Applying the Lanchester equations to model conflicts involving tunnels

Abstract

Tunnels have become a significant factor in modern warfare, providing concealment, protection, and strategic advantages to combatants. The use of tunnels complicates warfare, making it challenging for military planners to analyze operations involving extensive tunnel networks. The study introduces a technique that uses a modified form of the Lanchester equations to model scenarios where one army is significantly leveraging tunnels. This technique uses agent-based combat models to estimate the attrition coefficients required for the model. A case study of the Gaza conflict is presented, where Israel confronts Hamas, a group extensively using tunnels. In this scenario, Israeli forces engage Hamas fighters in urban areas, discover a tunnel network, and proceed to clear the tunnel. Despite Israel’s numerical superiority, which ultimately allows them to win the conflict, the use of tunnels enabled Hamas to inflict considerably more damage. This case study underscores the substantial impact of tunnel warfare on conflict dynamics. The technique presented in this study also provides valuable insights into the operational and strategic effects of tunnels in contemporary military engagements.

Keywords

Lanchester equations agent-based modeling tunnel warfare

1. Introduction

The conflict between Israel and Hamas in the Gaza has highlighted the complexities of modern combat, especially when it involves fighting in urban areas. Despite being technologically and numerically superior, Israel has struggled against Hamas in the subterranean environment. Hamas has developed an extensive network of tunnel networks spanning 300 miles, from which they launch their military operations.¹ These military operations include offensive operations, covert battlefield maneuvers, uninterrupted resupply missions, and rocket attacks. While in the tunnels, much of the Israeli technology is neutralized, as they are forced to conduct disjointed, small-unit operations.2

This analysis sets out to analyze combat in scenarios similar to that in the Gaza Strip through the Lanchester equations, a set of differential equations commonly used for modeling conflicts. The Lanchester equations were modified to account for subterranean combat. The modified equations are then combined with agent-based combat simulation to determine the attrition coefficients needed in the Lanchester equations. The ensuing methodology is then applied to the conflict in the Gaza strip to understand the complications associated with conflict both on the surface and underground.

2. Background

2.1. Use of tunnels in armed conflict

Tunnel warfare has a rich history dating back centuries, with various civilizations employing tunnels for military purposes long before the 20th century. In medieval Europe, besieging armies used tunnels, or saps, to undermine castle walls and breach city defenses, leading to the development of specialized combat engineers known as sappers.³ During the Crusades, both Muslim and Christian forces used tunnels extensively, with notable examples including the Siege of Jerusalem and the Siege of Antioch in 1098, where defenders dug countermines to thwart attackers.⁴ In addition, ancient Chinese military strategy often involved tunneling for surprise attacks and infiltration, particularly during the Warring States period (475–221 BC).⁵ These historical examples underscore the enduring significance of tunnel warfare as a strategic tool in military conflicts prior to the 20th century.⁶

The 20th century witnessed significant advancements in tunnel warfare, with notable examples occurring during conflicts, such as the Vietnam War and the War in Afghanistan. In Vietnam, the Viet Cong guerrillas constructed extensive tunnel networks, most famously the Cu Chi tunnels near Saigon. These tunnels served as hidden bases, supply routes, and living quarters for the Viet Cong, allowing them to evade detection by American forces and launch surprise attacks.⁴ The Cu Chi tunnels highlighted the ingenuity and resilience of the Viet Cong, featuring booby traps, ventilation systems, and multiple levels extending over many kilometers. During the Soviet–Afghan War (1979–1989) and subsequent conflicts, tunnels and caves played a crucial role in guerrilla warfare against occupying forces.⁷ Afghan mujahideen fighters used natural cave complexes and dug new tunnels in the mountainous terrain to hide from Soviet troops, store weapons, and launch ambushes. Similarly, during the War in Afghanistan following the September ¹¹ attacks, Al-Qaeda operatives used tunnels and caves in the Tora Bora region to evade capture by coalition forces and continue their insurgency.

