Population dynamics model for aging.

The chronological age used in demography describes the linear evolution of the life of a living being. The chronological age cannot give precise information about the exact developmental stage or aging processes an organism has reached. On the contrary, the biological age (or epigenetic age) represents the true evolution of the tissues and organs of the living being. Biological age is not always linear and sometimes proceeds by discontinuous jumps. These jumps can be negative (we then speak of rejuvenation) or positive (in the event of premature aging), and they can be dependent on endogenous events such as pregnancy (negative jump) or stroke (positive jump) or exogenous ones such as surgical treatment (negative jump) or infectious disease (positive jump). The article proposes a mathematical model of the biological age by defining a valid model for the two types of jumps (positive and negative). The existence and uniqueness of the solution are solved, and its temporal dynamic is analyzed using a moments equation. We also provide some individual-based stochastic simulations.

A return-to-home model with commuting people and workers.

This article proposes a new model to describe human intra-city mobility. The goal is to combine the convection-diffusion equation to describe commuting people's movement and the density of individuals at home. We propose a new model extending our previous work with a compartment of office workers. To understand such a model, we use semi-group theory and obtain a convergence result of the solutions to an equilibrium distribution. We conclude this article by presenting some numerical simulations of the model.

Spectral Method in Epidemic Time Series: Application to COVID-19 Pandemic.

BACKGROUND: The age of infection plays an important role in assessing an individual's daily level of contagiousness, quantified by the daily reproduction number. Then, we derive an autoregressive moving average model from a daily discrete-time epidemic model based on a difference equation involving the age of infection. Novelty: The article's main idea is to use a part of the spectrum associated with this difference equation to describe the data and the model. RESULTS: We present some results of the parameters' identification of the model when all the eigenvalues are known. This method was applied to Japan's third epidemic wave of COVID-19 fails to preserve the positivity of daily reproduction. This problem forced us to develop an original truncated spectral method applied to Japanese data. We start by considering ten days and extend our analysis to one month. CONCLUSION: We can identify the shape for a daily reproduction numbers curve throughout the contagion period using only a few eigenvalues to fit the data.

Return-to-home model for short-range human travel.

In this work, we develop a mathematical model to describe the local movement of individuals by taking into account their return to home after a period of travel. We provide a suitable functional framework to handle this system and study the large-time behavior of the solutions. We extend our model by incorporating a colonization process and applying the return to home process to an epidemic.

Modeling Vaccine Efficacy for COVID-19 Outbreak in New York City.

In this article we study the efficacy of vaccination in epidemiological reconstructions of COVID-19 epidemics from reported cases data. Given an epidemiological model, we developed in previous studies a method that allowed the computation of an instantaneous transmission rate that produced an exact fit of reported cases data of the COVID-19 outbreak. In this article, we improve the method by incorporating vaccination data. More precisely, we develop a model in which vaccination is variable in its effectiveness. We develop a new technique to compute the transmission rate in this model, which produces an exact fit to reported cases data, while quantifying the efficacy of the vaccine and the daily number of vaccinated. We apply our method to the reported cases data and vaccination data of New York City.

What can we learn from COVID-19 data by using epidemic models with unidentified infectious cases?

The COVID-19 outbreak, which started in late December 2019 and rapidly spread around the world, has been accompanied by an unprecedented release of data on reported cases. Our objective is to offer a fresh look at these data by coupling a phenomenological description to the epidemiological dynamics. We use a phenomenological model to describe and regularize the reported cases data. This phenomenological model is combined with an epidemic model having a time-dependent transmission rate. The time-dependent rate of transmission involves changes in social interactions between people as well as changes in host-pathogen interactions. Our method is applied to cumulative data of reported cases for eight different geographic areas. In the eight geographic areas considered, successive epidemic waves are matched with a phenomenological model and are connected to each other. We find a single epidemic model that coincides with the best fit to the data of the phenomenological model. By reconstructing the transmission rate from the data, we can understand the contributions of the changes in social interactions (contacts between individuals) on the one hand and the contributions of the epidemiological dynamics on the other hand. Our study provides a new method to compute the instantaneous reproduction number that turns out to stay below 3.5 from the early beginning of the epidemic. We deduce from the comparison of several instantaneous reproduction numbers that the social effects are the most important factor in understanding the epidemic wave dynamics for COVID-19. The instantaneous reproduction number stays below 3.5, which implies that it is sufficient to vaccinate 71% of the population in each state or country considered in our study. Therefore, assuming the vaccines will remain efficient against the new variants and adjusting for higher confidence, it is sufficient to vaccinate 75-80% to eliminate COVID-19 in each state or country.

Clarifying predictions for COVID-19 from testing data: The example of New York State.

