# Towards a generic agent-based vector-host model: effects of carrying capacity and host mobility

Cyrine Chenaoui, Nicolas Marilleau, Slimane Ben Miled

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## To cite this version:

Cyrine Chenaoui, Nicolas Marilleau, Slimane Ben Miled. Towards a generic agent-based vector-host model: effects of carrying capacity and host mobility. Applied Network Science, 2024, 9 (1), pp.33. 10.1007/s41109-024-00629-z. pasteur-04973223

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Submitted on 2 Mar 2025

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# Towards a generic agent-based vector-host model: effects of carrying capacity and host mobility

Cyrine Chenaoui<fnref n="1" />,<fnref n="2" />,<fnref n="3" />*, Nicolas Marilleau<fnref n="3" /> and Slimane Ben Miled<fnref n="2" />

## Abstract

The aim of our work is to develop a generic conceptual agent-based model to formalize the interaction of vector and host given climate change. The model consists in creating a hypothetical example of a vector-host system. It simulates the vector's life cycle while considering interactions with hosts and the temperature. It is presented following the ODD protocol and based on parameters and processes to conceptualize the vector-host complexity. It could accommodate a broad spectrum of vector species and different biogeographic regions. Our model can be extended to more ecologically complex systems with multiple species and real-world landscape complexity to test different host and / or vector-targeted control strategies and identify practical approaches to managing vector population and movement patterns.

**Keywords:** Agent based models, Vector-host system, ODD protocol, Tick

## Introduction

Ectoparasites are vectors of pathogens that cause vector-borne diseases. Most ectoparasites are hematophagous species, such as insects and acarians. Ectoparasitic acarians (i.e. ticks) of vertebrates (e.g. Cattle, sheep, etc.) are vectors of pathogens responsible for zoonoses and significant economic losses for livestock (Sonenshine and Roe 2014; Randolph 2004). Understanding the role of vectors in the transmission of pathogens involves studying their population dynamics (Ogden et al. 2004, 2005). In this regard, the study of vector population dynamics and their interactions with the environment seems essential to surveillance vector-borne diseases (Wikel 2018; Estrada-Peña and Fuente 2014; Estrada-Peña et al. 2023). The most important factor is climate change (Ogden et al. 2018, 2021). In fact, variations in climate and temperature induce great alterations in the distribution of vectors in nature and, in particular, in ticks (Ogden et al. 2021; Nolzen et al. 2022; Wu et al. 2010; Medlock et al. 2013; Eisen et al. 2016).

The use of computer simulations to address complex ecological systems with explicit details while implementing individual-level interactions and spatial structure is promising (Tracy et al. 2018). Moreover, mathematical and simulation models were developed to further understand the ecological mechanisms and processes underlying the

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<footnote>*Correspondence:
cyrine.chenaoui@ird.fr</footnote>

<footnote n="1">¹ Ecole Nationale d'Ingénieurs de Tunis, Université Tunis El-Manar, Tunis, Tunisia</footnote>

<footnote n="2">² Laboratoire de bio-(informatique, mathématiques et statistique), Institut Pasteur de Tunis, Université Tunis El-Manar, Tunis, Tunisia</footnote>

<footnote n="3">³ UMI 209 Unité Mixte Internationale de Modélisation Mathématique et Informatique des Systèmes Complexes, Institut de Recherche pour le Développement, Sorbonne Université, Paris, France</footnote>
vector-host systems and had shown efficiency in answering problems that controlled experiments or field observations solely cannot fully address (Ogden et al. 2021; Ndongo et al. 2022). In fact, these models give a new degree of knowledge to support public health strategies for disease and vector control (e.g. Wu et al. (2010); Healy et al. (2020); Tosato et al. (2022)).

Simulation models as the Agent-Based Models (ABM) models are prominent tools to predict the relationships between biological processes, the environment, and ecological patterns across different scales, both for population dynamics and evolutionary scales (Tosato et al. 2022; Smajgl and Barreteau 2014; Mount and Haile 1989; Halsey and Miller 2018; Tyre et al. 2004; Gilbert et al. 2014).

Many ABM models have been developed to tackle whether host-vector (e.g. Healy et al. (2020); Yong et al. (2015), host-vector-pathogen (e.g. Halsey and Miller (2018); Tardy et al. (2022); Gaff and Nadolny (2013); Nadolny and Gaff (2018)), or host-pathogen systems (refer to Mori et al. (2023)). Simulation models are increasingly being used to solve problems and aid in decision making in agricultural systems (Kremmydas et al. 2018) and veterinary epidemiological systems (Healy et al. 2020) as well as in public health systems (Tracy et al. 2018). Our work aims to develop a generic agent-based model to formalize the interaction of vector and host with explicit consideration of temperature when describing the vector's life cycle since ABMs can integrate several scales, unlike ODE mathematical models (e.g. Ndongo et al. (2022)). The distinguishing feature of our model is that it explicitly incorporates all the life stages of the vector, as well as temperature, as an environmental factor in the development of the vector's life cycle. In addition, the integration of host movement and interactions with the vectors. Our objectives are (1) to develop an agent-based model that simulates the host-vector relationships while considering vector development as a temperature-dependent process, (2) to study the effects of host movement patterns (random versus herd movement) on vector population dynamics, and (3) study the effect of carrying capacity of the host on vector population dynamics.

This paper is organized as follows: In Sect. 2.1, we describe the background knowledge and modeling assumptions we considered. In Sect. 2.2, we explain our model using the overview, design concepts, and details (ODD) protocol (Grimm et al. 2010, 2020). Then, in Sect. 3 we present our simulation's results by comparing two movement approaches of Cattle agents. Finally, in Sect. 4, we discuss our results and point out some limits and perspectives to generalize and promote our model to fill the gap between understanding the ecology of the vector and hosts on the one hand and the effects of host movement on the other hand.

