# Exploring the Potential of Agent-Based Models for the Problem of Transhumance Path Exits in Sub-Saharan Africa: Chad's Routes as a Case Study

Mahamat Abdouna, Daouda Ahmat, Bertrand Cloez, Adrien Jean Cotil, Hazaël Jones

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Mahamat Abdouna, Daouda Ahmat, Bertrand Cloez, Adrien Jean Cotil, Hazaël Jones. Exploring the Potential of Agent-Based Models for the Problem of Transhumance Path Exits in Sub-Saharan Africa: Chad's Routes as a Case Study. AgriEngineering, 2025, 7 (3), pp.60. 10.3390/agriengineering7030060. [https://doi.org/10.3390/agriengineering7030060](https://doi.org/10.3390/agriengineering7030060)

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Article

# Exploring the Potential of Agent-Based Models for the Problem of Transhumance Path Exits in Sub-Saharan Africa: Chad's Routes as a Case Study

Mahamat Abdouna<fnref n="1" />, Daouda Ahmat<fnref n="1" />, Bertrand Cloez<fnref n="2" />,*, Adrien Cotil<fnref n="2" /> and Hazaël Jones<fnref n="2" />,<fnref n="3" /> ID

**Abstract:** Path exits for transhumant livestock are a major problem in many Sub-Saharan African countries. These problems contribute to many community conflicts. Several solutions are currently being studied, including dialogues between stakeholders. In this paper, we propose a numerical approach to address the problem. Based on a commonly accepted model of agent movement, we propose a path simulator to estimate and quantify the risk of exiting the path. This enables quantitative estimation of the exit rates of transhumant animals as a function of the geometric properties of the routes. This model is tested on real transhumance routes in Chad to evaluate the risks of exits along these routes. These new data allow us to better understand the geometric properties on real routes and to evaluate them in terms of exit risk, giving new information to this complex problem. Although our approach does not deal with the whole complexity of this problem, it opens the door to field experimentation with geolocation sensors.

**Keywords:** transhumance; community conflicts; agent-based model; numerical simulation; spatial path

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check for
updates

Academic Editor: Lilong Chai

Received: 6 December 2024

Revised: 28 January 2025

Accepted: 3 February 2025

Published: 27 February 2025

**Citation:** Abdouna, M.; Ahmat, D.; Cloez, B.; Cotil, A.; Jones, H. Exploring the Potential of Agent-Based Models for the Problem of Transhumance Path Exits in Sub-Saharan Africa: Chad's Routes as a Case Study. *AgriEngineering* 2025, 7, 60. https://doi.org/10.3390/agriengineering7030060

**Copyright:** © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
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Attribution (CC BY) license
(https://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

In Sub-Saharan Africa, conflicts between farmers and herders are growing and are frequently violent [1–5]. The number of victims of these conflicts could even exceed those of Boko Haram in some countries [6]. In fact, in Nigeria, 1300 people died for this reason from January to June 2018, roughly six times the number of civilians killed by Boko Haram during the same period. To give another concrete example, nine people died after a herd of cattle wandered into a private field during the week of 18 August 2023 in the Guéra province of Chad. More generally, in Chad in 2023, the Office for the Coordination of Humanitarian Affairs recorded 51 community conflicts (including not only livestock-related conflicts), resulting in 228 deaths and 452 injuries; see [7] for more details.

This increase in conflicts is due to the growth in agricultural land and population size [8], global warming [9,10], degradation of natural resources [10–12], and new mobilities [4,13], among many other causes. In particular, agricultural expansion on historic transhumance routes has become a major source of conflict [14]. In addition to herders who may leave their animals to feed in the fields, some farmers intentionally place their fields in transhumance corridors [15–17]. Various solutions are sought to reduce these conflicts. Some are political, such as improving the cohabitation between herders and farmers [18].

<footnote n="1">¹ Digital Technology Unit, University of N’Djamena, N’Djamena BP 5711, Chad; mahamat.abdouna@uvt.td (M.A.); daouda.ahmat@uvt.td (D.A.)</footnote>

<footnote n="2">² MISTEA, Université de Montpellier, INRAE, Institut Agro, 34000 Montpellier, France; hazael.jones@supagro.fr (H.J.)</footnote>

<footnote n="3">³ ITAP, Université de Montpellier, INRAE, Institut Agro, 34000 Montpellier, France</footnote>

<footnote>* Correspondence: bertrand.cloez@inrae.fr</footnote>
However, these solutions often suffer from conflicts of interest, as illustrated in [11,19,20], sometimes generating deeper crises [21,22].

