Evolutionary dynamics of incubation periods

Categorizing update mechanisms for graph-structured metapopulations

The structure of a population strongly influences its evolutionary dynamics. In various settings ranging from biology to social systems, individuals tend to interact more often with those present in their proximity and rarely with those far away. A common approach to model the structure of a population is evolutionary graph theory. In this framework, each graph node is occupied by a reproducing individual. The links connect these individuals to their neighbours. The offspring can be placed on neighbouring nodes, replacing the neighbours-or the progeny of its neighbours can replace a node during the course of ongoing evolutionary dynamics. Extending this theory by replacing single individuals with subpopulations at nodes yields a graph-structured metapopulation. The dynamics between the different local subpopulations is set by an update mechanism. There are many such update mechanisms. Here, we classify update mechanisms for structured metapopulations, which allows to find commonalities between past work and illustrate directions for further research and current gaps of investigation.

Asymptotic Absorption-Time Distributions in Extinction-Prone Markov Processes

We characterize absorption-time distributions for birth-death Markov chains with an absorbing boundary. For "extinction-prone" chains (which drift on average toward the absorbing state) the asymptotic distribution is Gaussian, Gumbel, or belongs to a family of skewed distributions. The latter two cases arise when the dynamics slow down dramatically near the boundary. Several models of evolution, epidemics, and chemical reactions fall into these classes; in each case we establish new results for the absorption-time distribution. Applications to African sleeping sickness are discussed.

Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality

Evolutionary graph theory (EGT) investigates the Moran birth-death process constrained by graphs. Its two principal goals are to find the fixation probability and time for some initial population of mutants on the graph. The fixation probability of graphs has received considerable attention. Less is known about the distribution of fixation time. We derive clean, exact expressions for the full conditional characteristic functions (CCFs) of a close proxy to fixation and extinction times. That proxy is the number of times that the mutant population size changes before fixation or extinction. We derive these CCFs from a product martingale that we identify for an evolutionary graph with any number of partitions. The existence of that martingale only requires that the connections between those partitions are of a certain type. Our results are the first expressions for the CCFs of any proxy to fixation time on a graph with any number of partitions. The parameter dependence of our CCFs is explicit, so we can explore how they depend on graph structure. Martingales are a powerful approach to study principal problems of EGT. Their applicability is invariant to the number of partitions in a graph, so we can study entire families of graphs simultaneously.

Quantifying SARS-CoV-2 Infection Risk Within the Google/Apple Exposure Notification Framework to Inform Quarantine Recommendations

Most early Bluetooth-based exposure notification apps use three binary classifications to recommend quarantine following SARS-CoV-2 exposure: a window of infectiousness in the transmitter, ≥15 minutes duration, and Bluetooth attenuation below a threshold. However, Bluetooth attenuation is not a reliable measure of distance, and infection risk is not a binary function of distance, nor duration, nor timing. We model uncertainty in the shape and orientation of an exhaled virus-containing plume and in inhalation parameters, and measure uncertainty in distance as a function of Bluetooth attenuation. We calculate expected dose by combining this with estimated infectiousness based on timing relative to symptom onset. We calibrate an exponential dose-response curve based on infection probabilities of household contacts. The probability of current or future infectiousness, conditioned on how long postexposure an exposed individual has been symptom-free, decreases during quarantine, with shape determined by incubation periods, proportion of asymptomatic cases, and asymptomatic shedding durations. It can be adjusted for negative test results using Bayes' theorem. We capture a 10-fold range of risk using six infectiousness values, 11-fold range using three Bluetooth attenuation bins, ∼sixfold range from exposure duration given the 30 minute duration cap imposed by the Google/Apple v1.1, and ∼11-fold between the beginning and end of 14 day quarantine. Public health authorities can either set a threshold on initial infection risk to determine 14-day quarantine onset, or on the conditional probability of current and future infectiousness conditions to determine both quarantine and duration.

Martingales and the characteristic functions of absorption time on bipartite graphs

Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald's martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.

Exit time as a measure of ecological resilience

Ecological resilience is the magnitude of the largest perturbation from which a system can still recover to its original state. However, a transition into another state may often be invoked by a series of minor synergistic perturbations rather than a single big one. We show how resilience can be estimated in terms of average life expectancy, accounting for this natural regime of variability. We use time series to fit a model that captures the stochastic as well as the deterministic components. The model is then used to estimate the mean exit time from the basin of attraction. This approach offers a fresh angle to anticipating the chance of a critical transition at a time when high-resolution time series are becoming increasingly available.