With the spread of COVID-19 across the world, a large amount of data on reported cases has become available. We are studying here a potential bias induced by the daily number of tests which may be insufficient or vary over time. Indeed, tests are hard to produce at the early stage of the epidemic and can therefore be a limiting factor in the detection of cases. Such a limitation may have a strong impact on the reported cases data. Indeed, some cases may be missing from the official count because the number of tests was not sufficient on a given day. In this work, we propose a new differential equation epidemic model which uses the daily number of tests as an input. We obtain a good agreement between the model simulations and the reported cases data coming from the state of New York. We also explore the relationship between the dynamic of the number of tests and the dynamics of the cases. We obtain a good match between the data and the outcome of the model. Finally, by multiplying the number of tests by 2, 5, 10, and 100 we explore the consequences for the number of reported cases.

Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data.

We model the COVID-19 coronavirus epidemic in China. We use early reported case data to predict the cumulative number of reported cases to a final size. The key features of our model are the timing of implementation of major public policies restricting social movement, the identification and isolation of unreported cases, and the impact of asymptomatic infectious cases.

Mathematical Parameters of the COVID-19 Epidemic in Brazil and Evaluation of the Impact of Different Public Health Measures.

A SIRU-type epidemic model is employed for the prediction of the COVID-19 epidemy evolution in Brazil, and analyze the influence of public health measures on simulating the control of this infectious disease. The proposed model allows for a time variable functional form of both the transmission rate and the fraction of asymptomatic infectious individuals that become reported symptomatic individuals, to reflect public health interventions, towards the epidemy control. An exponential analytical behavior for the accumulated reported cases evolution is assumed at the onset of the epidemy, for explicitly estimating initial conditions, while a Bayesian inference approach is adopted for the estimation of parameters by employing the direct problem model with the data from the first phase of the epidemy evolution, represented by the time series for the reported cases of infected individuals. The evolution of the COVID-19 epidemy in China is considered for validation purposes, by taking the first part of the dataset of accumulated reported infectious individuals to estimate the related parameters, and retaining the rest of the evolution data for direct comparison with the predicted results. Then, the available data on reported cases in Brazil from 15 February until 29 March, is used for estimating parameters and then predicting the first phase of the epidemy evolution from these initial conditions. The data for the reported cases in Brazil from 30 March until 23 April are reserved for validation of the model. Then, public health interventions are simulated, aimed at evaluating the effects on the disease spreading, by acting on both the transmission rate and the fraction of the total number of the symptomatic infectious individuals, considering time variable exponential behaviors for these two parameters. This first constructed model provides fairly accurate predictions up to day 65 below 5% relative deviation, when the data starts detaching from the theoretical curve. From the simulated public health intervention measures through five different scenarios, it was observed that a combination of careful control of the social distancing relaxation and improved sanitary habits, together with more intensive testing for isolation of symptomatic cases, is essential to achieve the overall control of the disease and avoid a second more strict social distancing intervention. Finally, the full dataset available by the completion of the present work is employed in redefining the model to yield updated epidemy evolution estimates.

Unreported Cases for Age Dependent COVID-19 Outbreak in Japan.

We investigate the age structured data for the COVID-19 outbreak in Japan. We consider a mathematical model for the epidemic with unreported infectious patient with and without age structure. In particular, we build a new mathematical model and a new computational method to fit the data by using age classes dependent exponential growth at the early stage of the epidemic. This allows to take into account differences in the response of patients to the disease according to their age. This model also allows for a heterogeneous response of the population to the social distancing measures taken by the local government. We fit this model to the observed data and obtain a snapshot of the effective transmissions occurring inside the population at different times, which indicates where and among whom the disease propagates after the start of public mitigation measures.

A cell-cell repulsion model on a hyperbolic Keller-Segel equation.

In this work, we discuss a cell-cell repulsion model based on a hyperbolic Keller-Segel equation with two populations, which aims at describing the cell growth and dispersion in the co-culture experiment from the work of Pasquier et al. (Biol Direct 6(1):5, 2011). We introduce the notion of solution integrated along the characteristics, which allows us to prove the existence and uniqueness of solutions and the segregation property for the two species. From a numerical perspective, we also observe that our model admits a competitive exclusion principle which is different from the classical competitive exclusion principle for the corresponding ODE model. More importantly, our model shows the complexity of the short term (6 days) co-cultured cell distribution depending on the initial distribution of each species. Through numerical simulations, we show that the impact of the initial distribution on the proportion of each species in the final population lies in the initial number of cell clusters and that the final proportion of each species is not influenced by the precise distribution of the initial distribution. We also find that a fast dispersion rate gives a short-term advantage while the vital dynamics contributes to a long-term population advantage. When the initial condition for the two species is not segregated, the numerical simulations suggest that asymptotic segregation occurs when the dispersion coefficients are not equal for two populations.

Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions.

We develop a mathematical model to provide epidemic predictions for the COVID-19 epidemic in Wuhan, China. We use reported case data up to 31 January 2020 from the Chinese Center for Disease Control and Prevention and the Wuhan Municipal Health Commission to parameterize the model. From the parameterized model, we identify the number of unreported cases. We then use the model to project the epidemic forward with varying levels of public health interventions. The model predictions emphasize the importance of major public health interventions in controlling COVID-19 epidemics.