In the present work, our objective is to propose a new numerical solution. Our aim is to (1) numerically test the feasibility of estimating trajectory exit risks using a multi-agent model and (2) to be able to derive from these simulations decision-making tools for the choice of transhumance routes. This approach complements existing approaches and adds to the range of solutions to be deployed. Based on the modelling of interactions between animals and with the herder, we provide a quantitative evaluation of the risk of a path by looking at its geometric characteristics (width, length of sections, angle between sections, etc.). We make the assumption here that the herder does not voluntarily let his animals enter the fields, and propose solutions to quantify the risk of this happening, based on various assumptions and/or using additional sensors. Our simulator is based on classical mathematical models of collective movement based on the first principles of swarming [23–26]. These agent-based models take into account forces of alignment, attraction, and repulsion to describe the speeds and positions of individuals/agents. In the context of breeding, they have been empirically validated in [27] on small samples of sheep and have proven to be particularly effective. From a more practical point of view, they have also been used in [28] to model the dynamics of a cattle herd in Sub-Saharan Africa using geolocation data from a sub-sample of animals to assess the risk of contact with wild buffalo. Our approach is therefore quite similar to that of the latter article, even if our goals clearly differ. In particular, we begin by using this path simulator to quantify the risk on theoretical paths and then on real transhumance paths. This evaluation is simplistic, as it overlooks the characteristics of the land adjacent to the paths and their appeal in terms of water and food. However, it represents an initial approach that warrants further development.

Our approach is not unrelated to [29], which evaluates how robots could replace sheepdogs for flock movements. More generally, these proposals for the use of sensors and algorithms are part of digital tools for monitoring and managing livestock in pastures. On this subject, one can read the survey [30] on geolocation technologies, virtual fences, etc., and their uses for animal welfare and herd management.

Finally, our study focuses on Chad, a predominantly rural and agro-pastoral country. During the past 15 years, its population has increased by 1.5 times, exacerbating issues of territorial management and conflicts between stakeholders. The Office for the Coordination of Humanitarian Affairs report [7] shows that the south of the country is particularly affected by these conflicts. Consequently, we focus our study on this southern area. For this case study, we give an initial assessment of how dangerous 12 real transhumance routes are.

## 2. Materials and Methods

### 2.1. *Study Area Description*

Of Chad's 23 regions, our study is focused on the main regions of southern Chad. These are the regions of Batha, Chari Baguirmi, Guéra, Logone-Occidental, Mandoul, and Salamat. We obtained the coordinates of 12 different transhumance routes from the Chad Ministry of Livestock. These tracks are official transhumance routes marked and managed by the government and are available in our Supplementary Materials. They are physically marked by pairs of concrete blocks to the left and right of the route, spaced about 1 km apart (see Figure 1). See also Figure 2, which illustrates the pre-marking of the official route before the installation of the perennial concrete blocks. Hereafter, we will denominate by "corridor", each part of route delimited by 4 blocks. Transhumance routes are located in the 6 regions mentioned above, two of which (Chari Baguirmi and Salamat) are illustrated in Figure 3. A summary table of these routes is given in Table 1.
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Photo of a concrete block delimiting a transhumance corridor. As shown in the bottom right-hand corner, sometimes these concrete blocks are less visible in rainy seasons due to vegetation.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Photo showing the preliminary marking. Before marking out the route, it is necessary to carry out preliminary marking to indicate the locations where the markings will be put in place.</figcaption></figure>

<table><caption>**Table 1.** Location of the 12 transhumance routes in our dataset.</caption><thead><tr><td>Province</td><td>Department</td><td>Sub-Prefecture</td><td>Canton</td><td>Corridor Number</td></tr></thead><tbody><tr><td>Batah</td><td>Batha ouest</td><td>Koudjourou</td><td>Kouka</td><td>2</td></tr><tr><td>Chari-Baguirmi</td><td>Baguirmi</td><td>Massenya</td><td>Massenya</td><td>1</td></tr><tr><td>Chari-Baguirmi</td><td>Baguirmi</td><td>Billi</td><td>Bulli</td><td>1</td></tr><tr><td>Guerra</td><td>Guera</td><td>Baro</td><td>Mégami</td><td>3</td></tr><tr><td>Logone-Occidental</td><td>Nya Pendé</td><td>Doba rural</td><td>Nancian</td><td>1</td></tr><tr><td>Salamat</td><td>Aboudeïa</td><td>Aboudeïa</td><td>Aboudeïa</td><td>3</td></tr><tr><td>Mandoul</td><td>Haraze-Mangueigne</td><td>Haraze</td><td>Haraze</td><td>1</td></tr></tbody></table>

