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. Secondly, a nonlinear S-I-R epidemic model is studied when the contact rate fluctuates rapidly. The final size of the epidemic is close to that obtained by replacing the contact rate with its average. An approximation of the correction can be calculated analytically when the basic reproduction number of the epidemic is close to 1. The correction term, which can be either positive or negative, is proportional to both the period of oscillations and the initial fraction of infected people.

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. The relative stability defined in the normalized system and the absolute (orbital) stability in the original system are examined. For the boundary exponential solutions corresponding to infection-free or totally infected status, we formulate the stability condition using reproduction numbers. We show that bi-unstability of boundary exponential solutions is one of conditions which guarantee the existence of coexistent exponential solutions.

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. A particular attention is set on the case of two environments, which leads to Riemann's differential equation.

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.

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. When a(t) is a cosine function with small amplitude or high (resp. low) frequency, approximate formulas for C can be obtained analytically following the method used in Assaf et al. (Phys Rev E 78:041123, 2008). These results are illustrated by numerical simulations.

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. The proof uses results concerning the asymptotic behavior of cooperative systems of differential equations. In epidemiology the extinction probability may be used as a time-periodic measure of the epidemic risk. As an example we consider a linearized SEIR epidemic model and data from the recent measles epidemic in France. Discrete-time models with potential applications in conservation biology are also discussed.

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 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. Its position with respect to 1 always determines population growth or decay in simulations, unlike another parameter suggested in a recent article (Hernandez-Suarez et al., Theor Popul Biol, doi: 10.1016/j.tpb.2012.05.004 , 2012). The position of the latter with respect to 1 determines growth or decay of the population's expectation. R0 is easily computed in the case of scalar population models without any structure. The main emphasis is on discrete-time models but continuous-time models are also considered.

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. So the 1906 epidemic is clearly not a good example of epidemic stopping because the number of susceptible humans has decreased under a threshold, as suggested by Kermack and McKendrick, but an example of epidemic driven by seasonality. We present a seasonal model for the plague in Bombay and compute the type reproduction numbers associated with rats and fleas, thereby extending to periodic models the notion introduced by Roberts and Heesterbeek.

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. In epidemiology, it is the asymptotic ratio of total infections in two successive generations of the infection tree. This result is compared with other recent work.

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.

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. As in the study of structured epidemic models in a constant environment, there is a need to emphasize the biological meaning of this spectral radius. In this paper we show that R (0) for periodic models is still an asymptotic per generation growth rate. We also emphasize the difference between this theoretical R (0) for periodic models and the "reproduction number" obtained by fitting an exponential to the beginning of an epidemic curve. This difference has been overlooked in recent studies of the H1N1 influenza pandemic.

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. Many observers argued that an annual testing rate was very difficult in practice. Small scale trials are nevertheless in preparation. In this paper, we use a more realistic age-structured model, which suggests that the recent high levels of reported condom use could already lead to a long-term decline of HIV in South Africa. The model therefore suggests that trials with for example 20% of the population tested each year would also be interesting. They would have similar (though smaller) advantages in terms of reduction of mortality and incidence, would be much easier to generalize to larger populations, and would not lead to long term persistence of HIV. Our model simulations also suggest that the age distribution of incidence has changed considerably over the past 20 years in South Africa. This raises some concern about an assumption presently used in EPP/Spectrum, the software used by UNAIDS for its estimates.

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. Some inequalities between lambda and R0 proved by Cushing and Zhou are also generalized to the periodic case. Finally, we add some remarks on Demetrius' notion of evolutionary entropy H and its relationship to the growth rate lambda in the periodic case.

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. But like in a constant environment, the final epidemic size tends to 0 when R(0) < 1 and the initial fraction of infected people tends to 0. When R(0) > 1, the final epidemic size is bigger than the fraction 1 - 1/R(0) of the initially nonimmune population. In summary, the basic reproduction number R(0) keeps its classical threshold property but many other properties are no longer true in a seasonal environment. These theoretical results should be kept in mind when analyzing data for emerging vector-borne diseases (West-Nile, dengue, chikungunya) or air-borne diseases (SARS, pandemic influenza); all these diseases being influenced by seasonality.

[On the extinction of populations with several types in a random environment]. / Sur l'extinction des populations avec plusieurs types dans un environnement aléatoire.

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.

Growth rate and basic reproduction number for population models with a simple periodic factor.

For continuous-time population models with a periodic factor which is sinusoidal, both the growth rate and the basic reproduction number are shown to be the largest roots of simple equations involving continued fractions. As an example, we reconsider an SEIS model with a fixed latent period, an exponentially distributed infectious period and a sinusoidal contact rate studied in Williams and Dye [B.G. Williams, C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci. 145 (1997) 77]. We show that apart from a few exceptional parameter values, the epidemic threshold depends not only on the mean contact rate, but also on the amplitude of fluctuations.

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. In the so-called subcritical case where the population goes for sure to extinction, there is an explicit formula for the speed of extinction. In some sense, environmental stochasticity slows down population extinction.

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. Models were calibrated with data on tuberculosis incidence and mortality in 2012. Representatives from national tuberculosis programmes and the advocacy community provided distinct country-specific intervention scenarios, which included screening for symptoms, active case finding, and preventive therapy. FINDINGS: Aggressive scale-up of any single intervention scenario could not achieve the post-2015 End TB Strategy targets in any country. However, the models projected that, in the South Africa national tuberculosis programme scenario, a combination of continuous isoniazid preventive therapy for individuals on antiretroviral therapy, expanded facility-based screening for symptoms of tuberculosis at health centres, and improved tuberculosis care could achieve a 55% reduction in incidence (range 31-62%) and a 72% reduction in mortality (range 64-82%) compared with 2015 levels. For India, and particularly for China, full scale-up of all interventions in tuberculosis-programme performance fell short of the 2025 targets, despite preventing a cumulative 3·4 million cases. The advocacy scenarios illustrated the high impact of detecting and treating latent tuberculosis. INTERPRETATION: Major reductions in tuberculosis burden seem possible with current interventions. However, additional interventions, adapted to country-specific tuberculosis epidemiology and health systems, are needed to reach the post-2015 End TB Strategy targets at country level. FUNDING: Bill and Melinda Gates Foundation.

Fuelwood harvesting in Niger and a generalization of Faustmann's formula.

In some forests of Niger where 'controlled rural markets' have been organized, fuelwood is harvested following a policy of the form: every T year, cut the dead trees and those live trees which have a diameter greater than D. Dead trees generally form the main part of the harvest. In this paper, we present a simple continuous time model for the management of these uneven-aged stands subject to a high natural death rate alpha, and we derive a formula for the cycle length and the diameter optimizing the discounted income over an infinite horizon. Faustmann's classical formula for even-aged stands corresponds to the limit alpha --> 0 and D = 0 (clear-cut).