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In the framework of population dynamics, the basic reproduction number ℛ0 is, by definition, the expected number of offspring that an individual has during its lifetime. In constant and time periodic environments, it is calculated as the spectral radius of the so‐called next‐generation operator. In continuously structured populations defined in a Banach lattice X with concentrated states at birth, one cannot define the next‐generation operator in X. In the present paper, we present an approach to compute the basic reproduction number of such models as the limit of the basic reproduction number of a sequence of models for which ℛ0 can be computed as the spectral radius of the next‐generation operator. We apply these results to some examples: the (classical) size‐dependent model, a size‐structured cell population model, a size‐structured model with diffusion in structure space (under some particular assumptions), and a (physiological) age‐structured model with diffusion in structure space. ; This work has been partially supported by the project MTM2017-84214-C2-2-P of the Spanish government

Tuberculosis (TB) will be a serious threat if not handled quickly and appropriately. The relatively long treatment in time and the high risk of death is a challenge in controlling the spread of this disease. The DOTS (Directly Observed Treatment Short-course) strategy is considered capable of controlling the spread of TB because of the high success rate, reaching 91%. The mathematical model of the spread of TB has been widely studied to determine the potential for the spread of this disease in an area. The purpose of this study is to build a model of tuberculosis spread in Lamongan to determine the rate of its spread and to predict whether it will be endemic or not. Using disease spread mathematical model type SEITR, this research has examined based on 2018 and 2019 data from the Lamongan health office then simulates the result. The research begins with the construction of the model followed by a stability analysis of the model by determining the basic reproduction number ( ), which is simulated after the parameter approach was carried out. From the simulation results, the result shows that means that TB will not endemic in Lamongan. Besides, the results of the effect of parameter ω on I(t) were obtained, which concluded that seeking and treating active TB alone would only reduce infected individuals but not reduce the length of time TB spread. Identification of the effect of parameter φ on I(t) has also been carried out which results in the conclusion that more treatment for susceptible individuals, in addition to reducing the number of infected individuals, will also reduce the length of time TB spreads in that area. This result will be a good suggestion for the government to deal faster with tuberculosis transmission.

A comprehensive investigation on the variation of the Covid-19 pandemic adequate reproduction number (R) in four critical European countries is provided in this study as a function of the variations related to the weather, mobility, government responses, and epidemiology. In this study, an open data set about the Covid-19 pandemic is used for the analysis. The data contains newly recorded components with different metrics since the appearance of the COVID19 pandemic. The aim is to investigate the impact of the weather, mobility, and government restriction on the R values and pandemic. The appropriate statistical analyses are used to reveal the association and the effect of the various attributes to the R. In addition, multiple predictive models were applied. The results show significant differences between studied countries and have different factors that influence the virus's spread.

Throughout the past six months, no number has dominated the public media more persistently than the reproduction number of COVID-19. This powerful but simple concept is widely used by the public media, scientists, and political decision makers to explain and justify political strategies to control the COVID-19 pandemic. Here we explore the effectiveness of political interventions using the reproduction number of COVID-19 across Europe. We propose a dynamic SEIR epidemiology model with a time-varying reproduction number, which we identify using machine learning. During the early outbreak, the basic repro6.33duction number was 4.22±1.69, with maximum values of and 5.88 in Germany and the Netherlands. By May 10, 2020, it dropped to 0.67±0.18, with minimum values of 0.37 and 0.28 in Hungary and Slovakia. We found a strong correlation between passenger air travel, driving, walking, and transit mobility and the effective reproduction number with a time delay of 17.24±2.00 days model provides the flexibility to simulate various outbreak. Our new dynamic SEIR control and exit strategies to inform political decision making and identify safe solutions in the benefit of global health.

Due to the particularities of SARS-CoV-2, public health policies have played a crucial role in the control of the COVID-19 pandemic. Epidemiological parameters for assessing the stage of the outbreak, such as the Effective Reproduction Number (R-t), are not always straightforward to calculate, raising barriers between the scientific community and non-scientific decision-making actors. The combination of estimators ofR(t)with elaborated Machine Learning-based forecasting techniques provides a way to support decision-making when assessing governmental plans of action. In this work, we develop forecast models applying logistic growth strategies and auto-regression techniques based on Auto-Regressive Integrated Moving Average (ARIMA) models for each country that records information about the COVID-19 outbreak. Using the forecast for the main variables of the outbreak, namely the number of infected (I), recovered (R), and dead (D) individuals, we provide a real-time estimation ofR(t)and its temporal evolution within a timeframe. With such models, we evaluateR(t)trends at the continental and country levels, providing a clear picture of the effect governmental actions have had on the spread. We expect this methodology of combining forecast models for raw data to calculateR(t)to serve as valuable input to support decision-making related to controlling the spread of SARS-CoV-2. ; Centre for Biotechnology and Bioengineering-CeBiB (PIA project, Conicyt, Chile) FB0001 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) 21181435

