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We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.

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Inter- and intraspecific competition is most important during the immature life stage for many species of interest, such as multiple coexisting mosquito species that act as vectors of diseases. Mortality caused by competition that occurs during maturation is explicitly modelled in some alternative formulations of the Lotka-Volterra competition model. We generalise this approach by using a distributed delay for maturation time. The kernel of the distributed delay is represented by a truncated Erlang distribution. The shape and rate of the distribution, as well as the position of the truncation, are found to determine the solution at equilibrium. The resulting system of delay differential equations is transformed into a system of ordinary differential equations using the linear chain approximation. Numerical solutions are provided to demonstrate cases where competitive exclusion and coexistence occur. Stability conditions are determined using the nullclines method and local stability analysis. The introduction of a distributed delay promotes coexistence and survival of the species compared to the limiting case of a discrete delay, potentially affecting management of relevant pests and threatened species.

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Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a major simplification consists in assuming that the infectious period is exponentially distributed, then implying that the chance of recovery is independent on the time since infection. Here, we first attempt to investigate the consequences of relaxing this assumption on the performances of time-variant disease control strategies by using optimal control theory. In the framework of a basic susceptible-infected-removed (SIR) model, an Erlang distribution of the infectious period is considered and optimal isolation strategies are searched for. The objective functional to be minimized takes into account the cost of the isolation efforts per time unit and the sanitary costs due to the incidence of the epidemic outbreak. Applying the Pontryagin's minimum principle, we prove that the optimal control problem admits only bang-bang solutions with at most two switches. In particular, the optimal strategy could be postponing the starting intervention time with respect to the beginning of the outbreak. Finally, by means of numerical simulations, we show how the shape of the optimal solutions is affected by the different distributions of the infectious period, by the relative weight of the two cost components, and by the initial conditions.

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This paper introduced a study for a new system that consists of one unit with mixed standby units. The mathematical model for the system is constructed using semi-Markov model with regenerative point technique in two cases: the first case when there is preventive maintenance provided to the main unit and the second case when there is no preventive maintenance in the system. Life and repair times of the units in the system are assumed to be generally distributed with fuzzy parameters defined by the bell-shaped membership function. A Numerical application is introduced to compare the performance of the system in the two cases.

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The SARS-CoV-2 (COVID-19) pandemic has placed unprecedented demands on entire health systems and driven them to their capacity, so that health care professionals have been confronted with the difficult problem of ensuring appropriate staffing and resources to a high number of critically ill patients. In light of such high-demand circumstances, we describe an open web-accessible simulation-based decision support tool for a better use of finite hospital resources. The aim is to explore risk and reward under differing assumptions with a model that diverges from most existing models which focus on epidemic curves and related demand of ward and intensive care beds in general. While maintaining intuitive use, our tool allows randomized "what-if" scenarios which are key for real-time experimentation and analysis of current decisions' down-stream effects on required but finite resources over self-selected time horizons. While the implementation is for COVID-19, the approach generalizes to other diseases and high-demand circumstances.

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The SEIR (Susceptible-Exposed-Infected-Removed) model is widely used in epidemiology to mathematically model the spread of infectious diseases with incubation periods. However, the SEIR model prototype is generic and not able to capture the unique nature of a novel viral pandemic such as SARS-CoV-2. We have developed and tested a specialized version of the SEIR model, called SEAHIR (Susceptible-Exposed-Asymptomatic-Hospitalized-Isolated-Removed) model. This proposed model is able to capture the unique dynamics of the COVID-19 outbreak including further dividing the Infected compartment into: (1) "Asymptomatic", (2) "Isolated" and (3) "Hospitalized" to delineate the transmission specifics of each compartment and forecast healthcare requirements. The model also takes into consideration the impact of non-pharmaceutical interventions such as physical distancing and different testing strategies on the number of confirmed cases. We used a publicly available dataset from the United Arab Emirates (UAE) as a case study to optimize the main parameters of the model and benchmarked it against the historical number of cases. The SEAHIR model was used by decision-makers in Dubai's COVID-19 Command and Control Center to make timely decisions on developing testing strategies, increasing healthcare capacity, and implementing interventions to contain the spread of the virus. The novel six-compartment SEAHIR model could be utilized by decision-makers and researchers in other countries for current or future pandemics.

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The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0,∞) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

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An S L 1 L 2 I 1 I 2 A 1 A 2 R epidemic model is formulated that describes the spread of an epidemic in a population. The model incorporates an Erlang distribution of times of sojourn in incubating, symptomatically and asymptomatically infectious compartments. Basic properties of the model are explored, with focus on properties important in the context of current COVID-19 pandemic.

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We examine basic asymptotic properties of relative risk for two families of generalized Erlang processes (where each one is based off of a simplified Armitage and Doll multistage model) in order to predict relative risk data from cancer. The main theorems that we are able to prove are all corroborated by large clinical studies involving relative risk for former smokers and transplant recipients. We then show that at least some of these theorems do not extend to other Armitage and Doll multistage models. We conclude with suggestions for lifelong increased cancer screening for both former smoker and transplant recipient subpopulations of individuals and possible future directions of research.

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