While ancient tunnels focused on bypassing siege walls, modern tunnels serve a much broader range of purposes in military operations. Contemporary military tunnels are designed not only to facilitate the covert movement of soldiers across the battlefield but also to provide protection from indirect fires, such as artillery and airstrikes.⁸ These tunnels are often integrated into urban environments, complementing above-ground activities.⁹ For instance, a soldier might use a tunnel to evade enemy forces, only to emerge in a different location to launch a surprise attack. This dual functionality enhances the strategic versatility of military units, allowing them to maneuver undetected and maintain the element of surprise. Modern tunnels also serve logistical purposes, enabling the safe transport of supplies, equipment, and communications to above-ground elements.10

More recently, tunnel warfare has featured heavily in the current war in the Gaza between Israel and Hamas. The complex network of tunnels has created a tactical challenge for the Israeli Defense Force (IDF) since clearing the Gaza of Hamas fighters requires clearing their tunnels. Indeed, as shown in Figure 1, the complexities of the tunnels and their impact on the conflict have become a defining feature of the war.^11–13

Figure 1.

Recent news headlines about the use of tunnels by Hamas in their fight with Israel.^11–13

2.2. Modeling attrition in armed conflicts with the Lanchester equations

The Lanchester equations, developed by Frederick W. Lanchester in the early 20th century, are a set of mathematical models used to analyze and predict the outcomes of combat between military forces. These equations are particularly applicable to scenarios involving attrition warfare, where the goal is to gradually wear down the enemy’s resources over time.14

In essence, the Lanchester equations describe how the strength of opposing forces changes over time as they engage in combat. They account for factors, such as the effectiveness of weaponry, the morale of troops, and the tactics employed by each side. By inputting variables, such as the initial strengths of the opposing forces and the rates at which they inflict casualties on each other, analysts can use the equations to predict the eventual outcome of the conflict.

There are three primary forms of the Lanchester equations: first linear law, second linear law, and square law. They are typically generalized to be put in terms of Blue and Red agents, setting the attrition rate of each force to be a function of the number of Blue (B) and Red (R) agents.

The linear form is seldom used and simply models combat as a series of one-on-one duals, such that the attrition rate is constant. In this case, the equations are simply:

dB / dt = - β_{R}

(1)

dR / dt = - β_{B}

(2)

In these equations, R and B are the number of Red and Blue forces, $t$ is the time, and $β$ is the attrition coefficient. Note that the $β_{R}$ is the rate that the Red force kills Blue forces, and $β_{B}$ is the rate that the Blue forces agent kills Red forces.

The second linear law assumes unaimed fire such that the attrition rate is proportional to both the number of firers and number of targets, such that the equations are:

dB / dt = - β_{R} RB

(3)

dR / dt = - β_{B} RB

(4)

In this case, the attrition rate $β_{R}$ would be the probability of an individual Red agent killing an individual Blue agent.

The square law form is more commonly used and provides a more accurate representation of modern combat where forces are concentrating fires. For combat between Red and Blue forces, the form of the equations for the square law is:

dB / dt = - β_{R} R

(5)

dR / dt = - β_{B} B

(6)

For the square law, the attrition rate, $β_{R}$ , is the rate of Blue agents killed by an individual Red agent.

These equations have been used to analyze a range of historic conflicts. For example, the Lanchester equations were used to analyze the Battle of Kursk in the work by Sahni and Das.¹⁵ Another study by Anderson¹⁶ provided insight into American operations in Desert Shield and Desert Storm. More recently, the Lanchester equations have been applied to understanding the trends of the Russia-Ukraine war.17

3. Methodology

The Lanchester equations as discussed in the previous section are not readily applicable for modeling tunnel warfare. Rather, they are intended to model two forces engaging in open terrain. Furthermore, the results of the analysis are dependent on the attrition coefficients in the equations, which are difficult to calculate for conflicts, especially when they involve the complexity of tunnels.

Figure 2 displays the process used for this analysis, which leverages agent-based combat models with the Lanchester equations. The model uses agent-based combat modeling to determine the attrition coefficients following a technique outlined in the work by Mittal.¹⁸ This approach develops representative scenarios of the engagements between the two forces and determines the casualties inflicted by both sides. The casualties and time frame of the scenario are then used to approximate the attrition coefficients.

Figure 2.

Methodology used in this study to model out the attrition from combat where one side is extensively using tunnels.

These attrition coefficients are then combined with the size of each fighting force and used in a modified Lanchester equations which reflects tunnel warfare. The form of the Lanchester equation is similar to what is outlined in the work by Kress et al.,¹⁹ which looked at three forces fighting each other. In this case, one force is fighting against two forces that are split into above-ground and below-ground components.

This paper will outline both the modified version of the Lanchester equations that account for tunnel warfare and the use of agent-based models to capture the attrition coefficients.