# Materials and methods

## Background

Our model creates a hypothetical example of a semi-intensive livestock production system. This closed system comprises only three entities: ticks, Cattle and Rodents. We consider that Cattle are grazing and then return to a barn. We do not consider in our model the diurnal activity of ticks nor that of Rodents. The host population is closed, while the tick population is an open monospecific population whose recruitment is done through egg laying. It is stratified in cohorts of biological life stages: egg, larva, nymph and adult.
Interstadial development of ticks are temperature-dependent processes (Ogden et al. 2004, 2005). A blood meal is the only interaction between the hosts and the ticks.

The vector model in this study is a species of hard ticks. Ticks have four developmental stages: an egg and three active obligatory hematophagous ectoparasitic stages characterized by only one larval and nymphal stage before reaching the adult stage and discontinuous feeding, a single blood meal to get through the next life stage.

At the end of the embryo's development, the eggs hatch into larvae (Sonenshine and Roe 2014; Arthur 1948). The larva begins to search for a host. All instars can enter a behavioural quiescence state called “diapause” where development is paused. In our model, we consider diapause as a temperature-dependent state, which is activated when the temperature of the environment reaches a stage-dependent threshold (Gray et al. 2016; Belozerov and Naumov 2002; Dautel and Knülle 2010). Two known host-finding strategies are described in the literature (Lane et al. 2009; Mejlon and Jaenson 1997): a passive strategy called *questing* where the tick waits on vegetation for passing by of a possible host and an active seeking host strategy where the ticks move actively looking for a host.

## The conceptual model

The description of the model follows the “Overview, Design Concepts, and Details (ODD)” protocol developed by Grimm et al. (2020) to standardize the use of ABMs.

### Overview

*Purpose and patterns* The purpose of the model is to reproduce the tick life cycle, considering temperature-dependent processes and host movement patterns.

*Entities, state variables, and scales:* The model is made up of two agent types: the vector agent and host agents. The vector agent represents the hard tick; the host is either a Cattle or a Rodent. The interactions between agents are ruled by two ecological processes: (1) vector-host interaction and (2) host movement.

*The Vector agent:* The *Vector* agents are characterized mainly by the life stage attribute *state* respectively: egg, larva, nymph, and adult. The activity status of the vector agents is characterized by three behavioural states (1) *questing*: This is a generic term to designate the activity of the vector before attaching to a possible host and while it waits for the passing of a host or the actively seeking of the host, (2) *feeding*: while the vector attaches to the host for taking the blood meal, and finally (3) *moulting*: the behavioral state after completing the blood meal and detaching from the host (Figs. 1 and 2).

*The host agent:* The *Host* agent includes two sub-agents (Fig. 2): *Cattle* and *Rodent* agents: Rodent agents have a random movement in the environment characterized by a velocity randomly fixed at the beginning of the simulation. The Cattle agent moves following two behaviours. In the first movement behavior, the cattle agents are endowed with random movements. In the second movement behaviour, Cattle agents abide by a herd movement pattern and follow a dominant Cattle agent considered a leader. In both behaviors, cattle are characterized by diurnal activity, whether active or resting, by regaining the barn.
<figure><img src="image_5.png" /><figcaption>**Fig. 1** Conceptual view of the vector life stages (*state*) and behavioural states *BehState* attributes.In every *state*, the vector agent can have, at one time, a sole behavioural state. The egg state has no behavior. The adult state has only two behavioural states. The larva and the nymph state can have the behavioural state, respectively, questing, feeding and moulting</figcaption></figure>

**The environment** The environment is set up as a hypothetical square world with a dimension of 1 km x 1 km with unwrapping boundaries movement and does not cause Host agents to jump to the other side of the world.

The environment is considered a grazing area where all Host agents can move around. In the centre of the environment, a hypothetical barn of square geometric shape and a dimension of 50 m × 50 m (refer to Fig. 3). As for the climatic parameters, only the temperature is considered a climatic factor influencing the vector agents.

*Process overview and scheduling* The model simulates the interactions between the vector (tick) and the hosts (Cattle and Rodents) agents on the one hand and between environmental (temperature) and vector agents on the other hand. We consider six submodels (Fig. 4):

Host agents (Cattle and Rodents) move according to the *move_host* submodel every hour. Questing vector (tick) agents are stationary, waiting for the passing-by of a possible host within a specific distance *PERCEPTION_DISTANCE*. When finding a host (Rodent or Cattle), vector (tick) attach to the host based on the *attach* submodel and start a blood meal. Each day, vectors detach from hosts following the *detach* submodel after completing the blood meal. Furthermore, the life cycle processes of the vector population are the result of the *development* submodel compiled of the following five submodels that occur every day: *previposition*, *incubation*,*moulting*, and finally the *diapause*
submodel. The *death* submodel is also composed of two submodels: *natural_mortality* and *environment_induced_mortality*.

The temperature is updated every day based on the *SetEnvCdt* submodel.

## Design concepts

**Basic principles** The model considers that the life cycle of ticks consists of four stages, the duration of which depends on temperature. Moulting between stages depends on blood meals and is, therefore, closely linked to the presence of hosts. Adult vectors are all females and would give offspring, and mating is implicitly included in our model. We have considered the host population a closed system with a constant host population. Both scenarios do not include trophic resources for *Host* agents.

**Emergence** Abundance and stage distribution emerge from the properties of the relationships between temperature and moulting duration and also from host location and probability of attaching to a possible host, permitting the fulfilment of the transition into the next behavioural stage and life state. The host agents' and vector's combined actions and states in response to the environment produce the total number of vectors questing, feeding, or moulting in any particular location at any given moment.

