Andrew N. Hill

The critical vaccination fraction for heterogeneous epidemic models

Given a population with m heterogeneous subgroups, a method is developed for determining minimal vaccine allocations to prevent an epidemic by setting the reproduction number to 1. The framework is sufficiently general to apply to several epidemic situations, such as SIR, SEIR and SIS models with vital dynamics. The reproduction number is the largest eigenvalue of the linearized system round the local point of equilibrium of the model. Using the Perron-Frobenius theorem, an exact method for generating solutions is given and the threshold surface of critical vaccine allocations is shown to be a compact, connected subset of a regular (m-1)-dimensional manifold. Populations with two subgroups are examined in full. The threshold curves are either hyperbolas or straight lines. Explicit conditions are given as to when threshold elimination is achievable by vaccinating just one or two groups in a multi-group population and expressions for the critical coverage are derived. Specific reference is made to an influenza A model. Separable or proportionate mixing is also treated. Conditions are conjectured for convexity of the threshold surface and the problem of minimizing the amount of vaccine used while remaining on the threshold surface is discussed.

Modelling tuberculosis trends in the USA

We present a mathematical transmission model of tuberculosis in the USA. The model is calibrated to recent trends of declining incidence in the US-born and foreign-born populations and is used in assessing relative impacts of treatment of latently infected individuals on elimination time, where elimination is defined as annual incidence <1 case/million. Provided current control efforts are maintained, elimination in the US-born population can be achieved before the end of this century. However, elimination in the foreign-born population is unlikely in this timeframe even with higher rates of targeted testing and treatment of residents of and immigrants to the USA with latent tuberculosis infection. Cutting transmission of disease as an interim step would shorten the time to elimination in the US-born population but foreign-born rates would remain above the elimination target.

Tuberculosis Infection in the United States: Prevalence Estimates from the National Health and Nutrition Examination Survey, 2011-2012

Background Reexamining the prevalence of persons infected with tuberculosis (TB) is important to determine trends over time. In 2011–2012 a TB component was included in the National Health and Nutrition Examination Survey (NHANES) to estimate the reservoir of persons infected with TB. Methods Civilian, noninstitutionalized U.S. population survey participants aged 6 years and older were interviewed regarding their TB history and eligibility for the tuberculin skin test (TST) and interferon gamma release assay (IGRA) blood test. Once eligibility was confirmed, both tests were conducted. Prevalence and numbers of TST positive (10 mm or greater), IGRA positive, and both TST and IGRA positive were calculated by adjusting for the complex survey design after applying corrections for item nonresponse and digit preference in TST induration measurements. To examine TST positivity over time, data from NHANES 1999–2000 were reanalyzed using the same statistical methods. The TST was performed using Tubersol, a commercially available purified protein derivative (PPD), rather than PPD-S, which was the antigen used in NHANES 1999–2000. Prior patient history of TB vaccination was not collected in this study nor were patients examined for the presence of a Bacillus of Calmette and Guerin (BCG) vaccine scar. Results For NHANES 2011–2012, TST and IGRA results were available for 6,128 (78.4%) and 7,107 (90.9%) eligible participants, respectively. There was no significant difference between the percentage of the U.S. population that was TST positive in 2011–2012 (4.7% [95% CI 3.4–6.3]; 13,276,000 persons) compared with 1999–2000 (4.3%; 3.5–5.3). In 2011–2012 the percentage that was IGRA positive was 5.0% (4.2–5.8) and double TST and IGRA positivity was 2.1% (1.5–2.8). The point estimate of IGRA positivity prevalence in foreign-born persons (15.9%; 13.5–18.7) was lower than for TST (20.5%; 16.1–25.8) in 2011–2012. The point estimate of IGRA positivity prevalence in U.S.-born persons (2.8%; 2.0–3.8) was higher than for TST (1.5%; 0.9–2.6). Conclusions No statistically significant decline in the overall estimated prevalence of TST positivity was detected from 1999–2000 to 2011–2012. The prevalence of TB infection, whether measured by TST or IGRA, remains lower among persons born in the United States compared with foreign-born persons.