4. Modeling of tunnel operations

4.1. Modified Lanchester equations

Figure 3 displays a schematic for a conventional military fighting against an enemy force that combines above- and below-ground operations. In this scenario, the Blue force consists of a conventional military that must contend with Red forces that are above and below ground. The Red force is split up into two forces: above ground and below ground. For this analysis, the above-ground forces are indicated with an upward arrow and the below-ground forces are annotated with a downward arrow.

Figure 3.

Schematic for modeling the attrition of two forces, where one force has a substantial subterranean component.

The Blue force and Red force engage each other with direct fires, hence the use of Lanchester’s Square Law. The attrition coefficients can be considered to be the number of enemy forces killed by a single soldier in a set time frame. Given these assumptions, the following systems of differential equations can be established based on Lanchester’s Laws:

dB / dt = - β_{R ↓} R_{↓} - β_{R ↑} R_{↑}

(7)

d R_{↓} / dt = - β_{B ↓} B

(8)

d R_{↑} / dt = - β_{B ↑} B

(9)

In these equations, B is the size of the Blue force engaging against an adversarial force that is fighting both above and below ground. $β_{B ↓}$ and $β_{B ↑}$ are the average rates that a single Blue agent is killing Red agents in the tunnel and on the surface, respectively. Meanwhile, $R_{↓}$ is the size of the Red forces in the tunnels, and $R_{↑}$ is the number of Red forces on the surface. $β_{R ↓}$ and $β_{R ↑}$ are the associated rates that each Red agent kills Blue agents.

4.2. Determining attrition coefficients

Solving the modified Lanchester equations given in Equations (7)–(9) requires knowledge as to the attrition coefficients for all the fighting forces. These attrition coefficients are calculated using a process outlined in the work by Mittal¹⁸ through the use of agent-based modeling. This technique specifically uses small-unit combat models to determine the attrition coefficients associated with the Lanchester equations. Indeed, the clearing of tunnels, while often overseen at a high-level echelon, is typically executed through direct conflicts between small units.8

It is important to note that when using the Lanchester Square Law, the attrition coefficients are effectively the rate that a single agent is killing an enemy force. As such, the attrition coefficients can be calculated from running the small-unit agent-based model and determining the respective KIA using the equations:

β_{R ↑} = - \frac{1}{R_{↑}} \frac{d B_{↑}}{dt} = \frac{# of Blue KIA Above Ground}{Initial # of Red Above Ground} \times \frac{1}{time frame}

(10)

β_{R ↓} = - \frac{1}{R_{↓}} \frac{d B_{↓}}{dt} = \frac{# of Blue KIA In Tunnel}{Initial # of Red In Tunnel} \times \frac{1}{time frame}

(11)

β_{B ↑} = - \frac{1}{B} \frac{d R_{↑}}{dt} = \frac{# of Red KIA Above Ground}{Initial # of BLue} \times \frac{1}{time frame}

(12)

β_{B ↓} = - \frac{1}{B} \frac{d R_{↓}}{dt} = \frac{# of Red KIA In Tunnel}{Initial # of Blue} \times \frac{1}{time frame}

(13)

Since agent-based models are often stochastic, the number of Red and Blue KIA is often a distribution of values. While the attrition coefficients could be left as a range of values and then used in a stochastic form of the Lanchester equations, for this analysis, the mean values are used in capturing the Red and Blue casualties in Equations (10)–(13), allowing for a point estimate for each attrition coefficient. While simplifying the analysis, this requires running the combat simulation a large number of times to ensure a small confidence interval for the casualty counts.

5. Case study—Israel versus Hamas

The Gaza Strip has put subterranean warfare to the center stage of international media. Since October 7, 2024, the IDF has fought a well-organized Hamas fighting force that has effectively used tunnels, which effectively negate the IDF’s numerical and technical superiority. The agent-scale simulation for this project was designed to replicate a realistic scenario in the Gaza Strip if the IDF were to assault the tunnels with personnel.

This analysis uses the combat simulation, Infantry Warrior Simulation (IWARS), to model the tactical, small-unit mission of IDF forces clearing out the tunnels. The results are then scaled up using the Lanchester equations to model the attrition of the IDF (I) and Hamas (H) forces with time.

It is important to note that this analysis is not intended to fully replicate the full war between the IDF and Hamas. Indeed, the IDF is trying a number of strategies to clear the tunnels without sending their soldiers into them. This analysis is intended to show what would happen if the IDF attempted to clear the tunnels systematically with their soldiers. The results will provide insight into the combat multiplying effect of tunnels.

