A net reproductive number for periodic matrix models

We give a definition of a net reproductive number R ₀ for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing–Zhou definition of R ₀ in the autonomous case. The value of R ₀ determines whether the population goes extinct (R ₀<1) or persists (R ₀>1). We discuss the biological interpretation of this definition and derive formulas for R ₀ for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R ₀ with another definition given recently by Bacaër.     

Growth rate and basic reproduction number for population models with a simple periodic factor

For continuous-time population models with a periodic factor which is sinusoidal, both the growth rate and the basic reproduction number are shown to be the largest roots of simple equations involving continued fractions. As an example, we reconsider an SEIS model with a fixed latent period, an exponentially distributed infectious period and a sinusoidal contact rate studied in Williams and Dye [B.G. Williams, C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci. 145 (1997) 77]. We show that apart from a few exceptional parameter values, the epidemic threshold depends not only on the mean contact rate, but also on the amplitude of fluctuations.     

A sparse matrix method for renal models

A sparse matrix method for the numerical solution of nonlinear differential equations arising in modeling of the renal concentrating mechanism is given. The method involves a renumbering of the variables and equations such that the resulting Jacobian matrix has a block tridiagonal structure and the blocks above and below the main diagonal have a known set of complementary nonzero columns. The computer storage for the method is O(n). Results of some numerical experiments showing the stability of the method are given.     

Complexity in matrix population models: Polyvariant ontogeny and reproductive uncertainty

Linear matrix models of stage-structured population dynamics are widely used in plant and animal demography as a tool to evaluate the growth potential of a population in a given environment. The potential is identified with λ₁, the dominant eigenvalue of the projection matrix, which is compiled of stage-specific transition and fertility rates. Advanced botanical studies reveal polyvariant ontogeny in perennial plants, i.e., multiple different versions of individual development within a local population of a single species. This phenomenon complicates any standard, successive-stage, life cycle graph to a digraph defined on a 2D lattice in the age and stage dimensions, the pattern of projection matrix becoming more complex too. In a kind of experimental design, the transition rates can be calculated directly from the data for two successive time moments, but the age-stage-specific rates of reproduction still remain uncertain, adding more complexity to the calibration problem. Simple additional assumptions could technically eliminate the uncertainty, but they contravene the biology of a species in which polyvariant ontogeny is considered to be the major mechanism of adaptation. Given the data and expert constraints, the calibration can be reduced instead to a nonlinear maximization problem, yet with linear constraints. I prove that it has a unique solution to be attained at a vertex of the constraint polyhedral. To facilitate searching for the solution in practice, I use the net reproductive rate R₀, a well-known indicator for the principal property of λ₁ to be greater or less than 1. The method is exemplified with the calibration of a projection matrix in an age-stage-structured model (published elsewhere) for Calamagrostis canescens, a perennial herbaceous species with a complex (multivariant) life cycle that features unlimited growth when colonizing open areas.     

On net reproductive rate and the timing of reproductive output

Abstract: Understanding the relationship between life-history patterns and population growth is central to demographic studies. Here we derive a new method for calculating the timing of reproductive output, from which the generation time and its variance can also be calculated. The method is based on the explicit computation of the net reproductive rate (R0) using a new graphical approach. Using nodding thistle, desert tortoise, creeping aven, and cat's ear as examples, we show how R0 and the timing of reproduction is calculated and interpreted, even in cases with complex life cycles. We show that the explicit R0 formula allows us to explore the effect of all reproductive pathways in the life cycle, something that cannot be done with traditional analysis of the population growth rate (lambda). Additionally, we compare a recently published method for determining population persistence conditions with the condition R0 > 1 and show how the latter is simpler and more easily interpreted biologically. Using our calculation of the timing of reproductive output, we illustrate how this demographic measure can be used to understand the effects of life-history traits on population growth and control.     

On the reproductive value and the spectrum of a population projection matrix with implications for dynamic population models

The eigenvalues of a population projection matrix-except for the Lotka coefficient-are uniquely determined by the reproductive values and the survival. This relation (proposed earlier, but not really well known in western literature) follows from another useful relation between fertility, reproductive values, survival, and Lotka’s coefficient. These results are applied to provide demographic interpretations to the intrinsically dynamic and metastable population models by Schoen and co-workers.     

Some stochastic models for plasmid copy number

Some stochastic models for the copy number of plasmids in a cell line are studied. When considering the behavior of copy number in the whole cell line, the theory of multitype branching processes is appropriate. Attention is paid to the cure rate in the cell line, and the asymptotic fractions of cells containing a given number of plasmids. These quantities are used to compare the models numerically.     

