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AbstractChemotherapy is used to treat cancer through a cytotoxic therapeutic approach. Certain conditions may allow some cells to evade the treatment and spread again. Doctors resort to completing the treatment with radiation, which increases the patient’s stress during the treatment journey. In this work, we study the multivariate probability distribution of these stochastically escaped cells to estimate their number in an attempt to reduce the doses of chemotherapy and radiation together. We consider these escaping tumor cells increased independently according to a Poisson process. We obtain not just this distribution but also the variance and expected value of these escaped cells at any given time. This time is depending on the total number of infected cells in the system. Furthermore, we may calculate the expected value of the treatment cells by knowing the expected number of escaping cells. This information aids in determining the right dosage of chemotherapy, which in turn helps to reduce the patients’ pain. [ABSTRACT FROM AUTHOR]

Coexistence between humans and wildlife is necessary for many conservation goals but is difficult to achieve in landscapes with increasing human populations and species that are often wary of people and may also threaten human safety. In these contexts, coexistence may be enhanced by identifying geographic areas where animal movement is particularly important and changes to human use via trail design could support both wildlife conservation and human safety. We used camera trap data to monitor the spatial distribution of grizzly bears (Ursus arctos), grey wolves (Canis lupus), and humans within the central Canadian Rocky Mountains, where anthropogenic development and human activity have gradually encroached on limited wildlife habitat. We quantified spatial variation in human use and then incorporated this output into models for the detection rates of bears and wolves. We interpolated metrics of human use throughout the study area using inverse distance weighted averages of human detection rates from cameras. This approach supported a novel estimate of the cumulative effects of human use at all nearby trails on animal space use. We used our models to estimate the zone of influence of human use on bears and wolves, determining the distance at which human use on nearby trails no longer exhibited a measurable change in detection rates for each of grizzly bears and wolves. The negative effects of human use on wildlife declined steeply with distance such that 50% of the decrease in detection rates immediately adjacent to trails would be expected to occur at 267 m for grizzly bears and 576 m for wolves. Weak effects, 5% as strong as the effect adjacent to trails, extended up to 1.8 and 6.1 km for grizzly bears and wolves, revealing the importance of cumulative measures of human use. Synthesis and applications. Our work shows how human activity over entire landscapes can alter wildlife detection rates. Our results identify target buffer distances for protected areas near trails, and the modelling framework could be used by land managers to predict how altering trail networks and modifying human activity could affect wary wildlife species and advance coexistence. [ABSTRACT FROM AUTHOR]

In a finite buffer queuing system, the congestion mostly occurs due to the higher blocking probability. In this article, the author has presented a Modified Bulk Service (MBS) rule for the finite buffer queueing system under assumptions that the server can accept a customer during ongoing service if serving batch size is lower; however, the time spent in serving the lower batch size is elapsed. Various performance metrics are discussed. [ABSTRACT FROM AUTHOR]

A common problem seen in the ineffective execution of global multicenter trials is the frequent inability to recruit a sufficient number of patients. A myriad of barriers to recruiting enough patients timely exist and may include practical limits on the number of recruiting sites imposed by the various countries, as well as administrative, cost and unanticipated issues. The Poisson-gamma recruitment model is a widely accepted statistical tool used to predict and track recruitment at different levels of a trial and make inference. An optimal recruitment plan can be designed using mathematical optimization, however, the search is complicated with multiple nonlinear objectives and constraints that arise from regulatory and budgetary considerations. We review nature-inspired metaheuristic algorithms which are powerful, flexible, general purpose optimization tools and demonstrate their capability and utility in optimizing complex recruitment designs for global clinical trials using the Poisson-gamma model as an exemplary model. This research opens up new avenues for improving the efficiency and effectiveness of patient recruitment in clinical trials, thereby potentially accelerating the development of new medical treatments. [ABSTRACT FROM AUTHOR]