5.1. The macro-level conflict

The IDF moved into Gaza to clear it of Hamas fighters with a large force. While different reports vary, several news agencies have cited a number of 300,000 IDF fighters involved in the conflict.²⁰ These fighters include a combination of active duty and reserve soldiers, sailors, and airmen. Only the soldiers would be relevant for the clearing of the tunnels. Table 1 gives the composition of the IDF by component for the IDF.²¹ Assuming that the 300,000 IDF fighters include all 165,900 active duty warfighters and that the balance is composed of Reservist from all three balances based on the size of the Reserve, the IDF ground force consists of approximately 238,300 soldiers. However, the ground forces are engaged in a number of activities, ranging from artillery missions to providing logistics support, so the Israeli forces will only be able to allocate a set percentage of this combat force to clearing the Hamas tunnels.

Table 1.

Composition of the IDFs.²¹

	Total		Conflict
	Active duty	Reserve	Active duty	Reserve
Army	126,000	400,000	126,000	112,300
Air force	34,000	55,000	34,000	15,400
Navy	9500	10,000	9500	2800

Similarly, the exact number of Hamas fighters is not well known. However, the media often cites an Israeli military report that estimates that Hamas forces in the Gaza Strip consist of 30,000 fighters divided up until 5 brigades, subdivided into 24 battalions and 140 companies.²² Hamas tactics tend to use small units consisting of squad-sized elements using the tunnels for the base of operations. They move out of the tunnels to set ambushes and attack the IDF.

5.2. Scenario

Much of the fighting in Gaza, especially those related to clearing of tunnels, is conducted at the small-unit level. As such, this analysis used the IWARS software to create a scenario to capture a common combat situation experienced between IDF and Hamas fighters. IWARS is an agent-based combat simulation that is intended for performing systems analysis on small-unit operations.²³ Each agent, or group of agents, are placed on a 3D map, and given missions to execute. Underlying the scenario is a database that captures soldier metrics, such as shooting accuracy, movement speeds, and detection times.

An image of the scenario is shown in Figure 4. In Box 1 of Figure 4, an IDF infantry squad (blue force) mounts a vehicle, in this instance simulating the bulldozers that are used to clear through the rubble of the city. The vehicle begins pushing through the urban area and reveals an opening to a tunnel network. When the opening is revealed, the squad dismounts the vehicle and is immediately engaged by three above-ground Hamas combatants (red force); this is shown in Box 2 of Figure 4. The IDF squad returns fire and proceeds to make its way toward the tunnel entrance. It establishes security with one team outside the entrance and pushes the other team in to begin clearing the tunnels as shown in Box 3 in Figure 4. Inside, the IDF team encounters three Hamas combatants, two of which are hiding behind hostages. The simulation concludes once the tunnels have been cleared or all parties of either side have been killed.

Figure 4.

Schematic of scenario where a (1) Blue force moves into a town on a vehicle. (2) They uncover a tunnel, dismount the vehicle, and engage Red forces. (3) They then move into the tunnel to clear it off adversaries and rescue the hostages (3).

It is important to note that for this scenario, IWARS does not offer a landscape with tunnel networks. To replicate these tunnels, the study used one of the main structures on the map that had a variety of hallways built into it. The windows and doors were walled off, leaving only one entrance to the “tunnel” in the northwest corner of the building. The field of view (FOV) of the agents in the tunnel was modified within the database of the simulation to be more narrowed and force the subterranean combatants to have “tunnel vision.” As shown in Figure 5, the unmodified FOV is 150° horizontally and 95° vertically; meanwhile, the modified FOV is shrunk down to 30° horizontally and 45° vertically.

Figure 5.

FOV for a soldier that is above ground and one that is in the tunnel.

During the initial attack in October 2023, Hamas took around 200 Israeli citizens hostage. Many of these hostages remain under capture and some are held within the tunnel networks. To account for this in the simulation, civilians (identified by the green dots) are placed in the tunnels in front of a couple of the Hamas combatants. As the IDF enters and clears the tunnel, the presence of hostages heightens the positive identification required to engage combatants. To account for this in the simulation, there is an added “wait” node of one second in the engagement code for the Blue Force to replicate an initial pause that would be taken by the IDF before firing in the presence of civilians.

IWARS is a stochastic simulation, where the random number seeds change each time that the simulation is executed. This captures the uncertainty associated with different events. As such, the simulation must be run a number of times providing a distribution of results. The results captured from this scenario is the number of IDF (blue), Hamas (red), and civilian (green) agents that were killed.

