Mathematical model for pest-insect control using mating disruption and trapping

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Environmental management

Environmental management includes a wide range of interventions to reduce the abun-dance of pest-insects populations. A way to reduce the damages of a pest is by transforming the pest’s habitat through environmental modifications to produce unfavourable conditions for its establishment. In the case of fruit flies or moths, removing and destroying the fallen fruits and/or mummies allow to interrupt the development and growth of the eggs and lar-vae and limit the number of new emerging adults susceptible to cause damage [106, 232]. Mosquitoes are highly abundant in urban areas due to the availability of hosts for blood meals, and the wide range of potential breeding sites. Mosquito breeding sites are typically small and shallow with stagnant water which oﬀer a wide range of potential breeding sites sometime diﬃcult to find. Breeding sites can be natural (leafs retaining water, puddles, etc.) as well as domestic (plant saucers, tires, ornamental bird baths , buckets, clogged gutters, any objects susceptible of retaining water)(Figure 1.7) [122]. Measures consisting in reducing the number of breeding site have been recommended at least since the 1910’s in order to prevent transmission of Malaria [249, 210]. However, such a method cannot be eﬀective without the awareness and help of the local population to clean there own habitat. Thus, environmental management has the advantage of being environmentally friendly as it only aﬀects the pest itself, with no risk for the producers and consumers. However, modifying the environment is often diﬃcult to put into practice as it requires a considerable manpower. In addition, to obtain significant results, a good understanding on the ecology of the pest is essential, in particular its distribution and seasonal variations.

Physical control

Physical control aims at limiting contacts between the pest and its host. For example, mosquito nets can be set to prevent people from being bitten. To prevent fruit fly damages in crops, fruits can be bagged (Figure 1.8) to prevent females from accessing the hosts to lay eggs. Another way to reduce contact between the host and the pest is to adapt agricultural practices. For instance, early harvesting of the fruits can also prevent fruit fly damages, as for the olive fruit fly [208]. Although physical control presents no risks for the environment, its implementation is diﬃcult on a large scale due to its local action and no direct reduction in the pest population abundance.

Biological control

Biological control consists in involving other living organisms such as natural predators, parasites, or parasitoids to maintain the abundance of the pest population below acceptable (health and economical) level. Parasitoids are organisms that “live” on another one taking resources from a single host. While parasitoids usually kill there host, it is not necessarily the case for parasites [107]. For instance, weaver ants (Oecophylla longinoda) have been used in controlling mango fruit fly in Benin [243]. As for the control of the false codling moth, there are 25 known natural enemies present in Southern Africa [232]. Particular attention has been given to the egg parasitoid Trichogrammatoidea cryptophlebia for commercial biological con-trol [232]. Concerning the biological control of mosquitoes, predation on the aquatic stage is usually more eﬀective than on the adult stage due to its limited spatial displacements. The most important predators for aquatic mosquitoes are species of fish, such as the Gam-busia aﬃnis fish or the Poecilia reticulata fish, respectively native from Central America and South America. These have been used for decades for mosquito biological control [154]. To a smaller extent, some birds species use mosquito larvae as a source of food such as the Anas platyrhynchos duck. Few birds actually feed on mosquitoes as they are usually day-light hunters, thus their hunting periods typically do not coincide with mosquitoes activities. On the other hand, bat are night-hunters and represent a higher predation risk for mosquitoes. Species of amphibians, like newts and their larvae are also considered as important predators for immature mosquitoes. Other predators can be mentioned, like the flatworm Turbellaria, some spices of spiders and mites or crustaceans, some insects (dragonflies, heteroptera, tri-choptera, diptera etc.) and parasites (nematodes). Although macro organisms, like fish or nematodes, have been widely used for mosquito control, they are often diﬃcult to rear and can only survive in specific ecological environments which makes their use in biological control programs diﬃcult and expensive. Further, a wide variety of pathogens have been discovered over the past decades which represent important risk for the mosquito population. For more extensive details on the predators of mosquitoes, the reader shall refer to the book of Becker. [24, Chapter 16].
As part of biological control, one can mention the replacement of dengue-vector Aedes mosquito population by a population of mosquitoes unable to transmit the disease using the intracellular bacterium Wolbachia which interferes with the transmission of the disease pathogen. Alternatively, Wolbachia can also be used to control a population via Cytoplasmic Incompatibility (IC). In this case, males infected with Wolbachia become unable to repro-duce successfully with uninfected females or with females infected with another strain of Wolbachia [251]. The method consists in releasing large amounts of mosquitoes infected by Wolbachia which contaminate wild mosquitoes leading to a decline of the abundance of mosquitoes able to transmit the disease [129, 130]. Further, mathematical modelling ap-proaches have shown to provide encouraging results for the successful control of a vector population using Wolbachia (see [97, 151, 29, 213] and references therein).

