The diffusive spread of alleles in heterogeneous populations

The spread of genes and individuals through space in populations is relevant in many biological contexts. I study, via systems of reaction-diffusion equations, the spatial spread of advantageous alleles through structured populations. The results show that the temporally asymptotic rate of spread of an advantageous allele, a kind of invasion speed, can be approximated for a class of linear partial differential equations via a relatively simple formula, c = 2 square root of (rD), that is reminiscent of a classic formula attributed to R. A. Fisher. The parameters r and D represent an asymptotic growth rate and an average diffusion rate, respectively, and can be interpreted in terms of eigenvalues and eigenvectors that depend on the population's demographic structure. The results can be applied, under certain conditions, to a wide class of nonlinear partial differential equations that are relevant to a variety of ecological and evolutionary scenarios in population biology. I illustrate the approach for computing invasion speed with three examples that allow for heterogeneous dispersal rates among different classes of individuals within model populations.     

Persistence and spread of stage-structured populations in heterogeneous landscapes

Journal of Mathematical Biology - Conditions for population persistence in heterogeneous landscapes and formulas for population spread rates are important tools for conservation ecology and...     

Eradication of infectious diseases in heterogeneous populations

We present a model of infectious diseases in heterogeneous populations, which allows for variable intra- to intergroup contact ratios. We give necessary and sufficient conditions for disease eradication by means of vaccination. Smallpox is used as an illustrative example.     

The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population. Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibrium. The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.     

On the definition and the computation of the type-reproduction number T for structured populations in heterogeneous environments

In the context of mathematical epidemiology, the type-reproduction number (TRN) for a specific host type is interpreted as the average number of secondary cases of that type produced by the primary cases of the same host type during the entire course of infection. Here, it must be noted that T takes into account not only the secondary cases directly transmitted from the specific host but also the cases indirectly transmitted by way of other types, who were infected from the primary cases of the specific host with no intermediate cases of the target host. Roberts and Heesterbeek (Proc R Soc Lond B 270:1359–1364, 2003) have shown that T is a useful measure when a particular single host type is targeted in the disease control effort in a community with various types of host, based on the fact that the sign relation sign(R ₀ ? 1) = sign(T ? 1) holds between the basic reproduction number R ₀ and T. In fact, T can be seen as an extension of R ₀ in a sense that the threshold condition of the total population growth can be formulated by the reproduction process of the target type only. However, the original formulation is limited to populations with discrete state space in constant environments. In this paper, based on a new perspective of R ₀ in heterogeneous environments (Inaba in J Math Biol 2011), we give a general definition of the TRN for continuously structured populations in heterogeneous environments and show some examples of its computation and applications.     

An exact procedure for the analysis of heterogeneous kinetics in polyclonal antibody populations

Antibody populations with heterogeneous binding properties exhibit complex first-order dissociation kinetics. An analytical method has been developed to determine the average dissociation rate constant and the heterogeneity index of a specific antibody population. This procedure was based on Laplace transformation of the gamma distribution function, which yielded an exact, macroscopic rate law for the entire antibody population. Linearization of the macroscopic rate law is achieved by plotting data points versus their numerical derivatives using log-log axes. Linear regression of such plots yields the average dissociation rate constant from the Y-intercept, and heterogeneity index from the slope. This analytic method is transparent to the antibody system and kinetic assay employed, requiring only a programmable calculator to perform the necessary calculations. The usefulness of this analytic method was demonstrated by the evaluation of dissociation kinetics in murine monoclonal and rabbit polyclonal anti-fluorescyl-IgG antibody populations.     