Due to the particularities of SARS-CoV-2, public health policies have played a crucial role in the control of the COVID-19 pandemic. Epidemiological parameters for assessing the stage of the outbreak, such as the Effective Reproduction Number (R_t), are not always straightforward to calculate, raising barriers between the scientific community and non-scientific decision-making actors. The combination of estimators of R_t with elaborated Machine Learning-based forecasting techniques provides a way to support decision-making when assessing governmental plans of action. In this work, we develop forecast models applying logistic growth strategies and auto-regression techniques based on Auto-Regressive Integrated Moving Average (ARIMA) models for each country that records information about the COVID-19 outbreak. Using the forecast for the main variables of the outbreak, namely the number of infected (I), recovered (R), and dead (D) individuals, we provide a real-time estimation of R_t and its temporal evolution within a timeframe. With such models, we evaluate R_t trends at the continental and country levels, providing a clear picture of the effect governmental actions have had on the spread. We expect this methodology of combining forecast models for raw data to calculate R_t to serve as valuable input to support decision-making related to controlling the spread of SARS-CoV-2.

We show that the acceleration index, a novel indicator that measures acceleration and deceleration of viral spread (Baunez et al. 2020a,b), is essentially a test-controlled version of the reproduction number. As such it is a more accurate indicator to track the dynamics of an infectious disease outbreak in real time. We indicate a discrepancy between the acceleration index and the reproduction number, based on the infectivity and test rates and we provide a formal decomposition of this difference. When applied to French data for the ongoing COVID-19 pandemic, our decomposition shows that the reproduction number consistently underestimates the resurgence of the pandemic since the summer of 2020, compared to the acceleration index which accounts for the time-varying volume of tests. Because the acceleration index aggregates all the relevant information and captures in real time the sizeable time variation featured by viral circulation, it is a sufficient statistic to track the pandemic's propagation.

We show that the acceleration index, a novel indicator that measures acceleration and deceleration of viral spread (Baunez et al. 2020a,b), is essentially a test-controlled version of the reproduction number. As such it is a more accurate indicator to track the dynamics of an infectious disease outbreak in real time. We indicate a discrepancy between the acceleration index and the reproduction number, based on the infectivity and test rates and we provide a formal decomposition of this difference. When applied to French data for the ongoing COVID-19 pandemic, our decomposition shows that the reproduction number consistently underestimates the resurgence of the pandemic since the summer of 2020, compared to the acceleration index which accounts for the time-varying volume of tests. Because the acceleration index aggregates all the relevant information and captures in real time the sizeable time variation featured by viral circulation, it is a sufficient statistic to track the pandemic's propagation.

We show that the acceleration index, a novel indicator that measures acceleration and deceleration of viral spread (Baunez et al. 2020a,b), is essentially a test-controlled version of the reproduction number. As such it is a more accurate indicator to track the dynamics of an infectious disease outbreak in real time. We indicate a discrepancy between the acceleration index and the reproduction number, based on the infectivity and test rates and we provide a formal decomposition of this difference. When applied to French data for the ongoing COVID-19 pandemic, our decomposition shows that the reproduction number consistently underestimates the resurgence of the pandemic since the summer of 2020, compared to the acceleration index which accounts for the time-varying volume of tests. Because the acceleration index aggregates all the relevant information and captures in real time the sizeable time variation featured by viral circulation, it is a sufficient statistic to track the pandemic's propagation.

We show that the acceleration index, a novel indicator that measures acceleration and deceleration of viral spread (Baunez et al. 2020a,b), is essentially a test-controlled version of the reproduction number. As such it is a more accurate indicator to track the dynamics of an infectious disease outbreak in real time. We indicate a discrepancy between the acceleration index and the reproduction number, based on the infectivity and test rates and we provide a formal decomposition of this difference. When applied to French data for the ongoing COVID-19 pandemic, our decomposition shows that the reproduction number consistently underestimates the resurgence of the pandemic since the summer of 2020, compared to the acceleration index which accounts for the time-varying volume of tests. Because the acceleration index aggregates all the relevant information and captures in real time the sizeable time variation featured by viral circulation, it is a sufficient statistic to track the pandemic's propagation.