The locations of these transhumance corridors are generally defined according to water resources, grazing areas, pre-existing infrastructures (communication routes, protected areas, etc.) and areas at high risk of conflict. The dimensions of these transhumance corridors differ from region to region. Their geometric properties (widths, angles, etc.) should be of crucial importance in facilitating herd mobility and ensuring proper resource management. In particular, a sufficiently wide corridor should guarantee safe herd movements and prevent potential conflicts. For this reason, a detailed description and analysis of our transhumance route dataset is given in Section 3.1, and the impact of geometries on outputs is evaluated in Section 3.2.
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Regions are shown with their location in Chad. Corridors are represented by points on the study area, representing transhumance routes. This graphic is available in the Supplementary Materials in a larger size.</figcaption></figure>
## 2.2. Mathematical Modelling

Among the mathematical models of animal movement, several models describe individual behaviour, such as [31,32]. Nevertheless, the aim here is to describe animal movements based on their interactions and the movement of the herder. To this end, we use classical models based on the first principles of swarming, including the Vicsek or Cucker-Smale models [25,26]. In particular, the Vicsek model was used in [28] to study bovine movements in Sub-Saharan Africa. Coupled with telemetry data, they aimed to estimate interactions with wildlife. Their approach relies heavily on mechanistic movement modelling to provide decision-support tools. As our approach is similar, we use certain features of their model and make similar parameter choices. However, Vicsek’s model assumes that animals have a constant (scalar) velocity; only the choice of direction of animal movement depends on the position of other animals. With this assumption in place, we have found through numerical simulations that Vicsek’s model is not suitable for simulating a herd of cattle moving in the direction of a herder. The latter must adapt their speed to that of the herder. For this reason, we decided to use the Cucker-Smale model [26], which is very close to the Vicsek model. The latter model operates in continuous time, takes variable velocity into account, and is deterministic but remains very close to Vicsek’s model. Inspired by [33], we added a random part to this model to account for slight fluctuations in displacement. In addition, we have also added interaction with the herder.

The equations we have used to simulate the herd are therefore represented by the following system of (Euler-type discretization of) stochastic differential equations: Let $N$ be the number of animals, $x_i(t)$ be the position of animal $i \in \{1, \dots, N\}$ at time $t$, and $v_i(t)$ be its velocity vector. We also define $x_0(t)$ and $v_0(t)$, the herder’s position and velocity vector, respectively. We assume that the herder moves in a straight line at a constant speed $v_0^*$ along the middle of the transhumance corridor. Defining $h$ as the time step of the dynamics, chosen as small as possible within a reasonable simulation time (in the following $h = 0.1$ s), this is written mathematically as

$$x_0(t+h) = x_0(t) + h v_0(t), \quad \|v_0(t)\| = v_0^*$$

where $\|v_0(t)\|$ corresponds to the norm of $v_0(t)$. In the present work, the herder’s position is simulated. However, the idea remains to apply our model with different sensors. The data $(x_0(t), v_0(t))$ could be updated automatically using a smartphone carried by the herder.

For animals, we assume there are four different types of force: attraction/repulsion between animals, alignment between animals, attraction/repulsion with the herder, and alignment with the herder. Mathematically, the position is actualized through

$$x_i(t+h) = x_i(t) + h v_i(t),$$

and the velocity is actualized through

$$v_i(t+h) = v_i(t) + h \times \frac{1}{N} \left( a^* \sum_{j \neq i} \phi(\|x_i(t) - x_j(t)\|) (x_j(t) - x_i(t)) \right) \quad (1)$$

$$+ b^* \sum_{j \neq i} \psi(\|x_i(t) - x_j(t)\|) (v_j(t) - v_i(t)) \quad (2)$$

$$+ a_0^* \phi(\|x_i(t) - x_0(t)\|) (x_0(t) - x_i(t)) \quad (3)$$

$$+ b_0^* \psi(\|x_i(t) - x_0(t)\|) (v_0(t) - v_i(t)) \quad (4)$$

$$+ \sqrt{h} \times \sigma^* B_i(t). \quad (5)$$
Line (1) corresponds to the sum over all other individuals j of the attraction/repulsion forces received by the individual i under consideration. Line (3) represents the same force but exerted by the herder. In both cases, the term $\phi(\|x_i - x_j\|)$ accounts for the influence of distance on the force. We assume that it is the same function $\phi$ for animals and herder but with different multiplicative coefficients $a^*$ and $a_0^*$, respectively. In particular, the addition of Lines (3) and (4) in the Cucker–Smale equations to model interaction with the herder is a novel feature of our model. Indeed, to our knowledge, this has not yet been introduced in the literature. Following [34], we choose

$$ \phi(\|x_i(t) - x_j(t)\|) = 1 - \left( \frac{r^*}{\|x_i - x_j\|} \right)^2. $$