The sterile insect technique

Another branch of control goes back from the 1950’s with the work of Knipling [146, 148] as he discovered that the abundance of mosquitoes could be reduced if females mate with sterile males. In this context, the sterile insect technique (SIT) is a promising control method for pest-insect populations. SIT consists in releasing a large number of reared sterilized males to compete with wild males for female insemination. Females inseminated by sterile males produce non-viable oﬀspring, leading to the reduction of the mosquito population [147, 155]. SIT is of great interest as it targets exclusively the insect to control and has no detrimental eﬀect on the surrounding environment. SIT has shown to be a successful method in various cases [148], for example, in the control of the mosquito Aedes albopictus in Italy [25]. SIT has also been considered for the control of the false codling moth in South Africa [131], or for the control of Bactrocera dorsalis fruit fly [92]. SIT is a promising alternative as it enables producers to meet the strict sanitary international exportation requirements (no pesticides residues on fruits). However, to be successful, SIT requires an acute understanding on the insect ecology and biology.

Characteristics of spatio-temporal models

Diﬀerent aspects can be considered to categorize spatio-temporal models [44, 112]. We follow here the characterisation given by [44]. The first characteristics that can be men-tioned is on whether the space component is taken into account implicitly in the model, or explicitly [66, 112]. In implicit spatial models the population is assumed to be spatially structured, however the model does not incorporate topographic localisations. These models are relatively simple but do not allow to study the eﬀect of the structure of the environment on the dynamics of the population. On the other hand, in explicit models the individuals of the population are allocated a topographic localisation. This allows to combine the dynamics of the population to geographic data, such as maps of the habitat. Therefore, explicit models allow to account for the complexity of the environment and obtain a better understanding on the ecology of a population.
Such models can simulate the trajectories of the individuals in a population. These are Individual-Based Models (IBM). The trajectory of each individual is governed by specific rules to take into account the variability of the individuals interactions with other mem-bers and the environment. An extensive review of applications of IBM models can be found in [75]. In particular, it has been used to model the dynamics of the Large Blue Butterfly, Maculinea arion in the framework of species conservation [111]. IBMs can be very useful to model complex processes. It has been used and analysed for instance to investigate dynamics of a prey-predator system of populations structured in age and space [174]. However some disadvantages of such models are that they are computationally intensive and that it is neces-sary to have a very acute understanding and knowledge of the insect’s behaviour to program and simulate the individual dynamics. Indeed, not only they require specific instructions for each individuals, but a large number of individual trajectories need to be simulated to obtain information at the population level [26].
Alternatively, models can account for the whole population assuming homogeneity among individuals using population models, also known as mass-interaction models. Popula-tion models are analytical and do not account for variability among individuals. It can be assumed, for example, that all individuals are identical [26]. Thus, population models are suitable for the study of an averaged behaviour of the population. Such models can be useful to study interactions between populations or within a population [66, 237]. This approach is particularly suitable for the modelling of abundant populations, such as insects [83]. Models discussed below in sections 1.4.2.2 and

are examples of population models.

Further, models can be categorised as deterministic or stochastic. Deterministic models simulate the average behaviour of the population without taking into account variability in the parameters or in the functions describing the model. Such an approach is particularly suitable to model the dynamics at a population level. On the opposite, stochastic models the dynamics are governed by probability function to account for variability which makes them more appropriate to model dynamics of small populations [85]. Although stochastic models are more realistic, they are also more diﬃcult to study theoretically.
Another distinction that can be made among these models is on the discrete or continuous nature of the space variable. In discrete-space models, the space is divided in separate areas with an index referring to its position [84], while continuous-space models use the spatial coordinates of each location [66].
Considering the possible combinations of the characteristics presented above gives rise to a wide variety of models. The choice of the model depends on the population as well as on the aim of the model. We now give some examples of commonly used models with applications.

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Metapopulation models

When the space variable is taken into account implicitly, temporal models at specific lo-cations can be constructed with location-specific parameters and interactions between them, while in the explicit case, the model incorporates a space variable. This is the case of metapopulation models, first formulated by R. Levin in 1969, where the space is dis-cretized in distinct patches with their own specificities on which temporal dynamics are described [117, 118]. The spatio-temporal model is then a coupling of the temporal mod-els on each of the patches with interactions between them. Metapopulation models can be formulated mathematically by ODE or Diﬀerence Equations. Metapopulation models have been used, for example, to study host-parasit interactions in patchy environments [120]. In particular it has been used to describe the spatio temporal dynamics of the pathogen Wol-bachia (used for the control of Aedes mosquitoes) in a age-structured host population [116]. Metapopulation models have also been used to study the spread of Malaria in a vector-host population structured according to its infectious state [11, 10]. Another example of application of metapopuation model is to model the adaptation of fruit flies in a patchy environment [221].

