Deux modèles de population dans un environnement périodique lent ou rapide.

Two problems in population dynamics are addressed in a slow or rapid periodic environment. We first obtain a Taylor expansion for the probability of non-extinction of a supercriticial linear birth-and-death process with periodic coefficients when the period is large or small. If the birth rate is lower than the mortality for part of the period and the period tends to infinity, then the probability of non-extinction tends to a discontinuous limit related to a "canard" in a slow-fast system.

An age-structured epidemic model for the demographic transition.

In this paper, we formulate an age-structured epidemic model for the demographic transition in which we assume that the cultural norms leading to lower fertility are transmitted amongst individuals in the same way as infectious diseases. First, we formulate the basic model as an abstract homogeneous Cauchy problem on a Banach space to prove the existence, uniqueness, and well-posedness of solutions. Next based on the normalization arguments, we investigate the existence of nontrivial exponential solutions and then study the linearized stability at the exponential solutions using the idea of asynchronous exponential growth.

[On the extinction of populations with several types in a random environment].

This study focuses on the extinction rate of a population that follows a continuous-time multi-type branching process in a random environment. Numerical computations in a particular example inspired by an epidemic model suggest an explicit formula for this extinction rate, but only for certain parameter values.

Sur la vitesse d'extinction d'une population dans un environnement aléatoire.

This study focuses on the speed of extinction of a population living in a random environment that follows a continuous-time Markov chain. Each individual dies or reproduces at a rate that depends on the environment. The number of offspring during reproduction follows a given probability law that also depends on the environment.

Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire.

An explicit formula is found for the rate of extinction of subcritical linear birth-and-death processes in a random environment. The formula is illustrated by numerical computations of the eigenvalue with largest real part of the truncated matrix for the master equation. The generating function of the corresponding eigenvector satisfies a Fuchsian system of singular differential equations.

Feasibility of achieving the 2025 WHO global tuberculosis targets in South Africa, China, and India: a combined analysis of 11 mathematical models.

Background: The post-2015 End TB Strategy proposes targets of 50% reduction in tuberculosis incidence and 75% reduction in mortality from tuberculosis by 2025. We aimed to assess whether these targets are feasible in three high-burden countries with contrasting epidemiology and previous programmatic achievements.

Methods: 11 independently developed mathematical models of tuberculosis transmission projected the epidemiological impact of currently available tuberculosis interventions for prevention, diagnosis, and treatment in China, India, and South Africa.

Le modèle stochastique SIS pour une épidémie dans un environnement aléatoire.

The stochastic SIS epidemic model in a random environment. In a random environment that is a two-state continuous-time Markov chain, the mean time to extinction of the stochastic SIS epidemic model grows in the supercritical case exponentially with respect to the population size if the two states are favorable, and like a power law if one state is favorable while the other is unfavorable.

Assessment of epidemic projections using recent HIV survey data in South Africa: a validation analysis of ten mathematical models of HIV epidemiology in the antiretroviral therapy era.

Background: Mathematical models are widely used to simulate the effects of interventions to control HIV and to project future epidemiological trends and resource needs. We aimed to validate past model projections against data from a large household survey done in South Africa in 2012.

Methods: We compared ten model projections of HIV prevalence, HIV incidence, and antiretroviral therapy (ART) coverage for South Africa with estimates from national household survey data from 2012.

On the stochastic SIS epidemic model in a periodic environment.

In the stochastic SIS epidemic model with a contact rate a, a recovery rate b < a, and a population size N, the mean extinction time τ is such that (log τ)/N converges to c = b/a - 1 - log(b/a) as N grows to infinity. This article considers the more realistic case where the contact rate a(t) is a periodic function whose average is bigger than b. Then log τ/N converges to a new limit C, which is linked to a time-periodic Hamilton-Jacobi equation.

On linear birth-and-death processes in a random environment.

We study the probability of extinction for single-type and multi-type continuous-time linear birth-and-death processes in a finite Markovian environment. The probability of extinction is equal to 1 almost surely if and only if the basic reproduction number R(0) is ≤ 1, the key point being to identify a suitable definition of R(0) for such a result to hold.

On the probability of extinction in a periodic environment.

