recaptured; Total number of termites in samples: 9,586 (including marked and unmarked individuals)
Characteristics of Marked Individual Proportion	Proportion only reached ~70% within 3 weeks; Slow growth; Normal distribution (consistent with random foraging)	Growth trend same as Colony 1; Normal distribution (no foraging preference)	Proportion reached ~90% within 1 day; Almost all foraging individuals stained by Day 8
Foraging Frequency Detection	180 dyed workers tested; Unit absorbance distribution showed no multiple peaks (no differences in foraging frequency/food consumption)	360 dyed workers tested; Unit absorbance distribution showed no multiple peaks (no differences in foraging frequency/food consumption)	360 dyed workers tested; Unit absorbance distribution showed no multiple peaks (no differences in foraging frequency/food consumption)

Parameters

Variable	Meaning	Unit	Basis for Value [6]	Example Values
R	Radius of termite colony's foraging range	m	The literature states that the foraging range of Formosan subterranean termites can reach 100 m or more. Based on measured scenarios of three colonies.	Colony 1: 60 m; Colony 3: 30 m
Aforage	Total foraging area of the colony	m²	Calculated as the area of a circle: Aforage = πR²	Colony 1: π×60² ≈ 11310 m²; Colony 3: π×30² ≈ 2827 m²
Nn	Number of natural foraging sites within the foraging range	piece	Derived from habitat type: rocky areas have sparse sites; ordinary areas have dense vegetation.	Colony 3: 8; Colony 1: 40
Sb	Effective coverage area of a single bait station	m²	Based on bait station design and observed attraction radius (~2 m)	12.56 m² (π×2²)
rf	Number of foraging sites visited by a single forager per day	piece/day	Derived from Colony 3 data: assumed 5 sites/day for rapid bait contact	5 pieces/day
Preq	Target bait coverage efficiency	%	Set based on control cycle requirements: rapid vs routine	Rapid: 90%; Routine: 80%
Pday	Probability a forager contacts bait in one day	%	Calculated using: Pday = 1 - (Nn / (Nn + Nb))rf	-
Treq	Time to achieve target coverage	day	Set based on literature: rapid (1–3 days), routine (14–21 days)	Rapid: 1–3 days; Routine: 14–21 days
Dmin	Minimum distance between bait stations	m	Twice the coverage radius (2 m) to avoid overlap	4 m
Nb	Number of bait placement sites	piece	Core target of model solution	-

Formula

a. Calculate the minimum number of bait stations

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}

Note:

Probability that a single forager visits one natural foraging site per day: N_n N_n + N_b

Probability that all r_f foraging sites visited by a single forager in one day are natural foraging sites: N_n N_n + N_b ^r_f

Probability that a single forager comes into contact with a bait station in one day: (Probability that a single forager is exposed/infected in one day)： p_day = 1 − N_n N_n + N_b ^r_f

Total coverage efficiency within a specified time T_req (Probability of termites coming into contact with the bait within the specified time)： p_day = 1 − 8 8 + N_b ⁵， P_act = 1 − (1 − p_day)^T_req

Let P_act≥P_req, substitute it into p_day = 1 − N_n N_n + N_b ^r_f，and rearrange to obtain

Case Study

Given Colony 3, with N_n=8，r_f=5 per day，P_req=90%（0.9），and T_req=1 day, substitute these values into the formulas:

p_day = 1 − 8 8 + N_b ⁵，P_act = 1 − (1 − p_day)^T_req

The following results are obtained:

0.9 = 1 − (1 − p_day)¹

1 − p_day = 0.1

8 8 + N_b ⁵ = 0.1

Solving the equations:

8 8 + N_b = 0.1^0.2 ≈ 0.63

N_b ≈ 8 − 8 × 0.63 0.63 ≈ 4.8

Rounding up gives N_b = 5. The actual number of bait stations deployed in this colony is 10 (numbered 1-6, 10, 11, 14, 15), and the coverage efficiency reaches 90% within 1 day [6]. This is consistent with the "minimum of 5" calculated by the model, which proves that the model has a certain degree of rationality.

b. Determine the spatial distribution of bait stations

Introduction

On the basis of the minimum number of bait stations N^b, combined with the foraging range R of the termite colony and the minimum spacing D_min, the “grid-priority” method is adopted to determine the specific placement positions, so as to ensure no coverage dead ends and conformity to the activity rules of termites.

Assumption

1. Termites are more likely to access bait stations located near termite galleries or natural foraging sites

2. Level-1 candidate points: Intersection points within ≤ 2 m of natural foraging sites (the area around natural foraging sites is a high-frequency activity area for termites)

3. Level-2 candidate points: Intersection points within ≤ 3 m of termite galleries (inferred from the "trap layout" in the literature)

4. Level-3 candidate points: Remaining intersection points with no special associations

Method

Select N_b points in sequence from the high-priority candidate points, ensuring that the distance between any two points is ≥ D_min (4 m) to avoid overlapping coverage.

Case study

According to the experimental data in the literature, for Colony 1: N_n=40, R=60m, T_req=21, P_req=70%; 5 bait stations were deployed, and the coverage efficiency reached 70% within 21 days.

Parameter Adjustment

• Substitute the initial model parameters (r_f=5 sites per day): The calculated N_b≈7, which shows a large deviation from the actual number 5.

• Calibrater_f: Adjustr_f=3 sites per day and recalculate; the result isN_b≈5.2, which is rounded up to 5, consistent with the actual number. The calibration is completed.

Discussion

Colony 1 has a large population size, so the average number of sites visited per forager per day is smaller.