Cellular-automata models

In cellular automata models the space explicit and discretized in a regular grid of cells. Each cell is attributed a state chosen among a fine set of states which is defined as a function of the states of a finite number of neighbouring cells. Cellular automata models can be seen as an extension of the metapopulation model proposed by Levin where the cells are patches with topographic locations [237]. A cellular automata is suitable for IBM representation where a cell may contain at most a single organism and the model can simulate local dispersal to the neighbouring cells emitted from a specific release cell. The rules which define the state of each cell may be stochastic or not. FlySim is a cellular automata program developed to simulate life cycles of the olive fruit fly [199]. In [223], a cellular automata model is used for the control of Chagas disease by modelling the dynamics of its insects vector.

Advection-Diﬀusion-Reaction

In the prospect of modelling pest-insect population dynamics, a particular attention is given to Advection-Diﬀusion-Reaction (ADR) models governed by PDEs. These are deterministic mass-interaction models continuous in time and space, thus they account for an average behaviour of the population. In ADR models, the diﬀusion process accounts for the random dispersal of the individuals of the population, without considering any sort of stimulus to direct their movements [74, 190]. The advection process accounts for directional displacements due to a flow, like wind transport, or attractiveness to a point (food, breading site, attractive traps…). Finally the reaction process governs the demography. In addition, in order to account for some heterogeneity within the population, the individuals can be grouped according to their age class or development stage. In this case the models becomes a system of coupled ADR equations where the reaction process also accounts for interactions between the diﬀerent compartments.
ADR models are continuous in space (x) and time (t) and are formulated in terms of partial diﬀerential equations [112]. The general form of a ADR equation is given as follows: ∂u(t, x) ∂u(t, x) ∂2u(t, x) = a ) + D + f(t, x), (1.9).
where u(t, x) represents the population density at time t and position x, the parameters D and a are respectively the diﬀusion rate and the advection force, and the function f is the reaction term. The population ADR models can be derived from random walk processes at the individual level using Taylor’s expansions. This derivation is detailed in several publications [27, 188, 238]. Alternatively, it is worth mentioning that the diﬀusion processes can also be derived using flux considerations instead of random walk processes [87, 238]. In the latter case, the derivation is based on the physics conservation law and the Fick’s law of diﬀusion which connects fluxes of particles to their gradient [61]. The derivation of models from individual level to the population level is a challenge of its own as the dynamics of the individuals are typically more complicated than a simple non-isotropic random walk. For example, in [200], a chemotaxis population model is derived from individual behaviours, for two interacting populations where one is a stimulus of the other.

Advection-Diﬀusion-Reaction models

In the models of chapters 3 and 4, we consider the spatio-temporal variations of a concen-tration or abundance of insects governed by ADR equations. This concentration is denoted u(t, x) and defined on a time-space domain ΩT = [0, T ]×Ω, where T ∈ R+ and Ω is a domain in R2 with a piecewise smooth boundary = ∂Ω. The ADR equation can be written in the general form: ∂u + Lu = f, in ΩT , u(0, x) = u0, (2.6) ∂t with 2 di,j(t, x) + 2 X ∂ ∂u X ∂u i,j=1 ∂xj ∂xi ∂xi Lu = − i=1 ai(t, x) + c(t, x)u. (2.7).
Further, to complete the formulation of the problem, we provide information on the dy-namics at the boundary, , of the domain. We can distinguish two main types of boundary conditions:
• Dirichlet: u(t, x) = α(t, x), x ∈ . Here the value is defined on the boundary.
Example of application: if we consider a domain out of which individuals cannot survive, than we take α(t, x) = 0.
• Neumann: ∂n∂ u(t, x) = g(t, x), x ∈ , where n denotes the outward normal vector to . Here the flux in and out of the domain is determined. For example, if we consider an isolated domain with no movement of individuals in and out of the domain, than we take g(t, x) = 0.
However, other boundary conditions can be formulated. For instance the Robin boundary condition is a combination of Dirichlet and Neumann, where the flux at the boundary depends on the density of u at the boundary. It is also possible to consider diﬀerent types of boundary conditions on diﬀerent parts of .

