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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.

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.

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., (Copyright © 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.)

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.

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., (Copyright © 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.)

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., (Copyright © 2016 The Author(s). Published by Elsevier Ltd. This is an Open Access article under the CC BY license. Published by Elsevier Ltd.. All rights reserved.)

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.

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. Model projections for 2012 were made before the publication of the 2012 household survey. We compared adult (age 15-49 years) HIV prevalence in 2012, the change in prevalence between 2008 and 2012, and prevalence, incidence, and ART coverage by sex and by age groups between model projections and the 2012 household survey., Findings: All models projected lower prevalence estimates for 2012 than the survey estimate (18·8%), with eight models' central projections being below the survey 95% CI (17·5-20·3). Eight models projected that HIV prevalence would remain unchanged (n=5) or decline (n=3) between 2008 and 2012, whereas prevalence estimates from the household surveys increased from 16·9% in 2008 to 18·8% in 2012 (difference 1·9, 95% CI -0·1 to 3·9). Model projections accurately predicted the 1·6 percentage point prevalence decline (95% CI -0·3 to 3·5) in young adults aged 15-24 years, and the 2·2 percentage point (0·5 to 3·9) increase in those aged 50 years and older. Models accurately represented the number of adults on ART in 2012; six of ten models were within the survey 95% CI of 1·54-2·12 million. However, the differential ART coverage between women and men was not fully captured; all model projections of the sex ratio of women to men on ART were lower than the survey estimate of 2·22 (95% CI 1·73-2·71)., Interpretation: Projections for overall declines in HIV epidemics during the ART era might have been optimistic. Future treatment and HIV prevention needs might be greater than previously forecasted. Additional data about service provision for HIV care could help inform more accurate projections., Funding: Bill & Melinda Gates Foundation., (Copyright © 2015 Eaton et al. Open Access article distributed under the terms of CC BY-NC-ND. Published by Elsevier Ltd.. All rights reserved.)

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.

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.

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.

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.

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.

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 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.

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.

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. The model shows that universal PrEP coverage would have to be impractically high to have a significant effect on incidence reduction while ART coverage expands. PrEP targeted to 15-35-year-old women would avert 10%-25% (resp. 13%-28%) of infections in this group and 5%-12% (resp. 7%-16%) of all infections in the period 2014-2025 if baseline incidence is 0.5% per year at 2025 (resp. 0.8% per year at 2025). The cost would be $12,500-$20,000 per infection averted, depending on the level of ART coverage and baseline incidence. An optimistic scenario of 30%-60% PrEP coverage, efficacy of at least 90%, no behavior change among PrEP users and ART coverage less than three times its 2010 levels is required to achieve this result. Targeting PrEP to 25-35-year-old women (at highest risk of infection) improves impact and cost-effectiveness marginally. Relatively low levels of condom substitution (e.g., 30%) do not nullify the efficacy of PrEP, but reduces cost-effectiveness by 35%-40%., Conclusions/significance: PrEP can avert as many as 30% of new infections in targeted age groups of women at highest risk of infection. The cost-effectiveness of PrEP relative to ART decreases rapidly as ART coverage increases beyond three times its coverage in 2010, after which the ART program would provide coverage to more than 65% of HIV(+) individuals. To have a high relative cost-effective impact on reducing infections in generalized epidemics, PrEP must utilize a window of opportunity until ART has been scaled up beyond this level.

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.

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.

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.

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. Resonance occurs when the Euler-Lotka equation has a complex root with an imaginary part (i.e., a natural frequency) close to the angular frequency of the contact rate and a real part not too far from the Malthusian parameter. This is a kind of continuous-time analogue of work by Tuljapurkar on discrete-time population models, which in turn was motivated by the work by Coale on continuous-time demographic models with a periodic birth. We illustrate this resonance phenomenon on several simple epidemic models with contacts varying periodically on a weekly basis, and explain some surprising differences, e.g., between a periodic SEIR model with an exponentially distributed latency and the same model but with a fixed latency.

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. The impact of antiretroviral therapy on the incidence of HIV is unclear and depends on the extent to which it reduces sexual transmission. However, our analysis suggests that it will greatly reduce the TB notification rate.

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.

Heterogeneity in sexual behavior is known to play an important role in the spread of HIV. In 1986, a mathematical model based on ordinary differential equations was introduced to take into account the distribution of sexual activity. Assuming proportionate mixing, it was shown that the basic reproduction number R0 determining the epidemic threshold was proportional to M +V/M, where M is the mean and V the variance of the distribution. In the present paper, we notice that this theoretical distribution is different from the one obtained in behavioral surveys for the number of sexual partnerships over a period of length tau. The latter is a "mixed Poisson distribution" whose mean m and variance v are such that M = m/tau and V = (v-m)/tau(2). So M+V/M =(m+ v/m-1)/tau. This way, we improve the link between theory and data for sexual activity models of HIV/AIDS epidemics. As an example, we consider data concerning sex workers and their clients in Yunnan, China, and find an upper bound for the geometric mean of the transmission probabilities per partnership in this context.

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R(0) of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p(0) (1+epsilon cos(omegat - phi)) with epsilon << 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p(0). The maximum correction due to the second term is (epsilon(2)/8)% and always tends to decrease R(0). The basic reproduction number R(0) is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R(0) are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality.

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R (0) which is adapted to periodic environments. This R (0) is estimated numerically for the epidemic in Chichaoua; approximately 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor approximately 3.76.

This paper presents a mathematical model of the HIV/AIDS epidemic in Kunming, the provincial capital of Yunnan, China. The population is divided into several groups, with individuals possibly changing group. Two transmission routes of HIV are considered: needle sharing between injecting drug users (IDUs) and commercial sex between female sex worker (FSWs) and clients. The model includes male IDUs who are also clients and female IDUs who are also FSWs. Groups are split in two--risky and safe--according to condom use and needle sharing. A system of partial differential equations is derived to describe the spread of the disease. For the simulation, parameters are chosen to fit as much as possible data publicly available for Kunming. Some mathematical properties of the model--in particular the epidemic threshold R0 which determines the goal of public health interventions--are also presented. Though the model couples two transmission routes of HIV, the approximation R0 approximately = max[R0(IDU), R0(sex)], with closed formulas for R0(IDU) and R0(sex), appears to be quite good. The critical levels of condom use and clean needle use necessary to stop both the sexual transmission and the transmission among IDUs can therefore be determined independently.

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).