Note that when the distance $\|x_i - x_j\|$ between individuals i and j is lower than $r^*$, $\phi(\|x_i(t) - x_j(t)\|)$ is negative, and thus, the force is repulsive, while, when $\|x_i - x_j\|$ is greater than $r^*$, $\phi(\|x_i(t) - x_j(t)\|)$ is positive, and thus, the force is attractive. The parameter $r^*$ thus corresponds to the "comfort" radius of animals. For the alignment forces (Lines (2) and (4)), we also assume that the functions that take into account the distance between individuals are identical. Following [26], we have chosen

$$ \psi(\|x_i(t) - x_j(t)\|) = \frac{1}{(1 + \|x_i - x_j\|^2)^{\beta^*/2}}. $$

Note that this $\psi$ function is always positive, which implies that individuals always tend to align with each other. The parameter $\beta^*$ models the individual range of interaction of animals. Finally, Line (5) corresponds to the stochastic part of the model. Variables $\sigma^*B_i(t)$ are Gaussian random variables of mean 0 and of variance $(\sigma^*)^2$. Parameter $\sigma^*$ represents the natural fluctuation in animal movements. It is a biologically fixed parameter that can also vary depending on the situation, such as the stress levels of the animals.

Once again, this article is preliminary work on the potential of GPS data to address this problem. In our work, we describe the potential of this model for evaluating trajectory outputs on simulated and real data. This model could be used to predict the real-time movement of non-GPS-equipped animals by updating GPS-equipped animals with their true positions and speeds rather than simulated data. With no data to calibrate the model and no known parameters in the literature (except [28], which provided the ranges of values tested), we simulated a number of trajectories in order to find the parameters that best reflected coherent trajectories including no large fluctuations, a herd effect, and an alignment behind the herder.

In line with these priorities, we have chosen to use a discretization step $h = 0.1$, cow interaction force $a^* = 1$, alignment cow interaction force $b^* = 1$, herder interaction force $a_0^* = 0.5$, alignment herder interaction force $b_0^* = 4.5$, interaction radius $r^* = 10$, and alignment dispersion force $\beta^* = 0.5$. In particular, cows are similarly attracted to being close to the herder or to their other congeners, but give much more importance to following the same direction as the herder when the latter is within visual range. The parameters $\sigma^*$ and $v_0^*$ are not fixed and can take values of 0.5, 1, or 2 and 1, 2, and 6, respectively. For the simulations, we chose herds of $N = 50$ cows, which represents a good compromise between computation time and realism. To investigate the parameter-dependent behaviour of this model over certain ranges of other test parameter values, we simulate the model several times with certain parameters (chosen with the Latin hypercube sampling method) and interpolate these estimates classically using kernel methods. Finally, all computations were conducted in Python 3.12 version, and codes are available in the Supplementary Materials.
# 3. Results

## 3.1. Elementary Description of Transhumance Routes

Our dataset includes 12 different routes separated into 456 corridors for a total of around 327 km; see Table 2. The corridors average 86 m in width with a standard deviation of 53. This illustrates a large variation between corridors. Table 2 even illustrates the variability between routes. They are around 716 m long with a standard deviation of 430. Histograms are provided in Figure 4 to give a better overview of the distribution of angles and width. These results are based on relatively different route types. Summary statistics per route are given in Table 2.

To illustrate one route in particular, let us take a look at the characteristics of a transhumance route in the Chari-Baguirmi region which can be seen in Figure 3. The corridors of this route are on average 115 m wide, with a standard deviation of 34.5, and 1097.5 m long, with a standard deviation of 573.5. The angles between two corridors average 41.4 degrees with a standard deviation of 28.9. Histograms are given in Figure 4.

