The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Competing populations on fragmented landscapes with spatially structured heterogeneities: improved landscape generation and mixed dispersal strategies

Interactions between two species competing for space were studied using stochastic spatially explicit lattice-based simulations as well as pair approximations. The two species differed only in their dispersal strategies, which were characterized by the proportion of reproductive effort allocated to long-distance (far) dispersal versus short-distance (near) dispersal to adjacent sites. All population dynamics took place on landscapes with spatially clustered distributions of suitable habitat, described by two parameters specifying the amount and the local spatial autocorrelation of suitable habitat. Whereas previous results indicated that coexistence between pure near and far dispersers was very rare, taking place over only a very small region of the landscape parameter space, when mixed strategies are allowed, multiple strategies can coexist over a much wider variety of landscapes. On such spatially structured landscapes, the populations can partition the habitat according to local conditions, with one species using pure near dispersal to exploit large contiguous patches of suitable habitat, and another species using mixed dispersal to colonize isolated smaller patches (via far dispersal) and then rapidly exploit those patches (via near dispersal). An improved mean-field approximation which incorporates the spatially clustered habitat distribution is developed for modeling a single species on these landscapes, along with an improved Monte Carlo algorithm for generating spatially clustered habitat distributions.      

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.