Analyzing COVID-19 data presents a challenge in Bayesian computations of the Poisson process because the experimental conditions are not under control. This lack of homogeneity can lead to inconsistent model parameters, which violates the assumptions of Bayesian inference. In this paper, we study the multiple change-point detection problem from this viewpoint for a non-homogeneous sample path of the Poisson process as the response variable. The rate parameters are linked to some explanatory using a generalized linear model. The number of change-points is considered to be unknown as well as their locations. We introduce a Bayesian paradigm to estimate the number and location of change-points. We also present an adaptive RJMCMC algorithm to generate pseudo-random samples from the posterior distributions. We apply the proposed model to analyze the COVID-19 infection curves from different countries and identify patterns of cases. We also assess the efficacy of interventions, such as vaccination and public health emergency responses, implemented by different countries. The results of the analysis provide valuable insights into the spread of COVID-19 and the effectiveness of interventions. The proposed model can be used to inform public health decision-making and help to improve the management of the pandemic. [ABSTRACT FROM AUTHOR]

The probability of gas escapes from steel pipelines due to different types of corrosion is studied with real failure data from an urban gas distribution network. Both the design and maintenance of the network are considered, identifying and estimating (in a Bayesian framework) an elementary multinomial model in the first case, and a more sophisticated non-homogeneous Poisson process in the second case. Special attention is paid to the elicitation of the experts' opinions. We conclude that the corrosion process behaves quite differently depending on the type of corrosion, and that, in most cases, cathodically protected pipes should be installed. [ABSTRACT FROM AUTHOR]

We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders of k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein and cumulant methods we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants of any orders are provided as an online resource. [ABSTRACT FROM AUTHOR]

Many insurers have started to underwrite cyber in recent years. In parallel, they developed their first actuarial models to cope with this new type of risk. On the portfolio level, two major challenges hereby are the adequate modelling of the dependence structure among cyber losses and the lack of suitable data based on which the model is calibrated. The purpose of this article is to highlight the importance of taking a holistic approach to cyber. In particular, we argue that actuarial modelling should not be viewed stand-alone, but rather as an integral part of an interconnected value chain with other processes such as cyber-risk assessment and cyber-claims settlement. We illustrate that otherwise, i.e. if these data-collection processes are not aligned with the actuarial (dependence) model, naïve data collection necessarily leads to a dangerous underestimation of accumulation risk. We illustrate the detrimental effects on the assessment of the dependence structure and portfolio risk by using a simple mathematical model for dependence through common vulnerabilities. The study concludes by highlighting the practical implications for insurers. [ABSTRACT FROM AUTHOR]

We obtain a Poisson Limit for return times to small sets for product systems. Only one factor is required to be hyperbolic, while the second factor is only required to satisfy polynomial deviation bounds for ergodic sums. In particular, the second factor can be elliptic or parabolic. As an application of our main result, several maps of the form Anosov map $ \times $ another map are shown to satisfy a Poisson Limit Theorem at typical points, some even at all points. The methods can be extended to certain types of skew products, including $ T, T^{-1} $-maps of high rank. [ABSTRACT FROM AUTHOR]

The paper considers a Cox process where the stochastic intensity function for the Poisson data model is itself a non-homogeneous Poisson process. We show that it is possible to obtain the marginal data process, namely a non-homogeneous count process exhibiting over-dispersion. The model generates intensity functions which are non-decreasing. This is not a restriction in practice since observed data can always be transformed which guarantees a non-decreasing intensity function. We are able, for a specific choice of Cox process, to mathematically marginalize out the latent process leaving with a directly available likelihood function. This makes inference much simpler compared to algorithms which retain the latent process. [ABSTRACT FROM AUTHOR]

Rainfall is an important trigger for a large number of landslides. Physics-based quantitative risk assessment of rainfall-induced landslides requires modelling of rainfall uncertainty in addition to soil parameters. Explicit modelling of rainfall uncertainty should consider the uncertainties in rainfall intensity, duration, as well as the occurrence of rainfall events. Towards this aspect, this study proposes a stochastic rainfall model, where the joint distribution of rainfall duration and intensity is constructed using copula theory and the occurrence of rainfall events is described using Poisson process. Based on the proposed stochastic rainfall model, a computationally efficient reliability method with a machine learning-based surrogate model is developed for assessing the failure probability of a slope subjected to rainfall infiltration within a given time period. An illustrative example with real rainfall data from Singapore is utilised to demonstrate the proposed approach. The results suggest that the slope failure probability is less sensitive to the uncertainty in rainfall occurrence, but highly sensitive to the adopted inter-event time definition for characterisation of the rainfall event. Overall, this study provides a useful stochastic rainfall model and some practical guidelines for quantitative risk assessment of rainfall-induced landslides. [ABSTRACT FROM AUTHOR]