**Sensing** Moulting and diapause behaviours are temperature-dependent processes, *vector* agents sense the temperature variation and enter diapause when the daily temperature reaches a stage-dependent threshold. *larva* and *nymph* states sense the temperature when moulting. *Egg* state of vector agents sense the temperature when incubating, as for *Adult* state sense it when either in preoviposition or oviposition phase. Intraspecific interactions are not taken into account in our model; *Vector* agents perceive the *host* agent within the range of their distance of perception (refer to tab 2) and do not sense each other or communicate. Both *Cattle* and *Rodent* agents do not sense the temperature. Rodents do not sense each other nor sense *Cattle* agents. In the first scenario (S1), *Cattle* do not sense each other; meanwhile, in the second scenario (S2), they sense the movement of each other and move in a herd pattern while following a leader *Cattle* agent.

**Interaction** *Vector* agents interact with both *Cattle* and *Rodent* agents through blood meals, the result of the *attach* submodel. *host* agent, through a blood meal, can be a limiting factor as it is necessary for the moulting from one stage to another.

The *larva* state of the *vector* agent can only attach and have a blood meal from a *Rodent* agent. *Nymph* and *adult* state can attach to both *Rodent* and *Cattle* agents.

**Stochasticity** Stochasticity plays a crucial role in three submodels within the system. Firstly, in scenario (S1), the movement of Rodents and Cattle is governed by randomness. Their velocity direction is randomly determined at each time step.

Additionally, the environmental-induced mortality is a stochastic process. At every time step, a probability of death is computed based on the life state and the temperature, both of which contribute to the stochastic nature of the model.

Moreover, the *attach* submodel relies on the probability of attachment, and the selection of the host is determined randomly at each time step. The probability of attachment is influenced by the number of hosts within the perception distance of the vector agents.
**Observation** The main target observations are the vector's population and per-stage abundance. Additionally, we tracked the life-stage behavioural state (questing, feeding, moulting, and diapause).

We denote the total abundance of the tick population, at time $t$, $N(t)$ and $\bar{N}(t)$ the mean over the replication of the population abundance and $N_{state(i)}(t)$, the life stage population abundance, where $i$ is, respectively to “egg”, “larva”, “nymph” and “adult” stage.

$$ \bar{N}(t) = \sum_{i} \bar{N}_{state(i)}(t) \qquad (1) $$

## Details

**Initialization** Each simulation starts with an environment wherein *Cattle* are situated in the centre of the barn. In the second scenario (S2), the leader *Cattle* agent is selected randomly. *Rodent* agents are located randomly in the environment. All *Vector* agents are adults on day zero of the preoviposition stage. The initial date of the simulation is October 1st, 1990. The initial daily temperature corresponds to that of the initialization date. The initial values of all parameters are in table 2.

**Input data** The model is parameterized for the species *Ixodes scapularis*. Climate data used in this model are the Climatic Research Unit (CRU) TS Time Series datasets 4.04 (Harris et al. 2020). This dataset is monthly estimates of Temperature recorded between the years 1990 and 2000 with a spatial resolution of (0.5x0.5 degree) grids for the region of Wisconsin. Fitting data points generate the daily estimates to a polynomial function of the order 3.

The parameters used in this model come from bibliographic data and/or the modeler's expertise, as detailed in table 2.

## Submodels

**Host's agent submodels** *move_my_parasite* submodel

We assume that the hosts will move the attached hosts on themselves following their position according to the *move_my_parasite* submodel. This submodel is common for both *Rodent* and *Cattle* agents. *Vector* agents attached to a *host* agent change their location as per the host they are fixed on until they detach.

**move_Cattle submodel** The *move_Cattle* sub-model simulates the movement and activity patterns of Cattle sub-agents. Two movement behaviours are considered in the *Cattle* and *Rodent* agents move randomly according to the *move_host_random* sub-model. Unlike the first scenario, *Cattle* agents follow a leader agent according to the *follow_leader* process, which itself follows a predefined path according to the *follow_path* sub-model starting from the centre of the barn and returning to it. *Cattle* agents, besides the leader, are moving in a herd cohesion according to a boid movement composed of the “separation”, “alignment”, and “cohesion” processes as described by Reynolds (1987), *Cattle* agents flocks to the centre of a mass of agents within the *minimal_distance*, and avoiding other agents while trying to match the position of other *Cattle* agent. If a current hour of the day is in the grazing time range, the *leader* will move with a random
speed following the path. Otherwise, the leader Cattle agent will move to the centre of the barn BARN\_LOC.

### Vector's agent submodels

The attach submodel The attach submodel describes the attachment of the vector to a host agent within the PERCEPTION\_DISTANCE when its behaviour stage is quest-ing. Vectors which have their life state equal to "larva" attach to Rodent agent, nymph and adult attach both Cattle and Rodent. The attach submodel does not apply to egg since it is a free life stage.