Table of contents :

1 Introduction
1.1 General context
1.2 Some insect species of interest
1.3 Control methods
1.3.1 Chemical control
1.3.2 Environmental management
1.3.3 Physical control
1.3.4 Biological control
1.3.5 The sterile insect technique
1.3.6 Genetic control
1.3.7 Behaviour disruption
1.4 Modelling population dynamics
1.4.1 Structured population models
1.4.2 Spatio-temporal models
1.4.2.1 Characteristics of spatio-temporal models
1.4.2.2 Metapopulation models
1.4.2.3 Cellular-automata models
1.4.2.4 Advection-Diffusion-Reaction
1.5 Estimating insect population size
1.5.1 Sampling
1.5.2 Mark-Release-Recapture
1.5.2.1 Assumptions
1.5.2.2 Some common methods
1.5.3 Trapping model approach
1.6 Modelling trapping
1.6.1 Spatially explicit trapping
1.6.2 Spatially implicit trapping
1.7 Aim and objectives of this thesis
2 Mathematical models and methods
2.1 Introduction
2.2 General setting for the models
2.3 Advection-Diffusion-Reaction models
2.3.1 Preliminaries
2.3.2 Variational formulation
2.3.3 Existence and uniqueness
2.3.4 Regularity of the weak solution
2.3.5 The maximum principle
2.3.6 Properties of the weak solution
2.3.7 Asymptotic behaviour
2.3.7.1 Existence and uniqueness
2.3.7.2 Steady state solutions
2.3.8 Application to single PDE model of Chapter 3
2.3.8.1 Variational formulation
2.3.8.2 Existence and uniqueness
2.3.8.3 Positivity of the solution
2.3.9 Application to the system of PDEs of Chapter 4
2.3.9.1 Variational formulation
2.3.9.2 Study of c: the stationary problem
2.3.9.3 Study of c: the evolution problem
2.3.9.4 Study of uf
2.4 Dynamical systems defined by a system of ODEs
2.4.1 Asymptotic properties
2.4.2 Monotone dynamical systems
2.5 Numerical methods for differential equations
2.5.1 Introduction
2.5.2 The Method of Lines and basic concepts
2.5.3 Finite difference method
2.5.3.1 Approximation of the 1st and 2nd space derivatives .
2.5.3.2 The finite difference scheme
2.5.4 Finite element method
2.5.4.1 The Galerkin method
2.5.4.2 The Finite Elements
2.5.4.3 Application to the problem of Chapter 4
2.6 The time discretization
3 Parameter identification in population models for insects using trap data
3.1 Abstract
3.2 Introduction
3.3 The Insect Trapping Model: The Direct Problem
3.4 The Parameter Identification Problem
3.5 Description of the experiments
3.6 Results and Discussion
3.6.1 Do interfering trap-settings provide better results than noninterfering trap settings?
3.6.2 Interfering trap-setting strategies
3.7 Conclusion
4 Simulations and parameter estimation of a trap-insect model using a finite element approach
4.1 Abstract
4.2 Introduction
4.3 The model
4.3.1 Modelling the spread of the chemical attractant
4.3.2 Modelling the insects’ dynamics
4.3.3 The coupled model
4.4 The variational formulation of the problem
4.5 Some qualitative properties
4.6 The numerical scheme
4.6.1 The semi-discrete approximation
4.6.2 The full discretization
4.7 Numerical simulations
4.7.1 Description of the experiments and mesh generation
4.7.2 Numerical simulation for the attractant
4.7.3 Numerical simulation for the insects
4.7.3.1 Using u0 homogeneous
4.7.3.2 Using a heterogeneous distribution for u0
4.8 Application to parameter identification
4.8.1 The parameter identification protocol
4.8.2 Description of the numerical experiments and simulations
4.8.3 Results and discussion
4.8.3.1 STEP 1: Identification of u
4.8.3.2 STEP 2: Identification of and
4.8.3.3 STEP 3: Identification of U
4.9 Conclusion
5 Mathematical model for pest-insect control using mating disruption and trapping
5.1 Abstract
5.2 Introduction
5.3 The compartmental model for the dynamics of the insect
5.3.1 Theoretical analysis of the model
5.3.1.1 Case 1: Male abundance
5.3.1.2 Case 2: Male scarcity
5.3.1.3 Conclusions for model (5.1)
5.4 Modelling mating disruption and trapping
5.4.1 Mating disruption and trapping
5.4.2 The model
5.4.3 Theoretical analysis of the control model
5.4.3.1 Properties of the equilibria in the male abundance region ( M > Y + YP )
5.4.3.2 Properties of the equilibria in the male scarcity region ( M < Y + YP )
5.4.3.3 Global asymptotic stability of the trivial equilibrium for sufficiently large YP
5.5 Numerical Simulation and Discussion
6 Conclusion and perspectives
6.1 Overview
6.2 Major findings of the thesis
6.2.1 Construction of trap-insect models
6.2.2 Protocol for the estimation of population parameters
6.2.3 Identification of thresholds for mating disruption and trapping control
6.3 Limits and perspectives