Spectral analysis of transient amplifiers for death-birth updating constructed from regular graphs

A central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

Fixation probabilities in graph-structured populations under weak selection

A population's spatial structure affects the rate of genetic change and the outcome of natural selection. These effects can be modeled mathematically using the Birth-death process on graphs. Individuals occupy the vertices of a weighted graph, and reproduce into neighboring vertices based on fitness. A key quantity is the probability that a mutant type will sweep to fixation, as a function of the mutant's fitness. Graphs that increase the fixation probability of beneficial mutations, and decrease that of deleterious mutations, are said to amplify selection. However, fixation probabilities are difficult to compute for an arbitrary graph. Here we derive an expression for the fixation probability, of a weakly-selected mutation, in terms of the time for two lineages to coalesce. This expression enables weak-selection fixation probabilities to be computed, for an arbitrary weighted graph, in polynomial time. Applying this method, we explore the range of possible effects of graph structure on natural selection, genetic drift, and the balance between the two. Using exhaustive analysis of small graphs and a genetic search algorithm, we identify families of graphs with striking effects on fixation probability, and we analyze these families mathematically. Our work reveals the nuanced effects of graph structure on natural selection and neutral drift. In particular, we show how these notions depend critically on the process by which mutations arise.

"Stay nearby or get checked": A Covid-19 control strategy

This paper repurposes the classic insight from network theory that long-distance connections drive disease propagation into a strategy for controlling a second wave of Covid-19. We simulate a scenario in which a lockdown is first imposed on a population and then partly lifted while long-range transmission is kept at a minimum. Simulated spreading patterns resemble contemporary distributions of Covid- 19 across EU member states, German and Italian regions, and through New York City, providing some model validation. Results suggest that our proposed strategy may significantly reduce peak infection. We also find that post-lockdown flare-ups remain local longer, aiding geographical containment. These results suggest a tailored policy in which individuals who frequently travel to places where they interact with many people are offered greater protection, tracked more closely, and are regularly tested. This policy can be communicated to the general public as a simple and reasonable principle: Stay nearby or get checked.

Wald’s martingale and the conditional distributions of absorption time in the Moran process

Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.

Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs

Evolutionary graph theory models the effects of natural selection and random drift on structured populations of competing mutant and nonmutant individuals. Recent studies have found that fixation times in such systems often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, obtaining exact solutions in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and nonmutants. In contrast, on the complete graph, the fixation-time distribution is a fitness-weighted convolution of two Gumbel distributions. When fitness is neutral, the fixation-time distribution jumps discontinuously and becomes highly skewed on both the complete graph and the ring. Even on these simple networks, the fixation-time distribution exhibits a rich fitness dependence, with discontinuities and regions of universality. Extensions of our results to two-fitness Moran models, times to partial fixation, and evolution on random networks are discussed.

Universal Statistics of Incubation Periods and Other Detection Times via Diffusion Models

We suggest an explanation of typical incubation times statistical features based on the universal behavior of exit times for diffusion models. We give a mathematically rigorous proof of the characteristic right skewness of the incubation time distribution for very general one-dimensional diffusion models. Imposing natural simple conditions on the drift coefficient, we also study these diffusion models under the assumption of noise smallness and show that the limiting exit time distributions in the limit of vanishing noise are Gaussian and Gumbel. Thus, they match the existing data as well as the other existing models do. The character of our models, however, allows us to argue that the features of the exit time distributions that we describe are universal and manifest themselves in various other situations where the times involved can be described as detection or halting times, for example response times studied in psychology.

Stationary log-normal distribution of weights stems from spontaneous ordering in adaptive node networks

Experimental evidence recently indicated that neural networks can learn in a different manner than was previously assumed, using adaptive nodes instead of adaptive links. Consequently, links to a node undergo the same adaptation, resulting in cooperative nonlinear dynamics with oscillating effective link weights. Here we show that the biological reality of stationary log-normal distribution of effective link weights in neural networks is a result of such adaptive nodes, although each effective link weight varies significantly in time. The underlying mechanism is a stochastic restoring force emerging from a spontaneous temporal ordering of spike pairs, generated by strong effective link preceding by a weak one. In addition, for feedforward adaptive node networks the number of dynamical attractors can scale exponentially with the number of links. These results are expected to advance deep learning capabilities and to open horizons to an interplay between adaptive node rules and the distribution of network link weights.

The reachability of contagion in temporal contact networks: how disease latency can exploit the rhythm of human behavior

Background

The symptoms of many infectious diseases influence their host to withdraw from social activity limiting their potential to spread. Successful transmission therefore requires the onset of infectiousness to coincide with a time when the host is socially active. Since social activity and infectiousness are both temporal phenomena, we hypothesize that diseases are most pervasive when these two processes are synchronized.

Methods

We consider disease dynamics that incorporate behavioral responses that effectively shorten the infectious period of the pathogen. Using data collected from face-to-face social interactions and synthetic contact networks constructed from empirical demographic data, we measure the reachability of this disease model and perform disease simulations over a range of latent period durations.

Results

We find that maximum transmission risk results when the disease latent period (and thus the generation time) are synchronized with human circadian rhythms of 24 h, and minimum transmission risk when latent periods are out of phase with circadian rhythms by 12 h. The effect of this synchronization is present for a range of disease models with realistic disease parameters and host behavioral responses.

Conclusions

The reproductive potential of pathogens is linked inextricably to the host social behavior required for transmission. We propose that future work should consider contact periodicity in models of disease dynamics, and suggest the possibility that disease control strategies may be designed to optimize against the effects of synchronization.

Electronic supplementary material

The online version of this article (10.1186/s12879-018-3117-6) contains supplementary material, which is available to authorized users.