Inferring the effective reproductive number from deterministic and semi-deterministic compartmental models using incidence and mobility data

The effective reproduction number (ℜ_t) is a theoretical indicator of the course of an infectious disease that allows policymakers to evaluate whether current or previous control efforts have been successful or whether additional interventions are necessary. This metric, however, cannot be directly observed and must be inferred from available data. One approach to obtaining such estimates is fitting compartmental models to incidence data. We can envision these dynamic models as the ensemble of structures that describe the disease’s natural history and individuals’ behavioural patterns. In the context of the response to the COVID-19 pandemic, the assumption of a constant transmission rate is rendered unrealistic, and it is critical to identify a mathematical formulation that accounts for changes in contact patterns. In this work, we leverage existing approaches to propose three complementary formulations that yield similar estimates for ℜ_t based on data from Ireland’s first COVID-19 wave. We describe these Data Generating Processes (DGP) in terms of State-Space models. Two (DGP1 and DGP2) correspond to stochastic process models whose transmission rate is modelled as Brownian motion processes (Geometric and Cox-Ingersoll-Ross). These DGPs share a measurement model that accounts for incidence and transmission rates, where mobility data is assumed as a proxy of the transmission rate. We perform inference on these structures using Iterated Filtering and the Particle Filter. The final DGP (DGP3) is built from a pool of deterministic models that describe the transmission rate as information delays. We calibrate this pool of models to incidence reports using Hamiltonian Monte Carlo. By following this complementary approach, we assess the tradeoffs associated with each formulation and reflect on the benefits/risks of incorporating proxy data into the inference process. We anticipate this work will help evaluate the implications of choosing a particular formulation for the dynamics and observation of the time-varying transmission rate.     

Analysis of periodic growth-disturbance models

In this paper, I present and discuss a potentially useful modeling approach for investigating population dynamics in the presence of disturbance. Using the motivating example of wildfire, I construct and analyze a deterministic model of population dynamics with periodic disturbances independent of spatial effects. Plant population growth is coupled to fire disturbance to create a growth-disturbance model for a fluctuating population. Changes in the disturbance frequency are shown to generate a period-bubbling bifurcation structure and population dynamics that are most variable at intermediate disturbance frequencies. Similar dynamics are observed when the model is extended to include a seed bank. Some general conditions necessary for a rich bifurcation structure in growth-disturbance models are discussed.     

Spatially-explicit matrix models

This paper is concerned with mathematical analysis of the critical domain-size problem for structured populations. Space is introduced explicitly into matrix models for stage-structured populations. Movement of individuals is described by means of a dispersal kernel. The mathematical analysis investigates conditions for existence, stability and uniqueness of equilibrium solutions as well as some bifurcation behaviors. These mathematical results are linked to species persistence or extinction in connected habitats of different sizes or fragmented habitats; hence the framework is given for application of such models to ecology. Several approximations which reduce the complexity of integrodifference equations are given. A simple example is worked out to illustrate the analytical results and to compare the behavior of the integrodifference model to that of the approximations.     

A framework for studying transient dynamics of population projection matrix models

Empirical models are central to effective conservation and population management, and should be predictive of real-world dynamics. Available modelling methods are diverse, but analysis usually focuses on long-term dynamics that are unable to describe the complicated short-term time series that can arise even from simple models following ecological disturbances or perturbations. Recent interest in such transient dynamics has led to diverse methodologies for their quantification in density-independent, time-invariant population projection matrix (PPM) models, but the fragmented nature of this literature has stifled the widespread analysis of transients. We review the literature on transient analyses of linear PPM models and synthesise a coherent framework. We promote the use of standardised indices, and categorise indices according to their focus on either convergence times or transient population density, and on either transient bounds or case-specific transient dynamics. We use a large database of empirical PPM models to explore relationships between indices of transient dynamics. This analysis promotes the use of population inertia as a simple, versatile and informative predictor of transient population density, but criticises the utility of established indices of convergence times. Our findings should guide further development of analyses of transient population dynamics using PPMs or other empirical modelling techniques.     

Cyclic modes in neural net models

Neural network models with possible cross-coupling from every neuron to every other neuron are condisered. The all-or-none law is assumed for firing of neurons. A network is shown to have a set of latent cyclic modes. If the net is stimulated briefly, it will subsequently either return to quiescence or settle into periodic activity in one of its cyclic modes. Realization of cyclic modes and analysis of nets are discussed. A learning rule for adjustment of synaptic strengths is presented.     

Practical considerations for measuring the effective reproductive number,Rt

Estimation of the effective reproductive number R_t is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using R_t to assess the effectiveness of interventions and to inform policy. However, estimation of R_t from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of R_t, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting R_t estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in R_t estimation.     