The process of attachment has a probability ProbIndAttach dependent on the num-ber of neighbouring hosts NeighHosts within the PERCEPTION\_DISTANCE, computed as follows:

$$ProbIndAttach = 1 - (0.8)^{\text{NeighHosts}} \qquad (2)$$

Since the vector is more likely to attach to one of its hosts, the probability of attachment success "ProbIndAttach" is determined by the number of host in the range of the vector's perception distance (refer to Eq. 2). When applying a carrying capacity for the host, we assume that every host agent (both Cattle and Rodent agents) have a maximum number of vectors that can be attached to them, parasite\_max. The attach submodel is modified in a way that if the attribute VectorOfParasiteis more or equal parasite\_max, the Vector agent will not attach to the host.

detach submodel This sub-model describes the detachment of vectors from the host after a successful blood meal. A successful blood meal is controlled by a fixed dura-tion of the blood meal: AttachToDetach (refer to table 1). When this occurs, the Vec-tor detach from the host agent at a random proximate location, and it behaviour state, BehState updates to "moulting".

develop submodel The develop submodel considers the life stages transitions and diapause process. In this submodel, we consider the development of the vectors as daily temperature functions (Fig. 5). Also, the develop submodel is itself organized into five processes, respectively, diapause, moulting, preoviposition, and finally incu-bation. The transition between each life state is the result of the success of the attach, detach, and moult submodels.

moult, preoviposition and incubation submodels Based on the work of Ogden et al. (2004), moult, preoviposition and incubation are temperature-dependant processes (refer to Figure 6). The development is incremented every day. We denote f(t) the duration function for every life state, respectively, m\_larva→nymph, m\_nymph→adult, (Fig. 6 and equations 3) and each behaviour is structured by life state i, respectively i\_moult and i\_preoviposition.

We assume at every time step, the duration of moulting, respectively for larva into a nymph and nymph into an adult, the preoviposition and the incubation are computed according to, respectively, the equations 3, where m is, respectively, the moult-ing duration respectively for larva and nymph, preoviposition duration, and T\_current is the daily temperature (Fig. 5, Eq. 3).
<table><caption>**Table 1** Summary of attributes and variables with definitions and periods of update</caption><thead><tr><th>Attri.</th><th>Typ.</th><th>Dim.</th><th>Desc.</th><th>Peri.</th></tr></thead><tbody><tr><td><em>Vector</em></td><td></td><td></td><td></td><td></td></tr><tr><td>location</td><td>R<sup>2</sup></td><td>-</td><td>Position</td><td>1 H</td></tr><tr><td>speed</td><td>R<sup>2</sup></td><td>km/h</td><td>Speed of vector agent</td><td>1 H</td></tr><tr><td>BehState</td><td>Category</td><td>-</td><td>Behavioural state</td><td>1 D</td></tr><tr><td>State</td><td>Category</td><td>-</td><td>Life stages;egg, larva, nymph,adult</td><td>1 D</td></tr><tr><td>AttachDetachDate</td><td>Date</td><td>-</td><td>blood meal duration</td><td>1 D</td></tr><tr><td>LayDate</td><td>Date</td><td>-</td><td>Date of egg hatching</td><td>1 D</td></tr><tr><td>QuestDate</td><td>Date</td><td>-</td><td>Questing start date</td><td>1 D</td></tr><tr><td>NeighHosts</td><td>List</td><td>-</td><td>Neighbouring hosts</td><td>1 H</td></tr><tr><td>TargetHost</td><td>Host</td><td>-</td><td>Target host</td><td>1 H</td></tr><tr><td>LayingEgg</td><td>Boolean</td><td>-</td><td>Fitness of adults to lay eggs</td><td>1 H</td></tr><tr><td>Diapause</td><td>Boolean</td><td>-</td><td>Diapause</td><td>1 D</td></tr><tr><td>LaidToIncubation</td><td>R</td><td>-</td><td>Incubation duration</td><td>1 D</td></tr><tr><td>PreoviToOvi</td><td>R</td><td>-</td><td>Preoviposition duration</td><td>1 D</td></tr><tr><td>MoultingDuration</td><td>R</td><td>-</td><td>Moulting duration</td><td>1 D</td></tr><tr><td>oviposition</td><td>Boolean</td><td>-</td><td>Oviposition</td><td>1 D</td></tr><tr><td>moulted</td><td>Boolean</td><td>-</td><td>True to change the life state</td><td>1 D</td></tr><tr><td>incubation</td><td>Boolean</td><td>-</td><td>incubation</td><td>1 D</td></tr><tr><td>ProbIndAttach</td><td>R</td><td>-</td><td>Probability of attachment</td><td>1 H</td></tr><tr><td>counter</td><td>R</td><td>-</td><td>Counter for development rate</td><td>1 D</td></tr><tr><td><em>Hosts</em></td><td></td><td></td><td></td><td></td></tr><tr><td>location</td><td>R<sup>2</sup></td><td>-</td><td>Position of the agent</td><td>1 H</td></tr><tr><td>speed</td><td>R<sup>2</sup></td><td>km/h</td><td>Speed of host agent</td><td>1 H</td></tr><tr><td>VectorOfParasite</td><td>N<sup>0</sup></td><td>-</td><td>List of vectors</td><td>1 H</td></tr></tbody></table>

$$ 
\begin{aligned}
m_{j(state1 \to state2)} &= aT_{current}^{-b} \\
i_{j(state1 \to state2)} &= i_{j(state1 \leftarrow state2)} + f(t)
\end{aligned} 
\qquad (3) $$

where

$$f(t) = 1/m_{j(state1 \to state2)}$$

**diapause submodel** The *diapause* submodel describes the temperature-dependent process of the diapause. The *diapause* state is activated when the daily temperature reaches a life state-dependent threshold temperature (refer to table. 1). When the *diapause* state is activated, the development incremental function *f* approaches infinity *f* → ∞, the *develop* submodel is paused until the daily temperature surpasses the *diapause* temperature threshold, then the *diapause* submodel is deactivated, and the *develop* submodel in its turn is activated.

**death submodel** The *death* submodel describes mortality processes. Two types of mortality are considered in this submodel *natural mortality*, *environment-induced mortality*.

