R0R_0 and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission

In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Frechet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number $\mathcal{R}_0$

Optimal epidemic control by social distancing and vaccination of an infection structured by time since infection: the COVID-19 case study

Motivated by the issue of COVID-19 mitigation, in this work we tackle the general problem of optimally controlling an epidemic outbreak of a communicable disease structured by age since exposure, with the aid of two types of control instruments, namely social distancing and vaccination by a vaccine at least partly effective in protecting from infection. By our analyses we could prove the existence of (at least) one optimal control pair. We derived first-order necessary conditions for optimality and proved some useful properties of such optimal solutions. Our general model can be specialized to include a number of subcases relevant for epidemics like COVID-19, such as, e.g., the arrival of vaccines in a second stage of the epidemic, and vaccine rationing, making social distancing the only optimizable instrument. A worked example provides a number of further insights on the relationships between key control and epidemic parameters

Population dynamics and conservation biology of over-exploited Mediterranean red coral.

Abstract The main goal of ecologists is nowadays to foster habitat and species conservation. Life-history tables and Leslie-Lewis transition matrices of population growth can be powerful tools suitable for the study of age-structured over harvested and/or endangered species dynamics. Red coral (Corallium rubrum L 1758) is a modular anthozoan endemic to the Mediterranean Sea. This slow growing, long lived species has been harvested since ancient times. In the last decades harvesting pressure increased and the overall Mediterranean yield reduced by 2 3 . Moreover, mass mortality (putatively-linked to global warming) recently affected some coastal populations of this species. Red coral populations are discrete genetic units, gonochoric, composed by several overlapping generations and provided of a discrete (annual) reproduction. A population of this precious octocoral was studied in detail and its static life table was compiled. In order to simulate the trends overtime of the population under different environmental conditions and fishing pressures, a discrete, non-linear model, based on Leslie-Lewis transition matrix, was applied to the demographic data. In this model a bell-shaped curve, based on experimental data, representing the dependence of recruitment on adult colonies density was included. On these bases the stability of the population under different density, reproduction and mortality figures was analysed and simulations of the population trends overtime were set out. Some simulations were also carried out applying to the studied population the mortality values measured during the anomalous mass mortality event which really affected some red coral populations in 1999. The population under study showed high stability and a strong resilience capability, surviving to a 61% reduction of density, to a 27.7% reduction of reproduction rate and to an unselective harvesting affecting 95% of the reproductive colonies.

Asynchronous growth and competition in a two-sex age-structured population model

Asynchronous exponential growth has been extensively studied in population dynamics. In this paper we find out the asymptotic behaviour in a non-linear age-dependent model which takes into account sexual reproduction interactions. The main feature of our model is that the non-linear process converges to a linear one as the solution becomes large, so that the population undergoes asynchronous growth. The steady states analysis and the corresponding stability analysis are completely made and are summarized in a bifurcation diagram according to the parameter R0. Furthermore the effect of intraspecific competition is taken into account, leading to complex dynamics around steady states

Mathematical modeling of bacterial virulence and host–pathogen interactions in the Dictyostelium/Pseudomonas system

International audienceWe present some studies on the mechanisms of pathogenesis based on experimental work and on its interpretation through a mathematical model. Using a collection of clinical strains of the opportunistic human pathogen , we performed co-culture experiments with amoebae, to investigate the two organisms' interaction, characterized by a cross action between amoeba, feeding on bacteria, and bacteria exerting their pathogenic action against amoeba. In order to classify bacteria virulence, independently of this cross interaction, we have also performed killing experiments of bacteria against the nematode . A mathematical model was developed to infer how the populations of the amoeba-bacteria system evolve according to a number of parameters, taking into account the specific features underlying the interaction. The model does not fall within the class of traditional prey-predator models because not only does an amoeba feed on bacteria, but it is in turn attacked by them; thus the model must include a feedback term modeling this further interaction aspect. The model shows existence of multiple steady states and the resulting behavior of the solutions, showing bi-stability of the system, gives a qualitative explanation of the co-culture experiments

The basic approach to age-structured population dynamics: models, methods and numerics

This book provides an introduction to age-structured population modeling which emphasises the connection between mathematical theory and underlying biological assumptions. Through the rigorous development of the linear theory and the nonlinear theory alongside numerics, the authors explore classical equations that describe the dynamics of certain ecological systems. Modeling aspects are discussed to show how relevant problems in the fields of demography, ecology, and epidemiology can be formulated and treated within the theory. In particular, the book presents extensions of age-structured modelling to the spread of diseases and epidemics while also addressing the issue of regularity of solutions, the asymptotic behaviour of solutions, and numerical approximation. With sections on transmission models, non-autonomous models and global dynamics, this book fills a gap in the literature on theoretical population dynamics. The Basic Approach to Age-Structured Population Dynamics will appeal to graduate students and researchers in mathematical biology, epidemiology and demography who are interested in the systematic presentation of relevant models and mathematical methods