For a certain class of multi-type branching processes in a continuous-time periodic environment, we show that the extinction probability is equal to (resp. less than) 1 if the basic reproduction number R(0) is less than (resp. bigger than) 1.

On the basic reproduction number in a random environment.

The concept of basic reproduction number R0 in population dynamics is studied in the case of random environments. For simplicity the dependence between successive environments is supposed to follow a Markov chain. R0 is the spectral radius of a next-generation operator.

On the biological interpretation of a definition for the parameter R₀ in periodic population models.

An adaptation of the definition of the basic reproduction number R (0) to time-periodic seasonal models was suggested a few years ago. However, its biological interpretation remained unclear. The present paper shows that in demography, this R (0) is the asymptotic ratio of total births in two successive generations of the family tree.

Persistence in seasonally forced epidemiological models.

In this paper we address the persistence of a class of seasonally forced epidemiological models. We use an abstract theorem about persistence by Fonda. Five different examples of application are given.

The model of Kermack and McKendrick for the plague epidemic in Bombay and the type reproduction number with seasonality.

The figure showing how the model of Kermack and McKendrick fits the data from the 1906 plague epidemic in Bombay is the most reproduced figure in books discussing mathematical epidemiology. In this paper we show that the assumption of constant parameters in the model leads to quite unrealistic numerical values for these parameters. Moreover the reports published at the time show that plague epidemics in Bombay occurred in fact with a remarkable seasonal pattern every year since 1897 and at least until 1911.

Evaluating the cost-effectiveness of pre-exposure prophylaxis (PrEP) and its impact on HIV-1 transmission in South Africa.

Background: Mathematical modelers have given little attention to the question of how pre-exposure prophylaxis (PrEP) may impact on a generalized national HIV epidemic and its cost-effectiveness, in the context of control strategies such as condom use promotion and expanding ART programs.

Methodology/principal Findings: We use an age- and gender-structured model of the generalized HIV epidemic in South Africa to investigate the potential impact of PrEP in averting new infections. The model utilizes age-structured mortality, fertility, partnership and condom use data to model the spread of HIV and the shift of peak prevalence to older age groups.

Genealogy with seasonality, the basic reproduction number, and the influenza pandemic.

The basic reproduction number R (0) has been used in population biology, especially in epidemiology, for several decades. But a suitable definition in the case of models with periodic coefficients was given only in recent years. The definition involves the spectral radius of an integral operator.

An age-structured model for the potential impact of generalized access to antiretrovirals on the South African HIV epidemic.

A simple mathematical model (Granich et al., Lancet 373:48-57, 2009) suggested recently that annual HIV testing of the population, with all detected HIV(+) individuals immediately treated with antiretrovirals, could lead to the long-term decline of HIV in South Africa and could save millions of lives in the next few years. However, the model suggested that the long-term decline of HIV could not be achieved with less frequent HIV testing.

On the final size of epidemics with seasonality.

We first study an SIR system of differential equations with periodic coefficients describing an epidemic in a seasonal environment. Unlike in a constant environment, the final epidemic size may not be an increasing function of the basic reproduction number R(0) or of the initial fraction of infected people. Moreover, large epidemics can happen even if R(0) < 1.

Periodic matrix population models: growth rate, basic reproduction number, and entropy.

This article considers three different aspects of periodic matrix population models. First, a formula for the sensitivity analysis of the growth rate lambda is obtained that is simpler than the one obtained by Caswell and Trevisan. Secondly, the formula for the basic reproduction number R0 in a constant environment is generalized to the case of a periodic environment.

Resonance of the epidemic threshold in a periodic environment.

Resonance between some natural period of an endemic disease and a seasonal periodic contact rate has been the subject of intensive study. This paper does not focus on resonance for endemic diseases but on resonance for emerging diseases. Periodicity can have an important impact on the initial growth rate and therefore on the epidemic threshold.

Modeling the joint epidemics of TB and HIV in a South African township.

We present a simple mathematical model with six compartments for the interaction between HIV and TB epidemics. Using data from a township near Cape Town, South Africa, where the prevalence of HIV is above 20% and where the TB notification rate is close to 2,000 per 100,000 per year, we estimate some of the model parameters and study how various control measures might change the course of these epidemics. Condom promotion, increased TB detection and TB preventive therapy have a clear positive effect.