<table><caption>**Table 2.** Table summarizing the main characteristics of our 12 transhumance routes. Rtes designates the transhumance route, the column width ± Std shows the mean width ± the standard deviation (in meters) of the corridors of the routes. The column angles ± Std designates the mean of the angles ± the standard deviation (in degree) between corridors. Similarly, length ± Std designs the mean length ± the standard deviation (in meters) of corridors. The last two columns represent the total size (in km) and the number of road corridors.</caption><thead><tr><th>Rtes</th><th>Width ± Std</th><th>Angles ± Std</th><th>Length ± Std</th><th>Total Length</th><th>Corridor Nb</th></tr></thead><tbody><tr><td>1</td><td>115.01 ± 34.51</td><td>138.63 ± 28.86</td><td>1097.45 ± 573.47</td><td>65.85</td><td>60</td></tr><tr><td>2</td><td>77.89 ± 23.05</td><td>157.44 ± 19.79</td><td>932.69 ± 67.17</td><td>54.10</td><td>58</td></tr><tr><td>3</td><td>154.25 ± 174.77</td><td>170.5 ± 7.08</td><td>993.90 ± 136.33</td><td>21.87</td><td>22</td></tr><tr><td>4</td><td>72.64 ± 11.45</td><td>170.42 ± 9.92</td><td>978.82 ± 20.00</td><td>25.45</td><td>26</td></tr><tr><td>5</td><td>66.96 ± 16.71</td><td>173.79 ± 5.32</td><td>346.30 ± 103.35</td><td>24.59</td><td>71</td></tr><tr><td>6</td><td>68.45 ± 47.22</td><td>170.97 ± 7.04</td><td>263.54 ± 152.71</td><td>12.39</td><td>47</td></tr><tr><td>7</td><td>56.70 ± 9.72</td><td>171.15 ± 7.43</td><td>221.78 ± 37.77</td><td>3.55</td><td>16</td></tr><tr><td>8</td><td>61.84 ± 8.26</td><td>170.05 ± 11.53</td><td>213.57 ± 52.37</td><td>5.34</td><td>25</td></tr><tr><td>9</td><td>106.76 ± 20.39</td><td>176.61 ± 2.74</td><td>306.94 ± 54.81</td><td>8.59</td><td>28</td></tr><tr><td>10</td><td>68.36 ± 35.10</td><td>158.57 ± 18.13</td><td>992.69 ± 59.25</td><td>51.62</td><td>52</td></tr><tr><td>11</td><td>108.22 ± 32.53</td><td>161.08 ± 18.84</td><td>1038.21 ± 189.68</td><td>48.80</td><td>47</td></tr><tr><td>12</td><td>108.21 ± 8.63</td><td>145.59 ± 10.96</td><td>1093.82 ± 596.07</td><td>4.38</td><td>4</td></tr><tr><td>Total</td><td>86.01 ± 53.25</td><td>163.01 ± 19.97</td><td>716.02 ± 430.56</td><td>326.50</td><td>456</td></tr></tbody></table>

## 3.2. Path Exits on Synthetic Routes

Section 3.1 illustrates the various geometric properties of corridors in our dataset. Before describing the risks of path exits on real data in Section 3.3, by assessing the black spots on the path (from a purely mathematical point of view, under our clearly reductive assumptions), let us look here at the impact of corridor geometric properties on exit rates on synthetic paths. In all this section, we fix two corridors and look at the path exit of the herd according to some fixed parameters which depend on the numerical experiment. We nevertheless always consider routes with two corridors 100 m long.

Using the model developed in Section 2.2, we can simulate the displacement of the herd. See Figure 5 for a first hint in this way. Specifically, if the herder moves too quickly or the cows are too disorganized, then there are significant and frequent deviations in their trajectories (Figure 6). To make better use of these simulations, we repeat the model 50 times and evaluate the herd's behaviour on the basis of two different functionals. In Section 3.2.1, we study the mean distance to the herder, and in Section 3.2.2, we study more precisely the exit rates. As detailed below, our objective is to analyse the impact of the geometric properties of the corridors on path exits.
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Two summary statistics of all the transhumance routes of our dataset. (a) Histogram of the angles (in degrees) of path along all the transhumance routes of our dataset. (b) Histogram of the width (in meters) of corridors along all the transhumance routes of our dataset. In both cases (a,b), each histogram bar represents the exact number of corridors in the corresponding classes.</figcaption></figure>

All our computations are available in the Supplementary Materials. In what follows, we present only our main representative examples and highlight our main conclusions. These are supported by additional figures, which are available in the Supplementary Materials.

### 3.2.1. Evolution of the Distance to Herder

Fixing the width of the corridors and the angle between them, we calculate the mean distance (over the cows) between the cows and the herder as a function of the position of the herder in the route. In particular, the middle of the x-axis of Figures 7–9 presented in this section corresponds to the moment when the herder stakes out the corner between two corridors.
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Time evolution of the herd over two corridors. The bold line in the centre of the route represents the herder path. Blue lines correspond to the paths of the cows. We highlight the exit point of cows with a red point. We fix $v_0^* = 1.0$, $\sigma^* = 0.5$, and the width is 10 m. In the sub-figures, if $\theta$ is the angle (in degrees) between corridors, then other parameters are (a) $\theta = 10.0$, (b) $\theta = 45.0$, (c) $\theta = 135.0$, (d) $\theta = 180.0$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Time evolution of the herd over two corridors, where we can observe a numerous number of large exits (with red points). As in Figure 5 The bold line in centre of the route represents the herder path. Blue lines correspond to the paths of the cows. This arises when the herder goes too fast or when cows are disorganized (high $\sigma^*$). If $\theta$ is the angle (in degrees) between corridors, then the parameters of sub-figures are (a) $\sigma^* = 0.5$, $v_0^* = 2$, $\theta = 45$ (b) $\sigma^* = 0.5$, $v_0^* = 2.0$, $\theta = 90$ (c) $\sigma^* = 1.5$, $v_0^* = 1.0$, $\theta = 90$, (d) $\sigma^* = 1.5$, $v_0^* = 1.0$, $\theta = 15$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Evolution of the mean distance to the herder over time when $v_0^* = 1.0$, $\sigma^* = 0.5$, and the width is 10 m. The angle between corridors is (in degrees) (a) 10 (b) 45 (c) 135 (d) 180.</figcaption></figure>