We demonstrate that the recently introduced triple generally weighted moving average (GWMA) chart and its counterpart, the double GWMA chart, incorporate sub‐optimal weighting patterns that may assign more weight to certain historical data points at the expense of more recent ones. Moreover, these control charts, when compared to the exponentially weighted moving average (EWMA) chart, exhibit a substantial computational burden. Our findings underscore that a well‐designed EWMA chart offers superior overall performance in comparison to these control charts. [ABSTRACT FROM AUTHOR]

Ensuring an acceptable level of reliability stands as a primary imperative for any mission‐focused operation since it serves as a critical determinant of success. Inadequate reliability can lead to severe repercussions, including substantial expenses for repairs and replacements, missed opportunities, service disruptions, and in the worst cases, safety violations and human casualties. Within national defense organizations such as the USAF, the precise assessment and maintenance of system reliability play a pivotal role in ensuring the success of mission‐critical operations. In this research, our primary objective is to model the reliability of repairable subsystems within the framework of competing and complementary risks. Subsequently, we construct the overall reliability of the entire repairable system, utilizing day‐to‐day group‐censored maintenance data from two identical aircraft systems. Assuming that the lifetimes of subsystems follow non‐identical exponential distributions, it is theoretically justified that the system reliability can be modeled by homogeneous Poisson processes even though the number of subsystems of any particular type is unknown and the temporal order of multiple subsystem failures within a given time interval is uncertain due to interval censoring. Using the proposed model, we formulate the likelihood function for the mean time between failures of subsystems with different causes, and subsequently establish an inferential procedure for the model parameters. Given a considerable number of parameters to estimate, we explore the efficacy of a Bayesian approach, treating the contractor‐supplied estimates as the hyperparameters of prior distributions. This approach mitigates potential model uncertainty as well as the practical limitation of a frequentist‐based approach. It also facilitates continuous updates of the estimates as new maintenance data become available. Finally, the entire inferential procedures were implemented in Microsoft Excel so that it is easy for any reliability practitioner to use without the need to learn sophisticated programming languages. Thus, this research supports an ongoing, real‐time assessment of the overall mission reliability and helps early detection of any subsystem whose reliability is below the threshold level. [ABSTRACT FROM AUTHOR]

We present solutions to some discounted nonzero-sum optimal stopping games of two players related to the perpetual game-type contingent claims with payoffs representing linear functions of the running values of a geometric Brownian motion. It is assumed that the underlying process can be stopped by the both players only at certain random intervention times which coincide with the jump times of the two appropriate independent Poisson processes. The optimal stopping times forming a Nash equilibrium are shown to be the first times at which the underlying process is either below or above certain lower or upper constant boundaries at the jump times of the appropriate Poisson processes. The proof is based on the reductions of the original games to the associated coupled free-boundary problems and the solutions to the latter problems by means of the smooth-fit conditions at the optimal boundaries for every player. We show that the optimal stopping constant lower and upper boundaries are determined as (possibly multiple) solutions to the equivalent coupled systems of arithmetic equations. The obtained results can be interpreted as the rational valuation of some perpetual randomized Bermudian game-type contingent claims in the Black-Merton-Scholes model. [ABSTRACT FROM AUTHOR]

This paper introduces a class of stochastic processes making jumps around the circle. These circular processes are the wrapped versions of the Poisson, the negative binomial, the binomial processes and of extensions thereof obtained by compounding with a secondary frequency distribution. Their prevailing application is for modeling planar motions. These processes can be weakly stationary, can have uniform stationary distribution, can have independent or stationary circular increments. Their autocovariance functions, one-dimensional trigonometric moments and wrapped distributions are obtained. For the wrapped Poisson process, it is shown that the formula for the one-dimensional distribution can be obtained either by discrete Fourier transform or by wrapping the Poisson distribution around the circle and by simplifying it with the generalized hyperbolic function. Some numerical illustrations and comparisons are provided. [ABSTRACT FROM AUTHOR]