Sequence periodic pattern of HERV LTRs: A matrix simulation algorithm

Flanking regulatory long terminal repeats (LTRs) in Human endogenous retrovirus (HERV) is a kind of typical DNA repeat that is widespread in the human genome. Currently, many algorithms have been developed to detect the latent periodicity of a wide range of DNA repeats. However, no such attempt was made for HERV LTRs. The present study focused on the investigation of the possible sequence periodic patterns in the HERV LTRs and their regulatory mechanisms. We calculated the sequence periods of 5′, 3′ and combined LTRs in HERVs with our devised matrix simulation algorithm. It is interesting that 5′ and 3′ LTRs have the same period of 7, and combined LTRs have a period of 9. These results indicated that HERV LTRs have predominant periodic patterns. Based on the obtained sequence periodicity, we constructed periodic consensus sequences of 5′, 3′ and combined LTRs. As to 5′ and 3′ LTRs with the same period – 7, we manually scanned the nucleotide bases in the corresponding positions of their periodic consensus sequences, and found some positions have the nucleotide base unchanged, such as the 1st, 5th and 7th positions. These conservative nucleotide base positions represent critical binding sites of regulatory LTRs, and may be indicative of conserved regulatory mechanisms in LRT-participating regulatory networks.     

Hierarchical spatiotemporal matrix models for characterizing invasions

The growth and dispersal of biotic organisms is an important subject in ecology. Ecologists are able to accurately describe survival and fecundity in plant and animal populations and have developed quantitative approaches to study the dynamics of dispersal and population size. Of particular interest are the dynamics of invasive species. Such nonindigenous animals and plants can levy significant impacts on native biotic communities. Effective models for relative abundance have been developed; however, a better understanding of the dynamics of actual population size (as opposed to relative abundance) in an invasion would be beneficial to all branches of ecology. In this article, we adopt a hierarchical Bayesian framework for modeling the invasion of such species while addressing the discrete nature of the data and uncertainty associated with the probability of detection. The nonlinear dynamics between discrete time points are intuitively modeled through an embedded deterministic population model with density-dependent growth and dispersal components. Additionally, we illustrate the importance of accommodating spatially varying dispersal rates. The method is applied to the specific case of the Eurasian Collared-Dove, an invasive species at mid-invasion in the United States at the time of this writing.     

Importance of correlations among matrix entries in stochastic models in relation to number of transition matrices

Stochastic matrix models are used to predict population viability and the risk of extinction. Different stochastic methods require different amounts of estimation effort and may lead to divergent estimates. We used 16 transition matrices collected from ten populations of the perennial herb Primula veris to compare population estimates produced by different stochastic methods, such as selection of matrices, selection of vital rates, selection of matrix elements, and Tuljapurkar's approximation. Specifically, we tested the reliability of the methods using different numbers of transition matrices, and examined the importance of correlations among matrix entries. When correlations among matrix entries were included in the models, selection of vital rates produced the lowest and Tuljapurkar's approximation produced the highest estimates of mean population growth rates. Selection of matrices and matrix elements often produced nearly similar population estimates. Simulations based on incompletely estimated correlations among matrix entries considerably differed from those based on all correlations estimated, particularly when correlations were strong. The magnitude of correlations among matrix entries depended on the number of matrices, which made it difficult to generalize correlations within a species. Given that selection of vital rates or matrix elements is used, correlations among matrix entries should usually be included in the model, and they should preferably be estimated from the present data rather than according to other information of the species.     

A graph-theoretic method for the basic reproduction number in continuous time epidemiological models

In epidemiological models of infectious diseases the basic reproduction number is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for can be obtained from the digraphs in the reduction process.     

A model for estimating the minimum number of offspring to sample in studies of reproductive success

Molecular parentage permits studies of selection and evolution in fecund species with cryptic mating systems, such as fish, amphibians, and insects. However, there exists no method for estimating the number of offspring that must be assigned parentage to achieve robust estimates of reproductive success when only a fraction of offspring can be sampled. We constructed a 2-stage model that first estimated the mean (μ) and variance (v) in reproductive success from published studies on salmonid fishes and then sampled offspring from reproductive success distributions simulated from the μ and v estimates. Results provided strong support for modeling salmonid reproductive success via the negative binomial distribution and suggested that few offspring samples are needed to reject the null hypothesis of uniform offspring production. However, the sampled reproductive success distributions deviated significantly (χ(2) goodness-of-fit test p value < 0.05) from the known simulated reproductive success distribution at rates often >0.05 and as high as 0.24, even when hundreds of offspring were assigned parentage. In general, reproductive success patterns were less accurate when offspring were sampled from cohorts with larger numbers of parents and greater variance in reproductive success. Our model can be reparameterized with data from other species and will aid researchers in planning reproductive success studies by providing explicit sampling targets required to accurately assess reproductive success.