Looking at Figure 7, we can see the impact of the geometry on the distance to the herder. Indeed, as expected, the “simpler” the path is, i.e., with a flat or wide angle rather than an acute angle or a hairpin bend, the easier it is for the animals to follow the herder. At this stage, our assessment is qualitative and not very informative, but here, we illustrate that with some simple data, this model would enable us to refine this quantity and thus give a quantitative estimate of these risks.
<figure><img src="image_11.png" /><figcaption>**Figure 8.** Evolution of the mean distance to the herder over time when the angle is 45 degrees, $\sigma^* = 0.5$, and the width is 10 m. In the sub-figures, parameters are (a) $v_0^* = 1.0$, (b) $v_0^* = 2.0$, (c) $v_0^* = 4.0$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Evolution of the mean distance to the herder over time when $v_0^* = 2$, the angle is 135 degrees, and the width is 10 m. Parameters of the sub-figures are (a) $\sigma^* = 0.5$, (b) $\sigma^* = 1.0$, (c) $\sigma^* = 1.5$.</figcaption></figure>

Using Figure 8, we observe that with our model, the average distance between cows increases linearly as a function of the herder's speed. We can indeed see the evolution of the scale of the y-axis Once again, this result is expected, so the model and its outputs are consistent. Unlike the angle, which is relative to the path and on which we can act, or the speed of the herder, on which we can act in a different way, we also looked at the evolution of this metric (distance to the herder) as a function of $\sigma^*$, which is a physical parameter of the cow (see Figure 9). This parameter can be seen as fixed or as a state of concentration/stress of the animals. Here, contrary to what we might have expected, the more random the cows' behaviour, i.e., their tendency to wander a little without following their attraction paradigm to the core, the less distance they have to the herder. The impact of the corner is also weaker for higher $\sigma^*$ values. This is rather informative, but does not allow us to describe a possible action.

### 3.2.2. Exit Rate and Escape Zone

Fixing two corridors of length 100 m and width 10 m, we illustrate here both the impact of geometric properties on exit rates, and the points of attention on routes.

In Figure 10, we can see, although less clearly than expected, that the angle affects exit rates. The flat path has a lower mean number of exits than the angled paths. We can see that the impact of the angles is more on the outside of the roads, rather at the level of the corners and afterwards.

In Figure 11, the influence of the herder's speed on the output can be seen. In addition to increasing (rather slightly) exit rates with the speeds chosen here, we can see that when the herder's speed is greater, so is the impact of changes of direction. It is more difficult for the herd to adopt a new direction, as it retains its inertia due to collective movement.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** This heat map illustrates the exit rate as a function of path location. The brighter the area, the greater the average number of exits in the simulations (illustrating a greater risk of exit). Parameters are $v_0^* = 1.0$, $\sigma^* = 0.5$, and the width is 10 m. If $\theta$ is the angle between corridors, then in sub-figures, we choose (a) $\theta = 10.0$, (b) $\theta = 45.0$, (c) $\theta = 135.0$, (d) $\theta = 180.0$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Evolution of the mean distance to the herder over time when the angle is 45 degrees, $\sigma^* = 0.5$, and the width is 10 m. In the sub-figures, parameters are (a) $v_0^* = 1.0$ and (b) $v_0^* = 2.0$.</figcaption></figure>

In Figure 12, we can observe the influence of the parameter $\sigma^*$ on the exit rate. First, by looking at the colour scale, we can see that this parameter has little influence on the total output rate. This confirms the results based on distance from the herder. However, these maps provide additional information. We can see that as the noise increases (curve (c)), the output points become more uniformly distributed. The angle only impacts trajectories for low $\sigma^*$ parameters.

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Evolution of the mean distance to the herder over time when $v_0^* = 2$, the angle is 135 degrees, and the width is 10 m. In the sub-figures, parameters are (a) $\sigma^* = 0.5$, (b) $\sigma^* = 1.0$, (c) $\sigma^* = 1.5$.</figcaption></figure>

To sum up, these curves all illustrate that the points to watch are, as you would expect, on the corners. Moreover, they give a certain quantitative vision. In fact, it should be remembered that the parameters chosen for the simulations were selected without any real
data, so the exit rates may not be an exact probability of exits. Nevertheless, we can rely on
the order-of-magnitude differences between the different configurations simulated to get
an idea of the differences in risk.