5.3. Simulation results

To better understand the variation of the forces and more appropriately model the scenario, the model was run with varying number of IDF agents. The IDF, given their military intelligence estimates of Hamas warfighters who are using the network of tunnels for their operations, can choose the number of forces that they wish to apply to clearing the tunnels. Given the nature of modern conflict, the offensive maneuver of clearing the tunnel would select a ratio of 3:1. However, for this analysis, ratios between 1:1, 1.5:1, 2:1, 2.5:1, 3:1, 3.5:1, and 4:1 were considered. With a ratio of 4:1, the IDF would be applying approximately half of their ground forces, 120,000 of 238,300 soldiers.

In every case, the IDF soldiers arrive, dismount their vehicle, and engage an above-ground Hamas force consisting of three agents. A team of IDF soldiers secure the entry to the tunnel, and the remaining IDF soldiers proceed into the tunnel to clear it of three additional Hamas fighters. The average of 100 runs for each simulation is given in Table 2.

Table 2.

Results from IWARS model of IDF engaging Hamas forces above ground and then in tunnels given different starting configurations.

Ratio	Initial IDF assault force	Initial Hamas agents	IDF KIA (surface)	IDF KIA (tunnels)	Hamas KIA (surface)	Hamas KIA (tunnels)
4:1	24	6	4.25	7.45	3.00	3.00
3.5:1	21	6	3.85	6.88	3.00	3.00
3:1	18	6	3.75	6.18	3.00	2.90
2.5:1	15	6	4.33	4.99	3.00	2.63
2:1	12	6	4.18	3.85	2.83	2.34
1.5:1	9	6	3.84	2.80	2.68	1.64
1:1	6	6	2.41	2.89	2.23	1.21

As expected, the IWARS simulations indicate that as the size of the IDF assault force increases, they are better able to neutralize all Hamas agents. As the assault force increases to 21, achieving a 3.5:1 ratio, the IDF is able to neutralize all Hamas agents. However, there is a trade-off in that as they increase the numbers, they take more initial casualties. As they dismount their vehicles, the close proximity of IDF soldiers results in them taking increased casualties. Meanwhile, the canalizing of forces in the tunnels results in the IDF taking increased casualties when they enter the tunnels.

Meanwhile, as the size of the IDF assault force decreases, they are less able to take on the Hamas agents. When there is a 1:1 ratio, the IDF sustain heavy losses above ground, with a number of simulation runs having all IDF forces being incapacitated prior to enter the tunnels. For those do enter the tunnels, the Hamas agents have a significantly better kill ratio than the IDF.

5.4. Expanding the simulation results

These data presented in Table 1 were then used to calculate the appropriate attrition coefficients based on the starting size of the IDF force relative to the Hamas force. The values for the attrition coefficients are given in Table 3. These attrition coefficients assume a unitless time frame of 1 in Equations (10)–(13). In this case, the time frame of 1 would be the complete amount of time required for the IDF to conduct the tunnel operations against the entire Hamas network.

Table 3.

Results from IWARS model of IDF engaging Hamas forces above ground and then in tunnels given different starting configurations.

Ratio (IDF: Hamas)	$β_{H ↓}$	$β_{H ↑}$	$β_{I ↓}$	$β_{I ↑}$
4:1	2.48	1.42	0.13	0.13
3.5:1	2.29	1.28	0.14	0.14
3:1	2.06	1.25	0.17	0.16
2.5:1	1.66	1.44	0.20	0.18
2:1	1.28	1.39	0.24	0.20
1.5:1	0.93	1.28	0.30	0.18
1:1	0.96	0.80	0.37	0.20

A numerical model was used to solve Equations (7)–(9), using the attrition coefficients given in Table 3, an initial Hamas fighting force of 30,000, and an initial IDF size as a multiple of that 30,000. The numerical model was developed in Microsoft Excel and used time steps of 0.001. At each time step, the values for $dI / dt$ , $d H_{↓} / dt$ , and $d H_{↑} / dt$ were calculated based on the size of each force. At the subsequent time steps, the number of IDF and Hamas agents was updated to reflect the losses in the previous time step.