**Natural mortality** Every day at the end of the day, each *vector* agent had a daily probability of mortality for every life state (refer to table 1). Also, all adult agents die after laying eggs.
<figure><img src="image_6.png" /><figcaption>**Fig. 2** UML (unified modelling language) class diagram represents our ABM, with two types of agents: Vector agent and Host agent. Both agents are located in the World</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 3** Screenshot of the Environment on Gama Platform. The central cell (in dark red) represents the barn. Triangles (in blue) represent the rodent agents. Circles (in red) represent the vector agents. The letter designs the behavioural state of every vector agent (q: questing, f: feeding, m: moulting)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4** Conceptual view of the submodels: the diagram represents an outline of the sequence of processes and the schedule of interactions between *Vector* and *Host* agents at each discrete time step</figcaption></figure>
<table><caption>**Table 2** Global Parameters and input values of the vector agent and host agent attributes. For the dimension, the following symbols are used: "-" indicates no dimensions</caption><thead><tr><th>Param.</th><th>Sym.</th><th>Val.</th><th>Dim.</th><th>Uni.</th><th>Ref.</th></tr></thead><tbody><tr><td colspan="6"><strong>World</strong></td></tr><tr><td>The current daily value of T°</td><td>CurrentTemp</td><td>-</td><td>R</td><td>°C</td><td>(Harris et al. 2020)</td></tr><tr><td>Maximum T°</td><td>MAX_TEMP</td><td>40</td><td>R</td><td>°C</td><td>-</td></tr><tr><td>Minimum T°</td><td>MIN_TEMP</td><td>-30</td><td>R</td><td>°C</td><td>-</td></tr><tr><td>Coeff.of Environment-induced mort.</td><td>ALPHA</td><td>0.2</td><td>R</td><td>-</td><td>-</td></tr><tr><td>Time step</td><td>step</td><td>2</td><td>R</td><td>hour</td><td>-</td></tr><tr><td>World size</td><td>shape</td><td>1</td><td>R</td><td>km</td><td>-</td></tr><tr><td>Posi. of the barn</td><td>BARN LOC</td><td>(500,500)</td><td>R<sup>2</sup></td><td>m</td><td>-</td></tr><tr><td colspan="6"><strong>Vector</strong></td></tr><tr><td>Initial num. of vectors</td><td>INI_NUM VECTOR</td><td>150</td><td>N</td><td>-</td><td>-</td></tr><tr><td>Distance of perception</td><td>PERCEPTION DISTANCE</td><td>7</td><td>N</td><td>m</td><td>-</td></tr><tr><td>Larva max. Lethal T°</td><td>LETHAL_TEMP Sup_L</td><td>40</td><td>R</td><td>°C</td><td>(Ogden et al. 2004)</td></tr><tr><td>Nymph max. Lethal T°</td><td>LETHAL_TEMP Sup_N</td><td>45</td><td>R</td><td>°C</td><td>(Ogden et al. 2004)</td></tr><tr><td>Egg max. Lethal T°</td><td>LETHAL_TEMP Sup_E</td><td>32</td><td>R</td><td>°C</td><td>(Ogden et al. 2004)</td></tr><tr><td>Adult max. Lethal T°</td><td>LETHAL_TEMP Sup_A</td><td>35</td><td>R</td><td>°C</td><td>(Vandyk et al. 1996)</td></tr><tr><td>Nymph min. Lethal T°</td><td>LETHAL_TEMP INF_N</td><td>-18</td><td>R</td><td>°C</td><td>(Brunner et al. 2012)</td></tr><tr><td>Adult min. Lethal T°</td><td>LETHAL_TEMP INF_A</td><td>-20</td><td>R</td><td>°C</td><td>(Brunner et al. 2012)</td></tr><tr><td>Larva min. Lethal T°</td><td>LETHAL_TEMP INF_L</td><td>-18</td><td>R</td><td>°C</td><td>(Daniels et al. 1996)</td></tr><tr><td>Egg min. Lethal T°</td><td>LETHAL_TEMP INF_E</td><td>-30</td><td>R</td><td>°C</td><td>(Dautel and Knülle 2010)</td></tr><tr><td>Threshold T° of diapause for L.</td><td>THRESHOLD T_L</td><td>8</td><td>R</td><td>°C</td><td>(Daniels et al. 1996)</td></tr><tr><td>Threshold T° of diapause for N.</td><td>THRESHOLD T_N</td><td>11</td><td>R</td><td>°C</td><td>(Brunner et al. 2012)</td></tr><tr><td>Threshold T° of diapause for E.</td><td>THRESHOLD T_E</td><td>8</td><td>R</td><td>°C</td><td>(Dautel and Knülle 2010)</td></tr><tr><td>Threshold T° of diapause for A.</td><td>THRESHOLD T_A</td><td>8</td><td>R</td><td>°C</td><td>(Vandyk et al. 1996)</td></tr><tr><td>Proba. to lay eggs</td><td>ProbToLay</td><td>1</td><td>[0, 1]</td><td>-</td><td>-</td></tr><tr><td>Larva Nat. Mortality Proba.</td><td>P_NAT MOR_N</td><td>0.006</td><td>[0, 1]</td><td>-</td><td>-</td></tr><tr><td>Nymph Nat. Mortality Proba.</td><td>P_NAT MOR_L</td><td>0.006</td><td>[0, 1]</td><td>-</td><td>-</td></tr><tr><td>Adult Nat. Mortality Proba.</td><td>P_NAT MOR_A</td><td>0.006</td><td>[0, 1]</td><td>-</td><td>-</td></tr><tr><td>Egg Natural Mortality Proba.</td><td>P_NAT MOR_E</td><td>0.006</td><td>[0, 1]</td><td>-</td><td>-</td></tr><tr><td>Proba. of attachment</td><td>P_ATTACH</td><td>0.9</td><td>[0, 1]</td><td>-</td><td>-</td></tr><tr><td>Duration of attachment</td><td>AttachToDetach</td><td>7</td><td>R</td><td>day</td><td>(Kocan et al. 2015)</td></tr><tr><td>Time to die when questing</td><td>TimeDieQuesting</td><td>167</td><td>R</td><td>day</td><td>-</td></tr><tr><td colspan="6"><strong>Host</strong></td></tr><tr><td>Initial num. of Cattle</td><td>INI_NUM Cattle</td><td>50</td><td>N</td><td>-</td><td>-</td></tr><tr><td>Initial num.of Rodents</td><td>INI_NUM Rodent</td><td>100</td><td>N</td><td>-</td><td>-</td></tr><tr><td>Min. Host speed</td><td>MIN H SPEED</td><td>0.02</td><td>R<sup>2</sup></td><td>km/h</td><td>-</td></tr><tr><td>Max. Host speed</td><td>MAX H SPEED</td><td>0.5</td><td>R<sup>2</sup></td><td>km/h</td><td>-</td></tr><tr><td>Cattle's activity start time</td><td>START ACTIVE TIME</td><td>9</td><td>R</td><td>hour</td><td>-</td></tr><tr><td>Cattle's activity end time</td><td>END ACTIVE TIME</td><td>16</td><td>R</td><td>hour</td><td>-</td></tr><tr><td>New position of boid movement</td><td>velocity</td><td>(0,0)</td><td>R<sup>2</sup></td><td>-</td><td>-</td></tr><tr><td>Resting status</td><td>rest</td><td>False</td><td>Boolean</td><td>-</td><td>-</td></tr><tr><td>Min. distance of perception</td><td>minimal_distance</td><td>50</td><td>N</td><td>meter</td><td>-</td></tr><tr><td>Carrying capacity</td><td>parasite_max</td><td>-</td><td>N</td><td>-</td><td>-</td></tr><tr><td>Movement behaviour</td><td>movement Beh</td><td>"R", "H"</td><td>list</td><td>Cat.</td><td>-</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>**Fig. 5** A schematic diagram for the duration incremental function *f*</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 6** The curve of the relationship of larva moulting duration with temperature variation</figcaption></figure>