Direct and indirect P-glycoprotein transfers in MCF7 breast cancer cells.

The efflux protein P-glycoprotein (P-gp) is over expressed in many cancer cells and has a known capacity to confer multi-drug resistance to cytotoxic therapies. We provide a mathematical model for the direct cell-to-cell transfer of proteins between cells and the indirect transfer between cells and the surrounding liquid. After a mathematical analysis of the model, we construct an adapted numerical scheme and give some numerical simulations. We observe that we obtain a better fit with the experimental data when we take into account the indirect transfer of the protein released in a dish. This quantity, usually neglected by the experimenters, seems to influence the results.

Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission.

A model of an epidemic outbreak incorporating multiple subgroups of susceptible and infected individuals is investigated. The asymptotic behavior of the model is analyzed and it is proved that the infected classes all converge to 0. A computational algorithm is developed for the cumulative final size of infected individuals over the course of the epidemic. The results are applied to the SARS epidemic in Singapore in 2003, where it is shown that the two-peak evolution of the infected population can be attributed to a two-group formulation of transmission.

The parameter identification problem for SIR epidemic models: identifying unreported cases.

A SIR epidemic model is analyzed with respect to identification of its parameters, based upon reported case data from public health sources. The objective of the analysis is to understand the relation of unreported cases to reported cases. In many epidemic diseases the ratio of unreported to reported cases is very high, and of major importance in implementing measures for controlling the epidemic. This ratio can be estimated by the identification of parameters for the model from reported case data. The analysis is applied to three examples: (1) the Hong Kong seasonal influenza epidemic in New York City in 1968-1969, (2) the bubonic plague epidemic in Bombay, India in 1906, and (3) the seasonal influenza epidemic in Puerto Rico in 2016-2017.

Competition for light in forest population dynamics: From computer simulator to mathematical model.

In this article we build a mathematical model for forest growth and we compare this model with a computer forest simulator named SORTIE. The main ingredient taken into account in both models is the competition for light between trees. The parameters of the mathematical model are estimated by using SORTIE model, when the parameter values of SORTIE model correspond to the ones previously evaluated for the Great Mountain Forest in USA. We see that the best fit of the parameters of the mathematical model is obtained when the competition for light influences only the growth rate of trees. We construct a size structured population dynamics model with one and two species and with spatial structure.

Special issue dedicated to the 70th birthday of Glenn F. Webb. Preface.

This special issue is dedicated to the 70th birthday of Glenn F. Webb. The topics of the 12 articles appearing in this special issue include evolutionary dynamics of population growth, spatio-temporal dynamics in reaction-diffusion biological models, transmission dynamics of infectious diseases, modeling of antibiotic-resistant bacteria in hospitals, analysis of Prion models, age-structured models in ecology and epidemiology, modeling of immune response to infections, modeling of cancer growth, etc. These topics partially represent the broad areas of Glenn's research interest.

A model of the 2014 ebola epidemic in west Africa with contact tracing.

A differential equations model is developed for the 2014 Ebola epidemics in Sierra Leone and Liberia. The model describes the dynamic interactions of the susceptible and infected populations of these countries. The model incorporates the principle features of contact tracing, namely, the number of contacts per identified infectious case, the likelihood that a traced contact is infectious, and the efficiency of the contact tracing process. The model is first fitted to current cumulative reported case data in each country. The data fitted simulations are then projected forward in time, with varying parameter regimes corresponding to contact tracing efficiencies. These projections quantify the importance of the identification, isolation, and contact tracing processes for containment of the epidemics.

Susceptible-infectious-recovered models revisited: from the individual level to the population level.

The classical susceptible-infectious-recovered (SIR) model, originated from the seminal papers of Ross [51] and Ross and Hudson [52,53] in 1916-1917 and the fundamental contributions of Kermack and McKendrick [36-38] in 1927-1932, describes the transmission of infectious diseases between susceptible and infective individuals and provides the basic framework for almost all later epidemic models, including stochastic epidemic models using Monte Carlo simulations or individual-based models (IBM). In this paper, by defining the rules of contacts between susceptible and infective individuals, the rules of transmission of diseases through these contacts, and the time of transmission during contacts, we provide detailed comparisons between the classical deterministic SIR model and the IBM stochastic simulations of the model. More specifically, for the purpose of numerical and stochastic simulations we distinguish two types of transmission processes: that initiated by susceptible individuals and that driven by infective individuals. Our analysis and simulations demonstrate that in both cases the IBM converges to the classical SIR model only in some particular situations. In general, the classical and individual-based SIR models are significantly different. Our study reveals that the timing of transmission in a contact at the individual level plays a crucial role in determining the transmission dynamics of an infectious disease at the population level.