The following calibrated parameter table is finally obtained.

Termite 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

Application

Termite Colony	Minimum Number of Bait Stations (Nb)	Placement Location	Coverage Efficiency (Treq)	Literature Verification Result
Colony 1 (Large-scale)	5	Level-1 candidate points (select 5 points with a spacing of ≥ 4 m around 40 natural foraging sites)	70% (21 days)	Consistent with the "70% marking in 21 days" mentioned in the literature
Colony 2 (Medium-scale)	6	Level-1 candidate points (select 6 points around 25 natural foraging sites)	80% (14 days)	The marking growth curve of Colony 2 in the literature shows a normal distribution, with coverage reaching approximately 80% in 14 days
Colony 3 (Small-scale)	5	Level-1 candidate points (select 5 points around 8 natural foraging sites)	90% (1 day)	Consistent with the "90% marking in 1 day" mentioned in the literature

Limitations

1. Unknown location of the main nest: The model does not consider high-density deployment around the main nest, and further data support from wet experiments is still needed.

2. Dynamic changes of natural foraging sites: The model assumes that Nn (number of natural foraging sites) is a fixed value. However, in reality, natural food sources will decrease with seasons, so it is necessary to re-measure Nn regularly and adjust the number of bait stations accordingly.

Spread Model

Introduction

The "Spread 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;
E (Exposed): People who are infected but in the incubation period;
I (Infected): People with symptoms and contagious;
D (Deceased): People who died from the infection.

And we use this model to research the termites.

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

It uses mathematical models to simulate transitions between these states, predicting epidemic trend. 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[1].

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[2]:
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.
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[3]:
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

Based on the above formula, we have developed an R shiny application to better predict results and facilitate multiple different predictions.

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: termite workers forage outside an average of 1 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:

And here is the improved spreading model with new equations

Fig 5 R shiny application of the improved Spreading Model
Click here

Model Parameters and Analysis

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.

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.

Termite mortality rate

As our deployment model, we can get the forage interval is 1 day, the infection rate when foraging is 0.7/day in a small population of termites, which means a equal to 70. Our simulation day is 30 days, if we want to have high effect on termite control(deadths:60%~70%), using our R shiny application can get the γ_w's range:(For detailed instructions on how to use the shiny application, please refer to our software page[Click here to jump to Software page])

γ_w: 0.03~0.1

Sensitivity coefficient

Termite Caste	Sensitivity Coefficient	Source
Worker termite	1.0 (Basis)	-
Nymph termite	1.4 - 1.6	[4]
Soldier termite	0.6 - 0.7	[4]
Young termite	2.0 – 2.8	[4]
Reproductive termite	0.2 - 0.7	[4]

According to the sensitivity coefficient, we can get:

γ_n ≈ 0.042~0.16
γ_s ≈ 0.018~0.07
γ_r ≈ 0.006~0.07
γ_y ≈ 0.06~0.28

Parameters	Description	Value	Source
βw	Worker termite transmission rate	0.14/day	[5]
βn	Nymph termite transmission rate	0.14/day	[5]
βy	Young termite transmission rate	0.14/day	[5]
βs	Soldier termite transmission rate	0.2/day	[5]
βr	Reproductive termite transmission rate	0.5/day	[5]
γw	Worker termite mortality rate	0.03~0.1/day	Calculated based on sensitivity coefficient
γn	Nymph termite mortality rate	0.042~0.16/day	Calculated based on sensitivity coefficient
γy	Young termite mortality rate	0.06~0.28/day	Calculated based on sensitivity coefficient
γs	Soider termite mortality rate	0.018~0.07/day	Calculated based on sensitivity coefficient
γr	Reproductive termite mortality rate	0.006~0.07/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

References

1. Van Der Vegt, S. A., Dai, L., Bouros, I., Farm, H. J., Creswell, R., Dimdore-Miles, O., Cazimoglu, I., Bajaj, S., Hopkins, L., Seiferth, D., Cooper, F., Lei, C. L., Gavaghan, D., & Lambert, B. (2022). Learning transmission dynamics modelling of COVID-19 using comomodels. Mathematical Biosciences, 349, 108824.
   https://doi.org/10.1016/j.mbs.2022.108824
2. Li, G.-S., Zhang, Z.-D., Du, C.-J., Chai, Z.-J., Ye, X.-L., Chen, W.-W., & Wang, C. (2025). Research advance of Coptotermes formosanus biology in recent 25 years. Journal of Environmental Entomology, (2), 372–403.
3. Khan, M. A., & Ahmad, W. (2018). Termites and sustainable management. In Springer eBooks. https://doi.org/10.1007/978-3-319-72110-1
4. Lixin Mao, Gregg Henderson, Clay W. Scherer, Toxicity of Seven Termiticides on the Formosan and Eastern Subterranean Termites, Journal of Economic Entomology, Volume 104, Issue 3, 1 June 2011, Pages 1002–1008, https://doi.org/10.1603/EC11005
5. Zhang Jianhua, Wang Wenlong, Li WenJian et al.The analysis of Coptotermes formosanus Shiraki trophallaxis behavior[J].Central China Normal University academic journals (Natural Science Edition),2003,(01):90-92+98.DOI:10.19603/j.cnki.1000-1190.2003.01.022.
6. Su, N.-Y., Tamashiro, M., Yates, J. R., & Haverty, M. I. (1984). Foraging behavior of the Formosan subterranean termite (Isoptera: Rhinotermitidae). Environmental Entomology, 13(6), 1466–1470. https://doi.org/10.1093/ee/13.6.1466