<table><caption>**Table 3** Experimental plan for the scenarios S1 and S2</caption><thead><tr><td>Parameter-Scenario</td><td>Scenario 1 (S1)</td><td>Scenario 2 (S2)</td></tr></thead><tbody><tr><td>Host movement behaviour</td><td>Random movement</td><td>Herd movement</td></tr><tr><td>Carrying Capacity</td><td>{2, 3, 4, 5, 6, 7, 8, 9, 10}</td><td>{2, 3, 4, 5, 6, 7, 8, 9, 10}</td></tr></tbody></table>

For both S1 and S2, we vary the carrying capacity parameter for these different values

*Environment-induced mortality process* the *Environment-induced mortality process* is temperature-dependent. Every life state has a daily *y* probability of death if the current daily temperature reaches the state-dependent lethal temperature threshold (refer to table 1).

## Results

The objective of our simulations is to study the effects of host movement patterns on vector population dynamics as well as the effect of the carrying capacity of the host on vector population dynamics. Simulations were performed using the Gama 1.8.2 (Tail-landier et al. 2019) platform. All simulations reproduce a duration of ten years from October 1st 1990 until September 30th 2000.

Two scenarios are tested for the movement behaviour of *Cattle* agent (refer to Table 3). First, we run the random movement scenario (S1) and then the Herd movement scenario (S2). Both simulation scenarios were run using the same parameter values and initial state. First, we run the simulations with no carrying capacity applied (thereafter, they are defined as, respectively, S1-NC and S2-NC), and then, we fix the carrying capacity for both S1 and S2 (refer to subsection 3.3). We analyze the total and per-stage population
and the total population abundance for all simulation sets. We run 30 replicas per simulation set.

## Effect of simulations replication

The replication effect is analyzed for both scenarios (S1 and S2) with no carrying capacity applied (S1-NC and S2-NC), we observe an annual fluctuation in the total population abundance per replication (refer to Figure 7a and 7b). Additionally, there is a noteworthy disparity in the magnitude of fluctuation between both scenarios and within the replications of each scenario, which becomes significant over the last four years of the simulation time. This is confirmed by computing the standard deviation $σ_i$ over the mean between the replication for both S1-NC and S2-NC (Fig. 7c and Fig. 7d).

## Random movement Versus Herd movement patterns

The total population abundance for both scenarios (S1 and S2) increases over time while no carrying capacity is fixed. Specifically in S1-NC (Fig. 8 and 8a). The mean population abundance $\bar{N}(t)$ grows exponentially from an initial value of 150 until reaching roughly 1200. In the second scenario, S2-NC, the mean population abundance knows a significant peak in the first year of the simulation, then, it recognizes a slight fluctuation in

<figure><img src="image_11.png" /><figcaption>(a) Population abundance per replica for S1-NC</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(c) Standard deviation $σ$ over the mean per replica for S1-NC</figcaption></figure>

<caption>(d) Standard deviation $σ$ over the mean per replica for S2-NC</caption>

**Fig. 7** Total population abundance and standard deviation $σ_i$, where *i* is the replica index, over the mean, per replica respectively for the first scenario with no carrying capacity applied (S1-NC) and for the second scenario with no carrying capacity applied (S2-NC)
<figure><img src="image_13.png" /><figcaption>(a) Mean total population abundance (S1-(b) Mean total population abundance (S2-NC)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c) Monthly per-stage population abundance</figcaption></figure>

**Fig. 8** Mean total population with no carrying capacity (NC) and Monthly per-stage population, respectively, for the first scenario with no carrying capacity applied (S1-NC) and for the second scenario with no carrying capacity applied (S2-NC)

the yearly maximum, after which it reaches two new peaks, respectively, in the ninth and tenth year of the simulation (Fig. 8b). For the first set of simulations (S1-NC and S2-NC), the total population has an increasing trend based on the Mann-Kendall Trend test ($p_{value} < 0.05$).