AbstractStatistical modeling of vehicle crashes leads to a better understanding of how and why such crashes occur. Due to the irregular network structure of roadways, analyses are typically confined to a single roadway rather than considering the entire network collectively. Here, we present methodology to model crash risk of vehicle crashes on irregular roadway networks and estimate how that risk varies with road characteristics. We model vehicle crashes observed on a road network as a Poisson point pattern with a piecewise linear intensity surface. Further, we combine Bayesian additive regression trees (BART) and spatial data analysis to accurately explain the intensity surface allowing inference on the effect of road characteristics on crash risk. We illustrate the methodology using a dataset of vehicle crashes on Interstate highways in Utah. [ABSTRACT FROM AUTHOR]

This article deals with the problem of estimating the service time distribution of the M t / G / ∞ queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure. [ABSTRACT FROM AUTHOR]

In this paper, we study a risk model with stochastic premium income and its impact on solvency risk management. It is assumed that both the premium arrival process and the claim arrival process are modelled by homogeneous Poisson processes, and that the premium amounts are modelled by independent and identically distributed random variables. While this model has been studied in the existing literature and certain explicit results are known under more restrictive assumptions, these results are relatively difficult to apply in practice. In this paper, we investigate the factors that differentiate this model and the classical risk model. After reviewing various known results of this model, we derive a simulation approach for obtaining the probability of ultimate ruin based on importance sampling, which does not require specific distributions for the premium and the claim. We demonstrate this approach first with examples where the distribution of the sampling random variable can be identified. We then provide additional examples where we use the fast Fourier transform to obtain an approximation of the sampling random variable. The simulated results are compared with the known results for the probability of ruin. Using the simulation approach, we apply this model to a real-life auto-insurance data set. Differences with the classical model are then discussed. Finally, we comment on the suitability and impact of using this model in the context of solvency risk management. [ABSTRACT FROM AUTHOR]

Copyright of Romanian Journal of Information Technology & Automatic Control / Revista Română de Informatică și Automatică is the property of National Institute for Research & Development in Informatics - ICI Bucharest and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, we present a novel methodology to perform Bayesian inference for Cox processes in which the intensity function is driven by a diffusion process. The novelty lies in the fact that no discretization error is involved, despite the non-tractability of both the likelihood function and the transition density of the diffusion. The methodology is based on an MCMC algorithm and its exactness is built on retrospective sampling techniques. The efficiency of the methodology is investigated in some simulated examples and its applicability is illustrated in some real data analyzes. for this article are available online. [ABSTRACT FROM AUTHOR]

A super-structure system for probability densities, covering not just typical types but also fractional ones, was developed using the time scale theory. From a mathematical point of view, we discover duals of the Poisson process on the time scale T = R for the time scales T = Z and T = q Z , evaluating ∇ - calculus and Δ - calculus. Also, we search the fractional extensions of the Poisson process on these time scales and detect duals of them. A simulation allows for comparing the nabla and delta types of the observed distributions, not just typical types but also fractional ones. As an application, we also propose new substitution boxes (S-boxes) using the proposed stochastic models and compare the performance of S-boxes created in this way. Given that the S-box is the core for confusion in Advanced Encryption Standard (AES), the formation of these new S-boxes represents an interesting application of these stochastic models. [ABSTRACT FROM AUTHOR]

This article studies the finite time stability (FNTS) of time-varying nonlinear stochastic systems with random impulses. We provide sufficient conditions for the FNTS or even fixed time stability (FXTS) for time-varying stochastic systems under two cases: (1) The impulse time is deterministic; (2) The impulse time is random. For case (1), we use the reverse average dwell time to investigate FNTS, but for case (2), we employ the Poisson process theory to solve the stability problem. In addition, the stability criteria capture the stabilising effect of stochastic noise in the FNTS problem. Finally, the correctness of the theoretical results is verified with two numerical examples. As far as the author knows, no one has studied the FNTS of time-varying stochastic systems with random impulses. [ABSTRACT FROM AUTHOR]

