From the Price equation to the selection gradient in class-structured populations: a quasi-equilibrium route

Metapopulation heterogeneities in host mobility, productivity, and immunocompetency always increase virulence and infectiousness

The epidemiology and evolution of diseases unfold in populations that are rarely homogeneous. Instead, hosts infected by pathogens often form metapopulations, in which local populations connected by the movement of hosts experience different demographic and epidemiological conditions. Here, we develop a general theory of the evolution of pathogens in heterogeneous metapopulations. We reveal the following key insights into the evolution of pathogen virulence and infectiousness: (1) When the mobility (movement rate), productivity (birth rate and carrying capacity), or immunocompetency (immunity-loss rate) differ among local populations, this variance always increases pathogen virulence and infectiousness (2) The increment of pathogen virulence caused by such heterogeneity is approximately proportional to the variance of the corresponding heterogeneous local conditions (3) This increment can be expressed as the covariance between the local selection pressures and the local reproductive values experienced by the pathogen (4) The reason why heterogeneity always increases pathogen virulence is explained by the positive correlation of local selection pressures with reproductive values (5) Combinations of multiple independent heterogeneities further increase virulence and infectiousness, even more so when their covariances are positive. Our key findings robustly hold for different epidemiological frameworks - including SI, SIS, SIR, and SIRS models, with both density- and frequency-dependent transmission as well as with superinfection. They provide insights into the risks of growing pathogen infectiousness in a world in which heterogeneity - caused, e.g., by the concentration of human populations in urban areas - is rising.

On the Interpretation of the Operation of Natural Selection in Class-Structured Populations

AbstractBiological adaptation is the outcome of allele-frequency change by natural selection. At the same time, populations are usually class structured as individuals occupy different states, such as age, sex, or stage. This is known to result in the differential transmission of alleles through nonheritable fitness differences called class transmission, which also affects allele-frequency change even in the absence of selection. How does one then isolate allele-frequency change due to selection from that due to class transmission? We decompose one-generational allele-frequency change in terms of effects of selection and class transmission and show how reproductive values can be used to reach a decomposition between any two distant generations of the evolutionary process. This provides a missing relationship between multigenerational allele-frequency change and the operation of selection. It also allows a measure of fitness to be defined summarizing the effect of selection in a multigenerational evolutionary process, which connects asymptotically to invasion fitness.

Metacommunities, fitness and gradual evolution

We analyze the evolution of a multidimensional quantitative trait in a class-structured focal species interacting with other species in a wider metacommunity. The evolutionary dynamics in the focal species as well as the ecological dynamics of the whole metacommunity is described as a continuous-time process with birth, physiological development, dispersal, and death given as rates that can depend on the state of the whole metacommunity. This can accommodate complex local community and global metacommunity environmental feedbacks owing to inter- and intra-specific interactions, as well as local environmental stochastic fluctuations. For the focal species, we derive a fitness measure for a mutant allele affecting class-specific trait expression. Using classical results from geometric singular perturbation theory, we provide a detailed proof that if the effect of the mutation on phenotypic expression is small ("weak selection"), the large system of dynamical equations needed to describe selection on the mutant allele in the metacommunity can be reduced to a single ordinary differential equation on the arithmetic mean mutant allele frequency that is of constant sign. This invariance on allele frequency entails the mutant either dies out or will out-compete the ancestral resident (or wild) type. Moreover, the directional selection coefficient driving arithmetic mean allele frequency can be expressed as an inclusive fitness effect calculated from the resident metacommunity alone, and depends, as expected, on individual fitness differentials, relatedness, and reproductive values. This formalizes the Darwinian process of gradual evolution driven by random mutation and natural selection in spatially and physiologically class-structured metacommunities.

The Price equation and reproductive value

The Price equation is widely recognized as capturing conceptually important properties of natural selection, and is often used to derive versions of Fisher's fundamental theorem of natural selection, the secondary theorems of natural selection and other significant results. However, class structure is not usually incorporated into these arguments. From the starting point of Fisher's original connection between fitness and reproductive value, a principled way of incorporating reproductive value and structured populations into the Price equation is explained, with its implications for precise meanings of (two distinct kinds of) reproductive value and of fitness. Once the Price equation applies to structured populations, then the other equations follow. The fundamental theorem itself has a special place among these equations, not only because it always incorporated class structure (and its method is followed for general class structures), but also because that is the result that justifies the important idea that these equations identify the effect of natural selection. The precise definitions of reproductive value and fitness have striking and unexpected features. However, a theoretical challenge emerges from the articulation of Fisher's structure: is it possible to retain the ecological properties of fitness as well as its evolutionary out-of-equilibrium properties? This article is part of the theme issue 'Fifty years of the Price equation'.