Invasion fitness,inclusive fitness,and reproductive numbers in heterogeneous populations

How should fitness be measured to determine which phenotype or “strategy” is uninvadable when evolution occurs in a group‐structured population subject to local demographic and environmental heterogeneity? Several fitness measures, such as basic reproductive number, lifetime dispersal success of a local lineage, or inclusive fitness have been proposed to address this question, but the relationships between them and their generality remains unclear. Here, we ascertain uninvadability (all mutant strategies always go extinct) in terms of the asymptotic per capita number of mutant copies produced by a mutant lineage arising as a single copy in a resident population (“invasion fitness”). We show that from invasion fitness uninvadability is equivalently characterized by at least three conceptually distinct fitness measures: (i) lineage fitness, giving the average individual fitness of a randomly sampled mutant lineage member; (ii) inclusive fitness, giving a reproductive value weighted average of the direct fitness costs and relatedness weighted indirect fitness benefits accruing to a randomly sampled mutant lineage member; and (iii) basic reproductive number (and variations thereof) giving lifetime success of a lineage in a single group, and which is an invasion fitness proxy. Our analysis connects approaches that have been deemed different, generalizes the exact version of inclusive fitness to class‐structured populations, and provides a biological interpretation of natural selection on a mutant allele under arbitrary strength of selection.     

Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations.

The spread of a potentially fatal infectious disease is considered in a host population that would increase exponentially in the absence of the disease. Taking into account how the effective contact rate C(N) depends on the population size N, the model demonstrates that demographic and epidemiological conclusions depend crucially on the properties of the contact function C. Conditions are given for the following scenarios to occur: (i) the disease spreads at a lower rate than the populations grows and does not modify the population growth rate: (ii) the disease initially spreads at a faster rate than the population grows and lowers the population growth rate in the long run and the following three subscenarios are possible: (iia) the population still grows exponentially, but at a slower rate; (iib) population growth is limited, but the population size does not decay; (iic) population increase is converted into population decrease.     

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.     

Models for the spread of universally fatal diseases

In the formulation of models of S-I-R type for the spread of communicable diseases it is necessary to distinguish between diseases with recovery with full immunity and diseases with permanent removal by death. We consider models which include nonlinear population dynamics with permanent removal. The principal result is that the stability of endemic equilibrium may depend on the population dynamics and on the distribution of infective periods; sustained oscillations are possible in some cases.     

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.     

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.     

On the 'rate of aging' in heterogeneous populations

It is well known that the rate of aging is constant for populations described by the Gompertz law of mortality. However, this is true only when a population is homogeneous. In this note, we consider the multiplicative frailty model with the baseline distribution that follows the Gompertz law and study the impact of heterogeneity on the rate of aging in this population. We show that the rate of aging in this case is a function of age and that it increases in (calendar) time when the baseline mortality rate decreases.     

A branching-process model for heterogeneous cell populations

A multitype branching-process model is introduced for the growth of heterogeneous cell populations. This model includes events representing mitosis, death, mutation, and conversion from one cell type to another. Formulas for conditioning on interim events, and generalizations allowing parameters to be functions of time or cell counts, are presented. The probability generating function (p.g.f.) is solved approximately in a way that is both accurate and efficient enough to solve important problems in tumor biology. The uses of the p.g.f. in the setting of clinical oncology are described.     

Comparing methods for measuring the rate of spread of invading populations

Measuring rates of spread during biological invasions is important for predicting where and when invading organisms will spread in the future as well as for quantifying the influence of environmental conditions on invasion speed. While several methods have been proposed in the literature to measure spread rates, a comprehensive comparison of their accuracy when applied to empirical data would be problematic because true rates of spread are never known. This study compares the performances of several spread rate measurement methods using a set of simulated invasions with known theoretical spread rates over a hypothetical region where a set of sampling points are distributed. We vary the density and distribution (aggregative, random, and regular) of the sampling points as well as the shape of the invaded area and then compare how different spread rate measurement methods accommodate these varying conditions. We find that the method of regressing distance to the point of origin of the invasion as a function of time of first detection provides the most reliable method over adverse conditions (low sampling density, aggregated distribution of sampling points, irregular invaded area). The boundary displacement method appears to be a useful complementary method when sampling density is sufficiently high, as it provides an instantaneous measure of spread rate, and does not require long time series of data.