Numerical analysis of generalised max-plus eigenvalue problems

AbstractThis paper is concerned with the deterministic discrete-time infinite horizon optimisation problem on a compact metric space with an average cost criterion involving two functions K (the “cost”) and T (the “time”). Firstly, we collect the different characterisations of the value λ in terms of generalised max-plus eigenvalue problem and in terms of linear programming. Secondly, we prove an error bound on λ when the space is discretised

Evaluating the Cost-Effectiveness of Pre-Exposure Prophylaxis (PrEP) and Its Impact on HIV-1 Transmission in South Africa

The original publication is available at http:/www.plosone.orgBackground: 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. © 2010 Pretorius et al.Publishers' Versio

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 eff ects 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 (diff erence 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 diff erential 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

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

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. 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

Le paramètre R0 pour la dynamique des populations dans un environnement périodique

Ce mémoire porte principalement sur la définition et les propriétés du paramètre appelé R0 en dynamique des populations dans le cas des modèles à coefficients périodiques en temps. Ce paramètre représente le taux auquel croissent les générations successives d’une population. Il est très utilisé en épidémiologie

Sur le modèle stochastique SIS pour une épidémie dans un environnement périodique

traduction de Bacaër, N. On the stochastic SIS epidemic model in a periodic environment. J. Math. Biol. 71, 491–511 (2015). https://doi.org/10.1007/s00285-014-0828-1In dem stochastichen SIS Modell für eine Epidemie mit einer Kontaktrate a, mit einer Heilungsrate b < a und mit einer Bevölkerungszahl N ist der Erwartungswert τ der Zeit bis zum Unterbruch der Epidemie so, daß (log τ) /N nach c = b/a−1−log(b/a) konvergiert wenn N unendlich wird. Dieser Artikel behandelt den mehr realistischen Fall, wo die Kontaktrate a (t) eine periodische Funktion ist, dessen Mittelwert größer als b ist. Dann konvergiert (log τ) /N nach einen neuen Grenzwert C, der mit einer periodischen Hamilton-Jacobi Gleichung gebunden ist. Wenn a (t) eine Kosinus-funktion mit einer kleinen Amplitude, mit einer großen Frequenz oder mit einer kleinen Frequenz ist, kann man annähernde analytische Formel für C erhalten, mit Hilfe der Methode von [Assaf et al. (2008) Phys Rev E 78, 041123]. Diese Resultate werden mit numerischen Simulationen veranschaulicht.Dans le modèle stochastique SIS pour une épidémie avec un taux de contact a, un taux de guérison b < a et une taille de population N , le temps moyen jusqu'à extinction τ est tel que (log τ)/N converge vers c = b/a − 1 − log(b/a) lorsque N tend vers l'infini. Cet article considère le cas plus réaliste où le taux de contact a(t) est une fonction périodique dont la moyenne est supérieure à b. Alors (log τ)/N converge vers une nouvelle limite C, qui est liée à une équation de Hamilton-Jacobi périodique en temps. Lorsque a(t) est une fonction cosinus avec une petite amplitude, avec une grande fréquence ou avec une fréquence très petite, on peut obtenir des formules approchées pour C de manière analytique en suivant la méthode utilisée par [Assaf et coll. (2008) Population extinction in a time-modulated environment. Phys Rev E 78, 041123]. Ces résultats sont illustrés par des simulations numériques

Quelques aspects de la disparition du français dans la recherche scientifique

International audienceDepuis plusieurs décennies, l'anglais a tendance à se substituer au français et aux autres langues dans les sciences « dures ». On aborde ici quelques aspects de ce problème. On recense d'abord les revues dont le nom en français a été remplacé par un nom en anglais ainsi que les revues qui, sans changer de nom, ont interdit les articles en français. On s'intéresse ensuite au pourcentage d'articles publiés en français dans les différentes sections des Comptes rendus de l'Académie des sciences entre 1981 et 2018. Puis on examine plus particulièrement le nombre d'articles de chimie en français dans la littérature scientifique. On étudie enfin les réactions des scientifiques et des institutions francophones à ces évolutions

Verhulst et l'équation logistique en dynamique des populations

European Communications in Mathematical and Theoretical Biology 10 (2008)Ce texte présente quelques remarques sur la postérité des travaux de Verhulst en dynamique des populations