Both simulation sets have roughly the same per-stage population distribution (refer to Fig. 8c). The egg population knows a peak in early summer as expected (Ogden et al. 2004, 2005). Larva population emerged after the rise of the egg population in midsummer, followed by a peak of the nymph population in late summer. The adult population increases in early autumn and stays steadily high until late spring.

## Host carrying capacity effect

We varied the carrying capacity of the host agent for both S1 and S2 and analysed the mean total population abundance over 10 years (Fig. 9). For every simulation set, the carrying capacity of host values is fixed. For low values of the parameter (refer to Fig. 9a
<figure><img src="image_15.png" /><figcaption>(a) S1- carrying capacity = 2</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(b) S2- carrying capacity= 2</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(c) S1- carrying capacity = 6</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(d) S2- carrying capacity= 6</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(e) S1- carrying capacity= 8</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(f) S2- carrying capacity= 8</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(g) S1- carrying capacity= 10</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(h) S2- carrying capacity= 10</figcaption></figure>

**Fig. 9** Mean population abundance for both scenarios S1 and S2 for the different carrying capacity parameter values. The evolution of the mean population abundance is inversely proportional to the carrying capacity
and 9b), representing a low number of vectors able to attach to the host, the average population decreases over the simulation time towards extinction. For higher values (refer to Fig. 9c and 9d), the population decreases less steeply. A stabilization begins to take hold for both scenarios when the carrying capacity is equal to 8 (refer to Fig. 9e and 9f). For the extreme value of the parameter, the population behaviour is similar to that of the scenario where the host's capacity limit is not applied in the first scenario (refer to Fig. 7).

### Scenario 1: Random pattern

We notice, as expected that the mean population abundance increases as the *parasite_max* values increase (Fig. 10). We fitted the output into a linear least square regression for the 30th replicas for every parameter value (refer to Figure 10a).

For S1, the least square coefficient increases gradually with the parameter value (refer to Figure 10b). Also, we notice the $R^2$ is the highest for the simulation set S2, as well as for the minor square coefficient, is the lowest for the carrying capacity is equal to 7 ($R^2$ = 0.004), then it is the value that best regulates our vector population (Figure 10c).

### Scenario 2: Herd movement pattern
The same approach used for the first scenario, S1, is followed in the second scenario (Fig. 10a and 11b). However, for the coefficient $R^2$, we notice the best value for the carrying capacity is equal to 10 (Figure 11c).

<figure><img src="image_23.png" /><figcaption>(a) Standard deviation of replicas for every parameter value</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(b) Least square coefficient for S1</figcaption></figure>

<caption>(c) $R^2$ for S1</caption>

<caption>**Fig. 10** The figure shows the standard deviation and the fitting of replicas for the first scenario S1.a Standard deviation of replicas for each parameter value for S1.b,c Least square fitting of replicas for each parameter value for S1</caption>
<figure><img src="image_25.png" /><figcaption>(a) Standard deviation of replicas for every parameter value</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(b) Least square coefficient for S2</figcaption></figure>

<caption>(c) Least square R² for S2</caption>

<caption>**Fig. 11** The figure shows the standard deviation and the fitting of replicas for the second scenario S2. **a)** Standard deviation of replicas for each parameter value for S2. **b, c)** Least square fitting of replicas for each parameter value for the second scenario S2</caption>

## Discussion

We developed an agent-based model to investigate the dynamics of tick populations. The model primarily aims to evaluate host movement patterns on tick population dynamics. We monitored annual and seasonal population fluctuations and population growth tendencies. We tested different host movement patterns while varying the carrying capacity of the host. The outcome of the different simulation scenarios suggests that the movement pattern of the host does not affect the seasonality of the vector agent, although it impacts its population abundance. Since the random movement pattern allows a higher likelihood for the host agent to be located within the perception distance of the vector agent, driving vectors, in a cascade effect, to execute the remaining processes and produce offspring. Furthermore, our model introduces an essential dimension by incorporating resource competition, mainly through host availability, as a significant regulatory factor influencing vector populations. This approach enhances the generalization and formalizes the intricate relationship between vectors and hosts, making our model adaptable to a broad spectrum of vector and host species across diverse biogeographic regions. The most obvious finding to emerge from analyzing the carrying capacity of the host, on the one hand, is the exponential growth of the population when increasing the capacity of the host to
attach vectors approaching the results of S1, and on the other hand, the extinction of the population when low carrying capacity is fixed, limiting the vector agents to continue other processes and fulfill the next life state. Over ten years of simulations, we note that the population abundance shows a decreasing behaviour for both S1 and S2 while applying low carrying capacity parameter values. While the carrying capacity of the host increases, the population knows a gradual increase and tends to stabilize over time for both S1 and S2 until reaching a turning point for the simulation, where the population behaves as the one in S1 when no carrying capacity is applied.

So far, our work has focused on host movement patterns' effect on vector population on the one hand, and on the other hand, it focused on the grooming effect of the host illustrated in our model by the carrying capacity parameter.