In contrast to Figures 10–12, which illustrate points of vigilance to watch out for on
each possible route, in Figure 13, we describe a synthesis of global output risks as a function
of road geometric parameters. This figure is probably the most informative. Sub-figure (a)
shows that exit risk decreases almost linearly with road width and angle. Once again, we
observe the obvious: it is preferable to have a wide road with a shallow angle. However,
our study quantifies this observation.

<figure><img src="image_16.png" /><figcaption>**Figure 13.** In these two figures, we simulate *n* different scenarios with different corridor widths (in the *x*-axis) and angles (in the *y*-axis) to represent exit rate in function to widths and angles. In the sub-figures, we fixed (a) $\sigma^* = 0.5$, $n = 500$ and (b) $\sigma^* = 1.0$ $n = 1000$.</figcaption></figure>

In sub-figure (b), we see what we said above: The more the herd tends to disperse (high sigma), the less important the angle, as the cows leave the herd at various points along the road. However, width has a definite impact on trajectory.

3.3. *Path Exits from Real Routes*

In Section 3.1, we have detailed the geometric properties of 12 real transhumance
routes. In particular, we saw that routes 1, 2, 10, 11, and 12 have lower mean angle than
the other routes and that routes 5, 6, 7, 8, and 10 have lower width. In Section 3.2, we have
shown the impact of low width and low angle on the exit rate. Consequently, as a first
approximation, one can expect that a route such as 10 is particularly more dangerous than
the other transhumance routes in our dataset.

To be more precise than the concatenation of these two results, we carried out several
simulations of possible trajectories along these routes and measured the proportion of time
per cow spent off the assigned path. These simulations are represented in Table 3. Due to
the cost of calculation, we have carried out few simulations, but this is not important, as
there are a large number of cows (i.e., 50), and as some unknown parameters are fixed, we
have to rely more on orders of magnitude than on the accuracy of the results.

In Table 3, routes can be grouped into three types, ranging from the most dangerous
to the best arranged. Routes 1, 5, 6, 7, and 12 appear to be particularly dangerous, with a
rate of exits from transhumance routes in the order of one per thousand. A second route
class includes Routes 2, 3, 8, 9, 10, and 11. These roads have an intermediate risk of running
off the route, of the order of 1 per 10,000. These risks are not negligible. Finally, Route 4 is
particularly safe, and is a model for official route choices, although it admits particularly
low width and large angle.
<table><caption>**Table 3.** Summary of time spent (in seconds) off-track for herds simulated on real data. These results are based on 5 simulations with 50 animals. Based on these estimates, we have classified these routes according to three danger levels, from the most dangerous (level 1) to the safest (level 3).</caption><thead><tr><th>Route</th><th>Mean Time Outside Route for All Cows</th><th>Mean Time Outside Route per Cow</th><th>Mean % Time Outside Route per Cow</th><th>Danger Level</th></tr></thead><tbody><tr><td>1</td><td>5078.05</td><td>101.56</td><td>3.15 · 10<sup>-1</sup></td><td>1</td></tr><tr><td>2</td><td>151.44</td><td>3.03</td><td>2.86 · 10<sup>-2</sup></td><td>2</td></tr><tr><td>3</td><td>419.86</td><td>8.40</td><td>7.93 · 10<sup>-2</sup></td><td>2</td></tr><tr><td>4</td><td>52.00</td><td>1.04</td><td>8.39 · 10<sup>-3</sup></td><td>3</td></tr><tr><td>5</td><td>2219.40</td><td>44.39</td><td>3.77 · 10<sup>-1</sup></td><td>1</td></tr><tr><td>6</td><td>1880.60</td><td>37.61</td><td>6.34 · 10<sup>-1</sup></td><td>1</td></tr><tr><td>7</td><td>127.59</td><td>2.55</td><td>1.48 · 10<sup>-1</sup></td><td>1</td></tr><tr><td>8</td><td>76.19</td><td>1.52</td><td>5.78 · 10<sup>-2</sup></td><td>2</td></tr><tr><td>9</td><td>69.71</td><td>1.39</td><td>3.39 · 10<sup>-2</sup></td><td>2</td></tr><tr><td>10</td><td>508.07</td><td>10.16</td><td>4.02 · 10<sup>-2</sup></td><td>2</td></tr><tr><td>11</td><td>121.30</td><td>2.43</td><td>1.03 · 10<sup>-2</sup></td><td>2</td></tr><tr><td>12</td><td>183.83</td><td>3.68</td><td>1.73 · 10<sup>-1</sup></td><td>1</td></tr></tbody></table>

## 4. Discussion

Our herd movement simulator quantifies the impact of route shape on the risk of exits. This was applied to real routes used for transhumance.

Although previous studies have investigated the socio-economic factors influencing herd movements [35], there remains a notable gap in quantitative analyses addressing the impact of physical route characteristics on movement patterns. This study addresses this gap by providing empirical data on the correlation between route geometry and exit risks. Furthermore, while models such as MSTRANS have simulated herd movements based on decision-making processes [36], our model introduces quantitative risk assessments related to route shapes, thereby adding a new dimension to the study of transhumance.

