Modelling arthropod active dispersal using Partial differential equations: the case of the mosquito Aedes albopictus
Graphical abstract
Introduction
Animal dispersal refers to movements away from the place of birth towards another location for reproduction. The main drivers of dispersal are related to the avoidance of kin competition and inbreeding and escaping deteriorating environmental conditions (Bowler and Benton, 2005). In the case of insects, assessing the active dispersal range might be of crucial importance particularly in the case of species which damage agricultural productions, unsettle ecosystems, and threat human health. Among more than 6 million species of insects known, only less than 100 are either important pests for major crops or relevant vectors of human and/or animal diseases. The deep knowledge of these species dispersal is instrumental to develop proper integrated pest management plans which maximize cost-effectiveness of interventions and protect the environment in a sustainable way.
Although many theoretical models are available, empirical studies are generally lacking due to the difficulties of linking observations to the quantification of dispersion (Tesson and Edelaar, 2013). Advanced satellite radio telemetry and acoustic are giving new opportunities to study dispersion of big size animals such as large mammals or sharks (Cagnacci et al., 2010; Spaet et al., 2020), but are less useful to quantify the dispersal of small animals. In the case of insects, Mark-release-recapture (MRR) is the most widely used technique for quantifying dispersal: specimens are first collected/reared and marked, then released from a single site and subsequently recaptured through traps placed at different distances in a given study area (Pollock et al., 1990).
The most common statistical approaches used to estimate dispersal from MRR data are regression techniques which aim to estimate the mean distance travelled (MDT) and the flight range (FRx) of a specified fraction x of the population, rather than the diffusion process of individual marked specimens. Other methods to estimate insect dispersal follow a Bayesian framework which explicitly models the diffusion process (Villela et al., 2015). For instance, the hierarchical Bayesian model proposed by Villela et al. (2015) for the mosquito Aedes aegypti takes advantage from the flexibility of the Bayesian approach and expands the frequentist approach by including three components: two probabilistic models, describing the spatial distribution of specimens and the daily survival of marked and native individuals, and an observation model describing the sampling process.
Partial Differential Equations (PDEs) represent a standard mathematical method to model diffusion processes, such as the gas dynamics and heat distribution (Borthwick et al., 2016). In ecology, PDEs have been applied to study spatial-temporal dispersal of animal populations in a continuous domain (Bassett et al., 2017; Kareiva et al., 1990), such as the home-range dynamics of meerkats (Suricata suricatta) (Bateman et al., 2015) and the dispersal of butterflies (Ovaskainen, 2004).
The aim of this work is to provide a PDE-based analytical method to estimate insect dispersion based on MRR field data. This method, compared to previously quoted ones, allows to estimate the dispersion taking into account the daily mortality of marked release insects and the days after release (time) in a single mathematical equation. In particular, we applied the proposed modeling framework to estimate the dispersal of blood-fed females of the tiger mosquito, Aedes albopictus, during the egg laying phase. This species represents a significant public health burden due to its capacity to transmit exotic arboviruses, such as dengue (DENV) and chikungunya (CHIKV), capable of induce serious diseases in humans (Zeller et al., 2016). The species was the primary responsible of the thousands of DENV cases recorded in the southwest of the Indian Ocean in 2015–2018 (Vincent et al., 2019) and of the first autochthonous cases of both viruses in Europe (Marrama Rakotoarivony and Schaffner, 2012), where it caused two large CHIKV outbreaks with hundreds of human cases (Italy 2007 and 2017; Caputo et al., 2020).
We expect that the proposed approach can be applied to quantify dispersal, and hence improve control of diseases transmitted by Ae. albopictus and by other mosquito species of global relevance (such as the major arbovirus vector, Ae. aegypti, and malaria vector species), as well as of agricultural pest species.
Section snippets
Modelling
Our model expands the one proposed in (Lutambi et al., 2013). Precisely, we included in the main equation the mortality and capture processes of the species of interest. Thus, the equation assumes the following form:Where μ is the mortality rate, is capture rate, (x, y) represent location coordinates as distance (in meters) along the x and y spatial axis from a given origin (x0, y0) respectively, t is the time (i.e., days or hours), M(x,y,t) is the density of the
Results
In this section we present: (i) the estimated values for the diffusion coefficients (D1, D2, D3 for MRR1, MRR2 and MRR3 respectively), the correction factor (ζ) and the daily mortality rate ( for the first, second and third semi-field experiments); (ii) the simulation of the dispersal process of the marked mosquitoes during five consecutive days after release; (iii) a validation of the mathematical-statistical model proposed here using the analytical solution of equation [2].
Discussion
Here we presented a PDE-based stochastic framework to estimate insect dispersal based on MRR data. We tested it in a specific case-study to overcome the limitations of the commonly used analytical approaches, i.e. regression analysis (Marini et al., 2019) and hierarchical Bayesian models (Villela et al., 2015). Indeed, the big advantage in the use of PDEs is the possibility of including time, space, and daily mortality in only one mathematical equation, thus providing a more realistic
Conclusions
The results here obtained are useful for the definition of the optimal buffer on which to focus emergency mosquito-borne virus control interventions (i.e. deployment of adulticides insecticides aimed at eliminating potentially infected mosquitoes in the area surrounding the residence of an arbovirus infected person). This information is crucial for public authorities, as it has already been shown that enlarging the size of the area to be treated and reducing the time interval between infective
Data availability statement
The R code and the data are available at https://github.com/Chia1992/Partial-Differential-Equation
CRediT Author Statement
Chiara Virgillito: Methodology, Data curation, Writing. Mattia Manica: Methodology, Data curation, Writing. Giovanni Marini: Methodology, Data curation, Writing. Beniamino Caputo: Collecting data, Writing. Alessandra della Torre: Collecting data, Writing-Reviewing. Roberto Rosà: Methodology, Data curation, Writing-Reviewing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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