arXiv subjects

Todd L. Parsons

Publications and source records attributed to Todd L. Parsons.

7 recordsLinked to original sources

arXiv2026

Epidemic "momentum" and a conservation law for infectious disease dynamics

Infectious disease outbreaks have precipitated a profusion of mathematical models. Epidemic curves predicted by these models are typically qualitatively similar, despite distinct model assumptions, but there is no theoretical explanation for this similarity in terms of any recognised common structure. We introduce a unifying concept of "epidemic momentum"---prevalence weighted by potential to infect---which is more informative than prevalence, yet analytically tractable. Epidemic momentum reveals a common underlying geometry in which outbreak trajectories always follow contours of a conserved quantity. This previously unrecognised conservation law constrains how epidemics can unfold and provides the mathematical basis for disentangling transmissibility from prior immunity using a single epidemic time series. Epidemic momentum also exposes the true final size of an outbreak and a universal phase-plane description that links generic renewal models to the classical SIR system.

q-bio.PE↗

arXiv2026

Jointly estimating transmissibility and prior immunity from epidemic time series

Infectious disease time series are often used to estimate a pathogen's basic reproduction number, $R_0$. However, fits of epidemic models to time series conflate pathogen transmissibility with pre-existing population immunity, so only the *effective* reproduction number, $R_{eff}$, can be inferred. This composite parameter is the product of the underlying $R_0$ and the pre-epidemic susceptible fraction, $x^-$. We show that a conservation law associated with epidemic momentum---prevalence weighted by potential to infect---makes it possible to disentangle transmissibility from prior immunity and to infer $R_0$ and $x^-$ separately from a single epidemic time series. We test the methodology using stochastic epidemic simulations, and illustrate the approach with a reappraisal of influenza transmissibility during the 1918 pandemic, estimating rather than assuming the degree of prior population immunity. For the autumn wave in Philadelphia, USA, we find $R_0\approx2.7$ and $x^-\approx0.8$, implying that about 20% of the population was already immune before that wave, plausibly as a result of infection during the spring 1918 herald wave.

q-bio.PE↗

arXiv2017

Invasion probabilities, hitting times, and some fluctuation theory for the stochastic logistic process

We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the large population limit, from which we obtain establishment probabilities and upper bounds for the process, as well as estimates for the waiting time to establishment and extinction. As a consequence, we show that conditional upon establishment, the stochastic logistic process will with high probability greatly exceed carrying capacity an arbitrary number of times prior to extinction.

q-bio.PE↗

arXiv2017

Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: a constructive approach with examples from theoretical biology

Systems composed of large numbers of interacting agents often admit an effective coarse-grained description in terms of a multidimensional stochastic dynamical system, driven by small-amplitude intrinsic noise. In applications to biological, ecological, chemical and social dynamics it is common for these models to posses quantities that are approximately conserved on short timescales, in which case system trajectories are observed to remain close to some lower-dimensional subspace. Here, we derive explicit and general formulae for a reduced-dimension description of such processes that is exact in the limit of small noise and well-separated slow and fast dynamics. The Michaelis-Menten law of enzyme-catalyzed reactions, and the link between the Lotka-Voltera and Wright-Fisher processes are explored as a simple worked examples. Extensions of the method are presented for infinite dimensional systems and processes coupled to non-Gaussian noise sources.

math.PR↗

arXiv2016

Evolutionary consequences of behavioral diversity

Iterated games provide a framework to describe social interactions among groups of individuals. Recent work stimulated by the discovery of "zero-determinant" strategies has rapidly expanded our ability to analyze such interactions. This body of work has primarily focused on games in which players face a simple binary choice, to "cooperate" or "defect". Real individuals, however, often exhibit behavioral diversity, varying their input to a social interaction both qualitatively and quantitatively. Here we explore how access to a greater diversity of behavioral choices impacts the evolution of social dynamics in finite populations. We show that, in public goods games, some two-choice strategies can nonetheless resist invasion by all possible multi-choice invaders, even while engaging in relatively little punishment. We also show that access to greater behavioral choice results in more "rugged " fitness landscapes, with populations able to stabilize cooperation at multiple levels of investment, such that choice facilitates cooperation when returns on investments are low, but hinders cooperation when returns on investments are high. Finally, we analyze iterated rock-paper-scissors games, whose non-transitive payoff structure means unilateral control is difficult and zero-determinant strategies do not exist in general. Despite this, we find that a large portion of multi-choice strategies can invade and resist invasion by strategies that lack behavioral diversity -- so that even well-mixed populations will tend to evolve behavioral diversity.

q-bio.PE↗

arXiv2016

Integrating Theory and Experiment to Explain the Breakdown of Population Synchrony in a Complex Microbial Community

We consider the extension of the `Moran effect', where correlated noise generates synchrony between isolated single species populations, to the study of synchrony between populations embedded in multi-species communities. In laboratory experiments on complex microbial communities, comprising both predators (protozoa) and prey (bacteria), we observe synchrony in abundances between isolated replicates. A breakdown in synchrony occurs for both predator and prey as the reactor dilution rate increases, which corresponds to both an increased rate of input of external resources and an increased effective mortality though washout. The breakdown is more rapid, however, for the lower trophic level. We can explain this phenomenon using a mathematical framework for determining synchrony between populations in multi-species communities at equilibrium. We assume that there are multiple sources of environmental noise with different degrees of correlation that affect the individual species population dynamics differently. The deterministic dynamics can then influence the degree of synchrony between species in different communities. In the case of a stable equilibrium community synchrony is controlled by the eigenvalue with smallest negative real part. Intuitively fluctuations are minimally damped in this direction. We show that the experimental observations are consistent with this framework but only for multiplicative noise.

q-bio.PE↗

arXiv2013

Limit Theorems for Competitive Density Dependent Population Processes

Near the beginning of the century, Wright and Fisher devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for mathematical, and not biological reasons - makes assumptions that are violated in most real-world populations. In this paper, I consider an alternative framework that merges P. A. P. Moran's continuous-time Markov chain model of allele frequency with the density dependent models of ecological competition proposed by Gause, Lotka and Volterra, that, unlike Moran's model allow for a stochastically varying -- but bounded -- population size. I require that allele numbers vary according to a density-dependent population process, for which the limiting law of large numbers is a dissipative, irreducible, competitive dynamical system. Under the assumption that this limiting system admits a codimension one submanifold of attractive fixed points -- a condition that naturally generalises the weak selection regime of classical population dynamics -- it is shown that for an appropriate rescaling of time, the finite dimensional distributions of the original process converge to those of a diffusion process on the submanifold. Weak convergence results are also obtained for a related process.

math.PR↗