Our findings provide insights that can significantly inform the design of transhumance corridors. By empirically demonstrating the correlation between route geometry and exit risks, our study allows for the creation of optimized routes that minimize these risks. By integrating quantitative risk assessments into corridor design, policymakers and planners can enhance the sustainability of transhumance practices, fostering a harmonious coexistence between agricultural and pastoral communities.

To our knowledge, we are the first to use a numerical approach to evaluate transhumance routes, which makes it difficult to compare our results with previous studies. However, our results are based on a number of assumptions, which lead to a lack of realism and direct application of our approach. Certain modelling assumptions, such as the voluntary actions of herders to let their flocks off the road to eat in other farmers' fields, seem to be difficult to include in our approach.

However, let us detail a few possible sources of improvement of our approach. The first source of improvement would be to equip a real herd of Sub-Saharan African cattle in real conditions to recover various parameters of interactions, attraction-repulsion, average speeds of the animals, noise determination, etc. GPS tags can be used to both determine the position of animals and herd (and then calculate the distance between them) as well as to estimate all parameters through a calibration method. In particular, it will also enable us to validate our variation of the Cucker-Smale model (with a herder). This is a costly operation at present, but there may be interpolations to be made between experiments in this context with few animals and more equipped Western herds. This approach is postponed to future work, as it requires further experimentation. A second source of improvement concerns the heterogeneity of landscapes. Having worked on real datasets, we should be able to exploit the heterogeneity of territories to really understand the risks of trajectory exits. One potential approach is to use Land Cover maps to identify the
fields surrounding the transhumance routes. Although some maps are available one the URL (https://oss-online.org/en/releases/REPSAHEL-Atlas_Tchad, https://www.spaceclimateobservatory.org/fr/eclat-lac-tchad#resources, accessed on 2 February 2025), they are not recent enough for our project. In addition, climate change, current agricultural transitions, and population growth are rapidly rendering this type of map obsolete. An alternative would be to work with Sentinel-2 data to characterize soils. Although some initial work has begun to characterize soils using such images [37], several challenges remain in quantifying the quality of forage and the attractiveness of livestock from satellite images [38] (accessed on 2 February 2025). Moreover, the results that we obtain on real data deserve to be compared with those of other databases. We have not found precise data on conflicts along the routes we studied. Relating our danger ratings to these data would enable us to better measure the impact of route shape on other possible factors. Furthermore, other sociological studies, such as surveys and collections of practices on these paths, would greatly contribute to addressing these conflicts issues.

Finally, this work, which merits further extension, presents promising results on potential approaches and paves the way for field experiments. It confirms that GPS technology, combined with other solutions (such as policies and awareness-raising), can contribute to addressing conflicts between farmers and herders. Additionally, although on a completely different scale, there are also conflicts between livestock farmers and other field users in the Occident, particularly in Western Europe [39–42]. The approach we are developing is therefore in synergy with the current deployment of GPS on livestock farms.

**5. Conclusions**

**Summary of Research Objective and Significance.** Exit paths for animals in Chad and other Sub-Saharan African countries represent a significant challenge. In this article, we have proposed a herd movement simulator for the evaluation of transhumance trajectory outputs. It is based on a new mathematical model, which is a slight variation of a model that has been validated both theoretically and experimentally.

**Main Contributions:**

1. Our simulation illustrated the potential of multi-agent mathematical models for interpolating knowledge from sensors measuring the position of one or more possible agents (herders, animals, etc.).

2. We have used this model to evaluate and test on both simulated and real transhumance routes.

3. This study confirms the expected results on the impact of route shape, where shallow roads and tight corners increase the risk of animals running off the road.

4. We have provided an initial quantitative estimate of the risk of exits on synthetic routes.

5. We applied our approach on 12 real transhumance routes in Chad, describing a new decision-support tool.

6. Roads have been identified and classified according to three degrees of danger in terms of the risk of exits.

**Broader Implications and Future Research.** Our exploration of agent-based models for transhumance path exits in Sub-Saharan Africa opens up exciting possibilities for future research. The findings offer crucial insights for designing transhumance corridors that can mitigate conflicts between farmers and herders. Furthermore, this research provides valuable guidelines for developing real-world applications, ensuring better route planning and enhanced safety for herding communities.

**Supplementary Materials:** The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/agriengineering7030060/s1.
**Author Contributions:** All authors have contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the French National Research Agency under the Investments for the Future Program, referred as ANR-16-CONV-0004.

**Data Availability Statement:** See the Supplementary Materials.

**Acknowledgments:** We thank the anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions.

**Conflicts of Interest:** The research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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