Estimating the future burden of multidrug-resistant and extensively drug-resistant tuberculosis in India, the Philippines, Russia, and South Africa: a mathematical modelling study

BACKGROUND Multidrug-resistant (MDR) and extensively drug-resistant (XDR) tuberculosis are emerging worldwide. The Green Light Committee initiative supported programmatic management of drug-resistant tuberculosis in 90 countries. We used estimates from the Preserving Effective TB Treatment Study to predict MDR and XDR tuberculosis trends in four countries with a high burden of MDR tuberculosis: India, the Philippines, Russia, and South Africa. METHODS We calibrated a compartmental model to data from drug resistance surveys and WHO tuberculosis reports to forecast estimates of incident MDR and XDR tuberculosis and the percentage of incident MDR and XDR tuberculosis caused by acquired drug resistance, assuming no fitness cost of resistance from 2000 to 2040 in India, the Philippines, Russia, and South Africa. FINDINGS The model forecasted the percentage of MDR tuberculosis among incident cases of tuberculosis to increase, reaching 12·4% (95% prediction interval 9·4-16·2) in India, 8·9% (4·5-11·7) in the Philippines, 32·5% (27·0-35·8) in Russia, and 5·7% (3·0-7·6) in South Africa in 2040. It also predicted the percentage of XDR tuberculosis among incident MDR tuberculosis to increase, reaching 8·9% (95% prediction interval 5·1-12·9) in India, 9·0% (4·0-14·7) in the Philippines, 9·0% (4·8-14·2) in Russia, and 8·5% (2·5-14·7) in South Africa in 2040. Acquired drug resistance would cause less than 30% of incident MDR tuberculosis during 2000-40. Acquired drug resistance caused 80% of incident XDR tuberculosis in 2000, but this estimate would decrease to less than 50% by 2040. INTERPRETATION MDR and XDR tuberculosis were forecast to increase in all four countries despite improvements in acquired drug resistance shown by the Green Light Committee-supported programmatic management of drug-resistant tuberculosis. Additional control efforts beyond improving acquired drug resistance rates are needed to stop the spread of MDR and XDR tuberculosis in countries with a high burden of MDR tuberculosis. FUNDING US Agency for International Development and US Centers for Disease Control and Prevention, Division of Tuberculosis Elimination.

An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing.

The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.

Estimating tuberculosis cases and their economic costs averted in the United States over the past two decades

BACKGROUND Following a concerted public health response to the resurgence of tuberculosis (TB) in the United States in the late 1980s, annual TB incidence decreased substantially. However, no estimates exist of the number and cost savings of TB cases averted. METHODS TB cases averted in the United States during 1995-2014 were estimated: Scenario 1 used a static 1992 case rate; Scenario 2 applied the 1992 rate to foreign-born cases, and a pre-resurgence 5.1% annual decline to US-born cases; and a statistical model assessed human immunodeficiency virus and TB program indices. We applied the cost of illness to estimate the societal benefits (costs averted) in 2014 dollars. RESULTS During 1992-2014, 368 184 incident TB cases were reported, and cases decreased by two thirds during that period. In the scenarios and statistical model, TB cases averted during 1995-2014 ranged from approximately 145 000 to 319 000. The societal benefits of averted TB cases ranged from US

Comparing Drivers and Dynamics of Tuberculosis in California, Florida, New York, and Texas.

3.1 to US

Progression from latent infection to active disease in dynamic tuberculosis transmission models: a systematic review of the validity of modelling assumptions

6.7 billion, excluding deaths, and from US

Modeling rates of infection with transient maternal antibodies and waning active immunity: Application to Bordetella pertussis in Sweden

6.7 to US

Computation of

14.5 billion, including deaths. CONCLUSIONS Coordinated efforts in TB control and prevention in the United States yielded a remarkable number of TB cases averted and societal economic benefits. We illustrate the value of concerted action and targeted public health funding.