Mathematical Modelling of Mosquito Dispersal in a Heterogeneous Environment.

Mosquito dispersal is a key behavioural factor that affects the persistence and resurgence of several vector-borne diseases. Spatial heterogeneity of mosquito resources, such as hosts and breeding sites, affects mosquito dispersal behaviour and consequently affects mosquito population structures, human exposure to vectors, and the ability to control disease transmission. In this paper, we develop and simulate a discrete-space continuous-time mathematical model to investigate the impact of dispersal and heterogeneous distribution of resources on the distribution and dynamics of mosquito populations. We build an ordinary differential equation model of the mosquito life cycle and replicate it across a hexagonal grid (multi-patch system) that represents two-dimensional space. We use the model to estimate mosquito dispersal distances and to evaluate the effect of spatial repellents as a vector control strategy. We find evidence of association between heterogeneity, dispersal, spatial distribution of resources, and mosquito population dynamics. Random distribution of repellents reduces the distance moved by mosquitoes, offering a promising strategy for disease control

Stochastic population growth in spatially heterogeneous environments: The density-dependent case

This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\infty)^n$. We prove that $r$, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter $r$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $r>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $r<0$, the population abundances of the patches converge almost surely to $0$ exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, $n=2$, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.Comment: 43 pages, 1 figure, edited according to the suggestion of the referees, to appear in Journal of Mathematical Biolog

A Bayesian space-time model for discrete spread processes on a lattice

Funding for this work was provided by GEOIDE through the Government of Canada’s Networks for Centres of Excellence program.In this article we present a Bayesian Markov model for investigating environmental spread processes. We formulate a model where the spread of a disease over a heterogeneous landscape through time is represented as a probabilistic function of two processes: local diffusion and random-jump dispersal. This formulation represents two mechanisms of spread which result in highly peaked and long-tailed distributions of dispersal distances (i.e., local and long-distance spread), commonly observed in the spread of infectious diseases and biological invasions. We demonstrate the properties of this model using a simulation experiment and an empirical case study - the spread of mountain pine beetle in western Canada. Posterior predictive checking was used to validate the number of newly inhabited regions in each time period. The model performed well in the simulation study in which a goodness-of-fit statistic measuring the number of newly inhabited regions in each time interval fell within the 95% posterior predictive credible interval in over 97% of simulations. The case study of a mountain pine beetle infestation in western Canada (1999-2009) extended the base model in two ways. First, spatial covariates thought to impact the local diffusion parameters, elevation and forest cover, were included in the model. Second, a refined definition for translocation or jump-dispersal based on mountain pine beetle ecology was incorporated improving the fit of the model. Posterior predictive checks on the mountain pine beetle model found that the observed goodness-of-fit test statistic fell within the 95% posterior predictive credible interval for 8 out of 10. years. The simulation study and case study provide evidence that the model presented here is both robust and flexible; and is therefore appropriate for a wide range of spread processes in epidemiology and ecology.PostprintPeer reviewe

Spreading Speed, Traveling Waves, and Minimal Domain Size in\ud Impulsive Reaction-diffusion Models

How growth, mortality, and dispersal in a species affect the species’ spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the populationat the end stage as a possibly nonmonotone function of the density of the population at the beginning of the stage. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species’ spreading speeds, traveling wave speeds, as well as and minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also determine an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results

Landscape attributes governing local transmission of an endemic zoonosis: rabies virus in domestic dogs

Landscape heterogeneity plays an important role in disease spread and persistence, but quantifying landscape influences and their scale dependence is challenging. Studies have focused on how environmental features or global transport networks influence pathogen invasion and spread, but their influence on local transmission dynamics that underpin the persistence of endemic diseases remains unexplored. Bayesian phylogeographic frameworks that incorporate spatial heterogeneities are promising tools for analysing linked epidemiological, environmental and genetic data. Here, we extend these methodological approaches to decipher the relative contribu- tion and scale-dependent effects of landscape influences on the transmission of endemic rabies virus in Serengeti district, Tanzania (area ~4,900 km2). Utilizing detailed epidemiological data and 152 complete viral genomes collected between 2004 and 2013, we show that the localized presence of dogs but not their density is the most important determinant of diffusion, implying that culling will be ineffec- tive for rabies control. Rivers and roads acted as barriers and facilitators to viral spread, respectively, and vaccination impeded diffusion despite variable annual cov- erage. Notably, we found that landscape effects were scale-dependent: rivers were barriers and roads facilitators on larger scales, whereas the distribution of dogs was important for rabies dispersal across multiple scales. This nuanced understanding of the spatial processes that underpin rabies transmission can be exploited for targeted control at the scale where it will have the greatest impact. Moreover, this research demonstrates how current phylogeographic frameworks can be adapted to improve our understanding of endemic disease dynamics at different spatial scales

Stochastic population growth in spatially heterogeneous environments

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^i\mu_i+\sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $\mu_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j \sigma_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_\infty$ as $t\to\infty$, and the stochastic growth rate $\lim_{t\to\infty}t^{-1}\log S_t$ equals the space-time average per-capita growth rate \sum_i\mu_i\E[Y_\infty^i] experienced by the population minus half of the space-time average temporal variation \E[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j] experienced by the population. We derive analytic results for the law of $Y_\infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into "ideal free" movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.Comment: 47 pages, 4 figure