This model complements studies e.g. Watts et al. (2018); Chen et al. (2022); O'Neill et al. (2023) by incorporating temperature-dependent vector developmental stages, host-vector interactions, and the influence of host movement on tick species characterized by passive host-seeking behaviour, exemplified by *Ixodes ricinus*. Our model's uniqueness lies in its ability to encompass the entire tick life cycle, representing development as a temperature-dependent process and the sole interaction between vector and host through blood meals. Our model's hypotheses exhibit a robust capacity to replicate the seasonal dynamics of the vector population. Remarkably, this is achieved by considering only temperature as the regulating factor governing transitions between developmental stages as Randolph (2004); Ogden et al. (2004) highlighted. While our model's assumptions are straightforward, they have proved essential in reproducing the tick's whole life cycle. Moreover, our model affords us a comprehensive descriptive framework for understanding the tick's life cycle, integrating physiological processes (illustrated by vector life state transitions and behavioural states of each instar) and ecological processes (captured through host availability for blood meals).

Also, our model seeks to explore the tangled mesh between ecological interactions at the individual level and their impact on physiological processes, particularly the development time of each life stage; the individual dependence on the temperature to develop will create a seasonality. This objective aligns with the distinction of various scales in ecology, as the scale of matter, from cellular processes to ecosystems, with individuals as the key link. This illustration of ecological scales facilitates the subsequent delineation of other dimensions, such as temporal and spatial scales ...(Maàroufi et al. (2014); Louati (2017); Ellison and Gotelli (2021)).

The issues related to changes in scale levels are relevant, and coupling different scales in modelling is a challenge. Prior studies have underscored the importance of agent-based modelling in elucidating the complexities of vector-host systems (e.g. Nolzen et al. (2022); Halsey and Miller (2018); Tardy et al. (2022)). These studies have predominantly focused on landscape characteristics, particular tick species, and host-specific models. Unlike its counterparts, our model does not rely on actual landscapes or the real movement of hosts; instead, it delves into a theoretical exploration. In doing so, it allows to examine intriguing ecological phenomena, such as the impact of temperature on developmental processes, in a controlled and hypothetical environment.

Our work utilizes empirical data outlined in Table 1 to enhance its adaptability across a diverse range of case studies and contexts. We take a comprehensive approach to
understanding the life cycle of tick species, considering their wide geographical distribu-
tion and varied phenological patterns. A key highlight of our contribution is its implicit
recognition of the role of host presence in influencing tick population dynamics, par-
ticularly in species with significant geographical expansion and diverse phenological
patterns. Our contribution underscores the significant role of host presence in shaping
tick population dynamics. This aspect is seamlessly integrated into our model through
the parameter of host carrying capacity, showcasing its adaptability in simulating the
complexities of tick populations and their interactions with hosts. This understanding
leads to another valuable application of our model: evaluating the capacity of hosts to
sustain tick populations. Such evaluations are particularly pertinent when considering
various control methods, both on-host and off-host, such as the application of acaricides
or host population management. These methods are aimed at reducing tick numbers,
especially nymphs and adult ticks that may pose risks to livestock. In addition, the pos-
sibility of extending the model to other tick species, it also has a potential for generaliza-
tion to include other vector species, such as the Anopheles mosquitoes responsible for
malaria transmission. The fundamental interactions between anopheles and their hosts
closely resemble those observed with ticks. However, a distinctive challenge arises when
selecting an appropriate scale representing mosquito movement. When addressing mos-
quitoes, we can describe them as individual clusters exhibiting synchronous movement,
collectively representing a single agent. Alternatively, we can model them as individual
mosquitoes acting as individual agents, a concept already outlined in our model.

Furthermore, a compelling application of our model is its capacity to assess the impact
of temperature variations as a primary driver influencing tick life cycles (see Sect. 2.2).
This includes predicting population densities and conducting comparative analyses
between outbreeding and inbreeding scenarios within stable livestock populations.

Conclusion

This work introduces an initial model designed to conceptualize the dynamics of the
vector-host system, specifically focusing on investigating the impact of both host move-
ment patterns and host carrying capacity on short-term vector population dynamics.
The model's hypotheses are aligned within the framework of a hypothetical landscape
encompassing host and vector populations while considering the vector physiological
development as a temperature-dependent process. The simulation experiments carried
out in our work substantiate the importance of host-carrying capacity in the dynamics
of the vector population and contribute to our understanding of these complex inter-
actions. One of the more significant conclusions to emerge from this work is that the
temperature can regulate vectors' life cycle and keep the vector population dynamics
seasonality. Moreover, the host movement behavior has no effect on the vector dynam-
ics. A natural progression of this model is to validate the findings with empirical data
and extend it to cover more environmental factors and include vector-borne diseases,
since our model possesses the versatility to explore a variety of host- and environment-
targeted control strategies, facilitating the identification of practical approaches for
managing vector population dynamics in a broader ecological context.
**Acknowledgements**

This work was carried out as part of a doctoral thesis funded by the International Doctoral Program in Modeling Complex Systems of the French Institute for Research and Development, "IRD". The authors would like to thank the "Mésocentre de calcul de Franche-Comté" for its support.

**Author contributions**

C.C. co-conceived the conceptualization and methodology and co-performed the computation, writing and the original draft preparation. N.M.: co-performed the computation, the validation and the reviewing and co-supervised the work S.B.M.: co-conceived conceptualization and methodology, co-performed the writing, reviewing and editing and co-supervised the work. All authors discussed the results and contributed to the final manuscript. All authors reviewed the manuscript.

**Funding**

This work was carried out as part of a doctoral thesis funded by the International Doctoral Program in Modeling Complex Systems of the French Institute for Research and Development, "IRD".

**Availability of data and materials**

Not applicable.

**Declarations**

**Conflict of interest**

All authors declare no Conflict of interest.

**Ethics approval and consent to participate**

Not applicable.

**Consent to participate**

Not applicable.

**Consent for publication**

All authors consent to the publication of the manuscript.

Received: 25 October 2023 Accepted: 18 May 2024

Published online: 08 July 2024

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