The manuscript exemplifies a retrial queueing-inventory system with a maximum inventory level of S (= a × n) units, where a and n are finite positive integers. It consists of an impatient purchaser’s retention, an obligatory vacation, and an ordinary vacation. It comprises a single Poisson arrival who demands exactly a single unit from inventory. Inventories are filled in accordance with the (s, Q) ordering policy. The server must take an obligatory vacation after serving each ‘n’ number of items, where ‘n’ is fixed. And the server takes an ordinary vacation once the server finds zero inventory after returning from its obligatory vacation. The purchasers may enter the orbit of infinite size with a prefixed probability when the server is on any kind of vacation. The impatient purchasers in the orbit have the decision of abandoning the orbit with a probability of p 1 or retaining it with a complimentary probability of q 1 . Vacations, replenishment, retention of impatient purchasers, and inter-retry duration are distributed exponentially. The stationary state probability vector is derived using the matrix geometric method. In the steady-state case, the joint probability distribution of the number of purchasers in the orbit and the inventory level is evaluated. Numerical computations have been used to determine the convexity of the overall expected cost rate for various ‘n’ values. It is based on the results of the cost factors. Some effects of retention and the obligatory vacation in the system are scrutinized. [ABSTRACT FROM AUTHOR]

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process. [ABSTRACT FROM AUTHOR]

Models of generalized counting processes time-changed by a general inverse subordinator are considered, their distributions are characterized, and governing equations for them are presented. The equations are given in terms of the generalized fractional derivatives, namely, convolution-type derivatives with respect to Bernštein functions. Some particular examples are presented. [ABSTRACT FROM AUTHOR]

We revisit the 'classical' delta-shock model and generalize it to the case of renewal processes of external shocks with arbitrary inter-arrival times and arbitrary distribution of the 'recovery' parameter delta. Our innovative approach is based on defining the renewal points for the model and deriving the corresponding integral equations for the survival probabilities of interest that describe the setting probabilistically. As examples, the cases of exponentially distributed and constant delta are analyzed. Furthermore, delta shock modeling for systems with protection and two shock processes is considered. The first process targets the defense system and can partially destroy it. In this case, the second process that targets the main, protected system can result in its failure. The damages of the defense system are recovered during the recovery time delta. As exact solutions of the discussed problems are rather cumbersome, we provide simple and easy approximate solutions that can be implemented in practice. These results are justified under the assumption of 'fast repair' when the recovery time delta is stochastically much smaller than the inter-arrival times of the shock processes. The corresponding numerical examples (with discussion) illustrate our findings. [ABSTRACT FROM AUTHOR]

From a continuous-time long memory stochastic process, a discrete-time randomly sampled one is drawn using a renewal sampling process. We establish the existence of the spectral density of the sampled process, and we give its expression in terms of that of the initial process. We also investigate different aspects of the statistical inference on the sampled process. In particular, we obtain asymptotic results for the periodogram, the local Whittle estimator of the memory parameter and the long run variance of partial sums. We mainly focus on Gaussian continuous-time process. The challenge being that the randomly sampled process will no longer be jointly Gaussian. [ABSTRACT FROM AUTHOR]

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory. [ABSTRACT FROM AUTHOR]

Recently, Freud and Rodriguez proposed a new counting process which is called the Bell–Touchard process and based on the Bell–Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are "degenerate versions" of the Bell–Touchard probability distributions and the Bell–Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process. [ABSTRACT FROM AUTHOR]

