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In this paper we revisit the notion of grouped dispersal that have been introduced by Soubeyrand and co-authors \cite{soubeyrand2011patchy} to model the simultaneous (and hence dependent) dispersal of several propagules from a single source in a homogeneous environment. We built a time continuous measure valued process that takes into account the main feature of a grouped dispersal and derive its infinitesimal generator. To cope with the mutligeneration aspect associated to the demography we introduce two types of propagules in the description of the population which is one of the main innovations here. We also provide a rigorous description of the process and its generator. We derive as well, some large population asymptotics of the process unveilling the degenerate ultra parabolic system of PDE satisfied by the density of population. Finally, we also show that such a PDE system has a non-trivial solution which is unique in a certain functional space.

The Stochastic Partial Differential Equation (SPDE) approach, now commonly used in spatial statistics to construct Gaussian random fields, is revisited from a mechanistic perspective based on the movement of microscopic particles, thereby relating pseudo-differential operators to dispersal kernels. We first establish a connection between L\'evy flights and PDEs involving the Fractional Laplacian (FL) operator. The corresponding Fokker-Planck PDEs will serve as a basis to propose new generalisations by considering a general form of SPDE with terms accounting for dispersal, drift and reaction. We detail the difference between the FL operator (with or without linear reaction term) associated with a fat-tailed dispersal kernel and therefore describing long-distance dependencies, and the damped FL operator associated with a thin-tailed kernel, thus corresponding to short-distance dependencies. Then, SPDE-based random fields with non-stationary external spatially and temporally varying force are illustrated and nonlinear bistable reaction term are introduced. The physical meaning of the latter and possible applications are discussed. Returning to the particulate interpretation of the above-mentioned equations, we describe in a relatively simple case their links with point processes. We unravel the nature of the point processes they generate and show how such mechanistic models, associated to a probabilistic observation model, can be used in a hierarchical setting to estimate the parameters of the particle dynamics.

Raw data on the cumulative number of deaths at a country level generally indicate a spatially variable distribution of the incidence of COVID-19 disease. An important issue is to determine whether this spatial pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we fit them to French mortality data with a mixed probabilistic-ODE approach. Using standard statistical criteria, we select the model with non-local transmission corresponding to a diffusion on the graph of counties that depends on the geographic proximity, with time-dependent contact rate and spatially constant parameters. This original spatially parsimonious model suggests that in a geographically middle size centralized country such as France, once the epidemic is established, the effect of global processes such as restriction policies, sanitary measures and social distancing overwhelms the effect of local factors. Additionally, this modeling approach reveals the latent epidemiological dynamics including the local level of immunity, and allows us to evaluate the role of non-local interactions on the future spread of the disease. In view of its theoretical and numerical simplicity and its ability to accurately track the COVID-19 epidemic curves, the framework we develop here, in particular the non-local model and the associated estimation procedure, is of general interest in studying spatial dynamics of epidemics.

The first cases of COVID-19 in France were detected on January 24, 2020. The number of screening tests carried out and the methodology used to target the patients tested do not allow for a direct computation of the real number of cases and the mortality rate.In this report, we develop a 'mechanistic-statistical' approach coupling a SIR ODE model describing the unobserved epidemiological dynamics, a probabilistic model describing the data acquisition process and a statistical inference method. The objective of this model is not to make forecasts but to estimate the real number of people infected with COVID-19 during the observation window in France and to deduce the mortality rate associated with the epidemic.Main results. The actual number of infected cases in France is probably much higher than the observations: we find here a factor x 15 (95%-CI: 4-33), which leads to a 5.2/1000 mortality rate (95%-CI: 1.5 / 1000-11.7/ 1000) at the end of the observation period. We find a R0 of 4.8, a high value which may be linked to the long viral shedding period of 20 days.

Insect-borne diseases are diseases carried by insects affecting humans, animals or plants. They have the potential to generate massive outbreaks such as the Zika epidemic in 2015-2016 mostly distributed in the Americas, the Pacific and Southeast Asia, and the multi-foci outbreak caused by the bacterium {\it Xylella fastidiosa} in Europe in the 2010s. In this article, we propose and analyze the behavior of a spatially-explicit compartmental model adapted to pathosystems with fixed hosts and mobile vectors disseminating the disease. The behavior of this model based on a system of partial differential equations is complementarily characterized via a theoretical study of its equilibrium states and a numerical study of its transitive phase using global sensitivity analysis. The results are discussed in terms of implications concerning the surveillance and control of the disease over a medium-to-long temporal horizon.

The movement of atmospheric air masses can be seen as a continuous and complex flow of particles hovering over our planet. It can however be locally simplified by considering three-dimensional trajectories of air masses connecting distant areas of the globe during a given period of time. In this paper, we present a mathematical framework to construct spatial and spatiotemporal networks where the nodes are the subsets of a partition of a geographical area and the links between these nodes are inferred from sampled trajectories of air masses passing over and across the nodes. We propose different estimators of link intensities relying on different bio-physical hypotheses and covering adjustable time periods. This approach leads to a new class of spatiotemporal networks characterized by adjacency matrices. We applied the approach in two real geographical contexts: the watersheds of the French region Provence-Alpes-C\^ote d'Azur and the coastline of the Mediterranean Sea. The analysis of the constructed networks allowed identifying a marked seasonal pattern in air mass movements in the study areas. These constructed networks can be used to investigate issues, e.g., in aerobiology and epidemiology of airborne plant pathogens. Similar networks could be estimated from other types of trajectories, such as animal trajectories.

For point patterns observed in natura, spatial heterogeneity is more the rule than the exception. In numerous applications, this can be mathematically handled by the flexible class of log Gaussian Cox processes (LGCPs); in brief, a LGCP is a Cox process driven by an underlying log Gaussian random field (log GRF). This allows the representation of point aggregation, point vacuum and intermediate situations, with more or less rapid transitions between these different states depending on the properties of GRF. Very often, the covariance function of the GRF is assumed to be stationary. In this article, we give two examples where the sizes (that is, the number of points) and the spatial extents of point clusters are allowed to vary in space. To tackle such features, we propose parametric and semiparametric models of non-stationary LGCPs where the non-stationarity is included in both the mean function and the covariance function of the GRF. Thus, in contrast to most other work on inhomogeneous LGCPs, second-order intensity-reweighted stationarity is not satisfied and the usual two step procedure for parameter estimation based on e.g. composite likelihood does not easily apply. Instead we propose a fast three step procedure based on composite likelihood. We apply our modelling and estimation framework to analyse datasets dealing with fish aggregation in a reservoir and with dispersal of biological particles.

Invasion of new territories by alien organisms is of primary concern for environmental and health agencies and has been a core topic in mathematical modeling, in particular in the intents of reconstructing the past dynamics of the alien organisms and predicting their future spatial extents. Partial differential equations offer a rich and flexible modeling framework that has been applied to a large number of invasions. In this article, we are specifically interested in dating and localizing the introduction that led to an invasion using mathematical modeling, post-introduction data and an adequate statistical inference procedure. We adopt a mechanistic-statistical approach grounded on a coupled reaction-diffusion-absorption model representing the dynamics of an organism in an heterogeneous domain with respect to growth. Initial conditions (including the date and site of the introduction) and model parameters related to diffusion, reproduction and mortality are jointly estimated in the Bayesian framework by using an adaptive importance sampling algorithm. This framework is applied to the invasion of \textit{Xylella fastidiosa}, a phytopathogenic bacterium detected in South Corsica in 2015, France.