Dynamics of an Impulsive Stochastic Predator–Prey System with the Beddington–DeAngelis Functional Response

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results.

A Likelihood-Ratio Test for Lumpability of Phylogenetic Data: Is the Markovian Property of an Evolutionary Process Retained in Recoded DNA?

In molecular phylogenetics, it is typically assumed that the evolutionary process for DNA can be approximated by independent and identically distributed Markovian processes at the variable sites and that these processes diverge over the edges of a rooted bifurcating tree. Sometimes the nucleotides are transformed from a 4-state alphabet to a 3- or 2-state alphabet by a procedure that is called recoding, lumping, or grouping of states. Here, we introduce a likelihood-ratio test for lumpability for DNA that has diverged under different Markovian conditions, which assesses the assumption that the Markovian property of the evolutionary process over each edge is retained after recoding of the nucleotides. The test is derived and validated numerically on simulated data. To demonstrate the insights that can be gained by using the test, we assessed two published data sets, one of mitochondrial DNA from a phylogenetic study of the ratites and the other of nuclear DNA from a phylogenetic study of yeast. Our analysis of these data sets revealed that recoding of the DNA eliminated some of the compositional heterogeneity detected over the sequences. However, the Markovian property of the original evolutionary process was not retained by the recoding, leading to some significant distortions of edge lengths in reconstructed trees.[Evolutionary processes; likelihood-ratio test; lumpability; Markovian processes; Markov models; phylogeny; recoding of nucleotides.]

Commuting in metapopulation epidemic modeling

The COVID-19 pandemic led authorities all over the world to imposing travel restrictions both on a national and on an international scale. Understanding the effect of such restrictions requires analysis of the role of commuting and calls for a metapopulation modeling that incorporates both local, intra-community infection and population exchange between different locations. Standard metapopulation models are formulated as markovian processes, and as such they do not label individuals according to their original location. However, commuting from home to work and backwards (reverse commuting) is the main pattern of transportation. Thus, it is important to be able to accurately model the effect of commuting on epidemic spreading. In this study we develop a methodology for modeling bidirectional commuting of individuals, without keeping track of each individual separately and with no need of proliferation of number of compartments beyond those defined by the epidemiologic model. We demonstrate the method using a city map of the state of Israel. The presented algorithm does not require any special computation resources and it may serve as a basis for intervention strategy examination in various levels of complication and resolution. We show how to incorporate an epidemiological model into a metapopulation commuting scheme while preserving the internal logic of the epidemiological modeling. The method is general and independent on the details of the epidemiological model under consideration.

Markovianization with approximate unitary designs

Memoryless processes are ubiquitous in nature, in contrast with the mathematics of open systems theory, which states that non-Markovian processes should be the norm. This discrepancy is usually addressed by subjectively making the environment forgetful. Here we prove that there are physical non-Markovian processes that with high probability look highly Markovian for all orders of correlations; we call this phenomenon Markovianization. Formally, we show that when a quantum process has dynamics given by an approximate unitary design, a large deviation bound on the size of non-Markovian memory is implied. We exemplify our result employing an efficient construction of an approximate unitary circuit design using two-qubit interactions only, showing how seemingly simple systems can speedily become forgetful. Conversely, since the process is closed, it should be possible to detect the underlying non-Markovian effects. However, for these processes, observing non-Markovian signatures would require highly entangling resources and hence be a difficult task.

Witnessing quantum memory in non-Markovian processes

We present a method to detect quantum memory in a non-Markovian process. We call a process Markovian when the environment does not provide a memory that retains correlations across different system-environment interactions. We define two types of non-Markovian processes, depending on the required memory being classical or quantum. We formalise this distinction using the process matrix formalism, through which a process is represented as a multipartite state. Within this formalism, a test for entanglement in a state can be mapped to a test for quantum memory in the corresponding process. This allows us to apply separability criteria and entanglement witnesses to the detection of quantum memory. We demonstrate the method in a simple model where both system and environment are single interacting qubits and map the parameters that lead to quantum memory. As with entanglement witnesses, our method of witnessing quantum memory provides a versatile experimental tool for open quantum systems.

Inferring epidemic containment probability from non-pharmaceutical interventions using a Gillespie-based epidemic model

The Gillespie algorithm, which is a stochastic numerical simulation of continuous-time Markovian processes, has been proposed for simulating epidemic dynamics. In the present study, using the Gillespie-based epidemic model, we focused on each single trajectory by the stochastic simulation to infer the probability of controlling an epidemic by non-pharmaceutical interventions (NPIs). The single trajectory analysis by the stochastic simulation suggested that a few infected people sometimes dissipated spontaneously without spreading of infection. The outbreak probability was affected by basic reproductive number but not by infectious duration and susceptible population size. A comparative analysis suggested that the mean trajectory by the stochastic simulation has equivalent dynamics to a conventional deterministic model in terms of epidemic forecasting. The probability of outbreak containment by NPIs was inferred by trajectories derived from 1000 Monte Carlo simulation trials using model parameters assuming COVID-19 epidemic. The model-based analysis indicated that complete containment of the disease could be achieved by short-duration NPIs if performed early after the import of infected individuals. Under the correctness of the model assumptions, analysis of each trajectory by Gillespie-based stochastic model would provide a unique and valuable output such as the probabilities of outbreak containment by NPIs.

Quasi-species evolution maximizes genotypic reproductive value (not fitness or flatness)

Growing efforts to measure fitness landscapes in molecular and microbial systems are motivated by a longstanding goal to predict future evolutionary trajectories. Sometimes under-appreciated, however, is that the fitness landscape and its topography do not by themselves determine the direction of evolution: under sufficiently high mutation rates, populations can climb the closest fitness peak (survival of the fittest), settle in lower regions with higher mutational robustness (survival of the flattest), or even fail to adapt altogether (error catastrophes). I show that another measure of reproductive success, Fisher’s reproductive value, resolves the trade-off between fitness and robustness in the quasi-species regime of evolution: to forecast the motion of a population in genotype space, one should look for peaks in the (mutation-rate dependent) landscape of genotypic reproductive values—whether or not these peaks correspond to local fitness maxima or flat fitness plateaus. This new landscape picture turns quasi-species dynamics into an instance of non-equilibrium dynamics, in the physical sense of Markovian processes, potential landscapes, entropy production, etc.