The models were set to run for one time period, the length of time allocated for tunnel clearing operations. The model terminates either at the end of the time period or when the IDF eliminates half of the Hamas fighters. At this point, the Hamas forces would not be combat-effective and lack the numbers necessary for holding their tunnel network, and they would need to change their tactics.²⁴ Furthermore, the Lanchester equations do not hold when one of the forces gets very small compared to the other forces since there are a number of external factors that would then come into play.²⁵ In particular, the continued fighting would be unproductive since the associated collateral damage could bolster Hamas’ recruitment efforts, potentially increasing their number of fighters.²⁶

The results of the model for each IDF starting ratio are shown in Figure 6. In every case, the IDF takes heavier losses than Hamas. This would be expected given the nature of tunnel warfare, where Hamas is in a well defended position. When the IDF to Hamas ratio is 1:1, the operation is not sufficient for meeting the IDF objectives since at the end of the operation, only a third of the Hamas forces are eliminated. As the ratio increases to 1.5:1, the IDF is able to achieve their objective, despite taking substantial losses.

Figure 6.

Israeli (solid) and Hamas (dashed) casualties as a function on time based on the ratio of starting forces.

The models were adjusted to run until the IDF task force is eliminated or until the Hamas forces lose half of their fighting force. The results are given in Table 4. When the IDF uses a 1:1 ratio, their entire task force is eliminated while only eliminating 12,122 Hamas warfighters. However, as the ratio increases, the IDF operation becomes successful. While the IDF takes heavy losses with a 1.5:1 ratio, they take fewer losses when the ratio is between 2:1 and 3:1. As the ratio continues to increase, the IDF starts to take increasing losses due to the canalizing of troops in the urban areas and in the tunnels. Furthermore, the results indicate that the time frame for the operation decreases as the ratio increases; however, diminishing returns set in, as increasing the ratio past 3:1 does not significantly reduce the time.

Table 4.

Hamas and IDF losses and overall outcome from the tunnel-clearing operation given a different number of initial IDF soldiers.

Ratio of IDF to Hamas	Hamas losses	IDF losses	Time frame	Outcome
4:1	15,000	24,438	0.56	IDF is successful
3.5:1	15,000	22,552	0.57	IDF is successful
3:1	15,000	21,563	0.58	IDF is successful
2.5:1	15,000	21,842	0.63	IDF is successful
2:1	15,000	21,200	0.72	IDF is successful
1.5:1	15,000	22,896	0.97	IDF is successful
1:1	12,122	30,000	1.55	IDF is unsuccessful

These results indicate that if the IDF had chosen to clear out the tunnels using soldiers, they would have needed to allocate a significant portion of their fighting force. However, they would have sustained heavy casualties. These casualties could have been reduced using a ratio of approximately 2:1 as compared to the more traditional 3:1. Furthermore, their decision to choose alternatives to entering the tunnel likely saved them a considerable number of casualties.

From the Hamas perspective, the tunnels provided a combat multiplier effect, where they inflicted significantly high losses on the IDF and would force them to allocate a significant amount of their forces to attempt to clear them. This effect has been noted qualitatively in a number of different studies on the effectiveness of tunnels.^1–5 Logically, these results indicate that if one force is facing a much larger fighting force, they should use tunnels to increase their combat effectiveness.

6. Conclusion

The strategic implications are clear: tunnels provide a formidable means of asymmetrical warfare, enabling smaller, less-equipped forces to inflict disproportionate damage and sustain their operations against larger military powers.

Tunnels have been used for centuries to provide a means of asymmetrical warfare, enabling smaller, less-equipped forces to inflict disproportionate damage and sustain their operations against larger military powers. As demonstrated by the Gaza conflict case study, the utilization of tunnels by Hamas allowed them to mitigate the numerical superiority of Israeli forces, significantly altering the conflict’s dynamics. The technique introduced in this study, leveraging a modified form of the Lanchester equations alongside agent-based combat models, offers a framework for analyzing and understanding these complex scenarios.

This method not only facilitates more accurate modeling of combat situations involving extensive tunnel networks but also highlights the increased attrition rates faced by forces combating tunnel-using adversaries. In the case of the Gaza conflict, if the Israeli forces chose to send their soldiers into the Hamas tunnels to clear them, they would have needed to have approximately a 2:1–3:1 ratio to clear the tunnels; even with superior numbers, they would have sustained a significant number of casualties.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Both authors are employees of the United States Army. The views presented are those of the authors and do not necessarily represent the views of the United States Army.

ORCID iD

Vikram Mittal

Author biographies

Jacques L Schold is an infantry officer in the US Army. He graduated with a BS in Systems Engineering from the US Military Academy in 2024.

Vikram Mittal is an associate professor in the Department of Systems Engineering at the US Military Academy, West Point, New York, USA.

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