The concurrency of edges, quantified by the number of edges that share a common node at a given time point, may be an important determinant of epidemic processes in temporal networks. We propose theoretically tractable Markovian temporal network models in which each edge flips between the active and inactive states in continuous time. The different models have different amounts of concurrency while we can tune the models to share the same statistics of edge activation and deactivation (and hence the fraction of time for which each edge is active) and the structure of the aggregate (i.e. static) network. We analytically calculate the amount of concurrency of edges sharing a node for each model. We then numerically study effects of concurrency on epidemic spreading in the stochastic susceptible-infectious-susceptible and susceptible-infectious-recovered dynamics on the proposed temporal network models. We find that the concurrency enhances epidemic spreading near the epidemic threshold, while this effect is small in many cases. Furthermore, when the infection rate is substantially larger than the epidemic threshold, the concurrency suppresses epidemic spreading in a majority of cases. In sum, our numerical simulations suggest that the impact of concurrency on enhancing epidemic spreading within our model is consistently present near the epidemic threshold but modest. The proposed temporal network models are expected to be useful for investigating effects of concurrency on various collective dynamics on networks including both infectious and other dynamics. [ABSTRACT FROM AUTHOR]

The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint distribution of a fixed number of upper order statistics. As applications of the result a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices. [ABSTRACT FROM AUTHOR]

We consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process. [ABSTRACT FROM AUTHOR]

Abstract Biodiversity monitoring is undergoing a revolution, with fauna observation data being increasingly gathered continuously over extended periods, through sensors like camera traps and acoustic recorders, or via opportunistic observations. These data are often analysed with discrete‐time ecological models, requiring the transformation of continuously collected data into arbitrarily chosen, non‐independent discrete‐time intervals. To overcome this issue, ecologists are increasingly turning to the existing continuous‐time models in the literature. Closer to the real detection process, they are lesser known than discrete‐time models, not always easily accessible and can be more complex. Focusing on occupancy models, a type of species distribution models, we asked ourselves: Should we dedicate time and effort to learning and using these continuous‐time models, or can we go on using discrete‐time models? We conducted a comparative simulation study using data generated within a continuous‐time framework. We assessed the performance of five static occupancy models with varying detection processes: discrete detection/non‐detection process, discrete count process, continuous‐time Poisson process and two types of modulated Poisson processes. Our goal was to assess their abilities to estimate occupancy probability with continuously collected data. We applied all models to empirical lynx data as an illustrative example. In scenarios with easily detectable animals, we found that all models accurately estimated occupancy. All models reached their limits with highly elusive animals. Variation in discretisation intervals had minimal impact on the discrete models' capacity to estimate occupancy accurately. Our study underscores that opting for continuous‐time models with an increased number of parameters, aiming to get closer to the sensor detection process, may not offer substantial advantages over simpler models when the sole aim is to accurately estimate occupancy. Model choice can thus be driven by practical considerations such as data availability or implementation time. However, occupancy models can encompass goals beyond estimating occupancy probability. Continuous‐time models, particularly those considering temporal variations in detection, can offer valuable insights into specific species behaviour and broader ecological inquiries. We hope that our findings offer valuable guidance for researchers and practitioners working with continuously collected data in wildlife monitoring and modelling.

Models of generalized counting processes time-changed by a general inverse subordinator are considered, their distributions are characterized, and governing equations for them are presented. The equations are given in terms of the generalized fractional derivatives, namely, convolution-type derivatives with respect to Bernštein functions. Some particular examples are presented.

The Susceptible–Infected–Susceptible (SIS) model in complex networks is one of the critical models employed in the modeling of virus spread. The study of the heterogeneous SIS model with a non-homogeneous nodal infection rate in finite-size networks has attracted little attention. Investigating the statistical properties of heterogeneous SIS epidemic dynamics in finite networks is thus intriguing. In this paper, we focus on the measure of variability in the number of infected nodes for the heterogeneous SIS epidemic dynamics in finite-size bipartite graphs and star graphs. Specifically, we investigate the metastable-state variance of the number of infected nodes for the SIS epidemic process in finite-size bipartite graphs and star graphs with heterogeneous nodal infection rates. We employ an extended individual-based mean-field approximation to analyze the heterogeneous SIS epidemic process in finite-size bipartite networks and star graphs. We derive the approximation solutions of the variance of the infected number. We verify the proposed theory by simulations. The proposed theory has the potential to help us better understand the fluctuations of SIS models like epidemic dynamics with a non-homogeneous infection rate. [ABSTRACT FROM AUTHOR]