Modeling the Impact of Alternative Immunization Strategies: Using Matrices as Memory Lanes

<div><p>Existing modeling approaches are divided between a focus on the constitutive (micro) elements of systems or on higher (macro) organization levels. Micro-level models enable consideration of individual histories and interactions, but can be unstable and subject to cumulative errors. Macro-level models focus on average population properties, but may hide relevant heterogeneity at the micro-scale. We present a framework that integrates both approaches through the use of temporally structured matrices that can take large numbers of variables into account. Matrices are composed of several bidimensional (time×age) grids, each representing a state (e.g. physiological, immunological, socio-demographic). Time and age are primary indices linking grids. These matrices preserve the entire history of all population strata and enable the use of historical events, parameters and states dynamically in the modeling process. This framework is applicable across fields, but particularly suitable to simulate the impact of alternative immunization policies. We demonstrate the framework by examining alternative strategies to accelerate measles elimination in 15 developing countries. The model recaptured long-endorsed policies in measles control, showing that where a single routine measles-containing vaccine is employed with low coverage, any improvement in coverage is more effective than a second dose. It also identified an opportunity to save thousands of lives in India at attractively low costs through the implementation of supplementary immunization campaigns. The flexibility of the approach presented enables estimating the effectiveness of different immunization policies in highly complex contexts involving multiple and historical influences from different hierarchical levels.</p></div

Cost-effectiveness (panels on the left represent incremental cost-effectiveness ratios (ICERs) in US$ per DALY averted) and percent change in mortality (panels on right) of alternative immunization strategy (S2, S3, S4) relative to baseline status quo immunization scenarios (S2/S3/S4 are described in Fig 2).

<p>For Latin America, ICERs represent the cost saved per DALY averted and mortality change as percent increases in mortality. For the other regions, mortality change represents the percent reduction in mortality. Bars are ordered according to MCV1 coverage.</p

Diagram summarizing the events each age cohort can experience in an annual cycle (it can be also understood as the outcomes an individual may experience each year).

<p>The cycle starts (left) and ends (right) with the proportions of susceptible and immune individuals. Events (vaccinations, infections, mortality) during the cycle change the proportions exported for the next cycle and generate intermediate outcomes of interest (proportions of cases and deaths). Each box represents a matrix of 104 age-groups × 71 years. Probabilities are depicted in the small white boxes: C: Vaccine coverage, E: Vaccine efficiency, F: Force of infection, D: Case fatality ratios, and W: waning Immunity probability. The large colored boxes are compartments, and sequences of letters represent sequences of events (e.g. SVSFI blue box: proportion of susceptible individuals S who were vaccinated–SV–but remained susceptible–SVS–were next infected–SVSF–and became immune SVSFI). N stands for non-vaccinated.</p

Measles immunization scenarios (Status quo, S2, S3, S4) simulated up to 2050.

<p>Vertical bars are proportional to coverage levels. First (MCV1) and second (MCV2) routine doses and supplementary immunization campaigns (SIAs) are represented in green, blue, and grey, respectively. Scenarios were selected by the WHO Strategic Advisory Group of Experts (SAGE). Dependence between MCV1 and MCV2 indicates the % of unvaccinated individuals targeted to receive MCV2. <u>Indian States</u>: Status quo: MCV1 dose at 9 months at current coverage (46%, 90%, 74%, 86%, and 95% in Bihar, Karnataka, Maharashtra, Orissa, and Tamil Nadu, respectively); S2: MCV1 coverage improved to 90% over 5 years; S3: status quo plus inclusion of MCV2 at 18 months, improving from 50% to 97% of MCV1 over 5 years (dependence between doses = 25%); S4: inclusion of SIAs from 2009 onwards targeted at ages 9 months to 5 years with 90% coverage. <u>Africa and Cambodia</u>: Status quo: MCV1 (coverages of 73%, 73%, 51%, 85%, 95%, and 79% in Cameroon, Democratic Republic of Congo, Equatorial Guinea, Ghana, Rwanda, and Cambodia, respectively) and SIAs every 2–4 years (target age: 9 months to 5 years, 90% coverage); S2: status quo and MCV2 at 18 months with coverage improving from 50% to 100% of MCV1 over 5 years (dependence between doses = 25%); S3: status quo and routine MCV2 to 7 year olds, MCV2 coverage improving from 50% to 100% of MCV1 over 5 years (dependence between doses = 25%). <u>Latin America</u>: Status quo: MCV1 (at 15 months) MCV2 (to 7, 4, 4, and 6 year olds in Costa Rica, El Salvador, Paraguay, and Mexico, respectively) at existing coverage (MCV1 and MCV2 coverage in Costa Rica, El Salvador, Paraguay, and Mexico, respectively: 89% and 94% of MCV1; 98% and 94% of MCV1; 88% and 66% of MCV1; 96% and 58% of MCV1) and SIAs implemented every 4 years (target age: 9 months to 5 years, 85% coverage in Costa Rica and 92% in remaining countries); S2: status quo and 100% dependence between MCV1 and MCV2; S3: status quo and elimination of SIAs; S4: status quo elimination of SIAs and 100% dependence between MCV1 and MCV2.</p

Estimates of periodicity and timing of influenza A (top panels) and B (bottom panels) epidemics in China.

<p>(Left) Timing of annual influenza peaks, in weeks. Timing is color coded by season. (Center) Amplitude of annual periodicity, ranging from low (yellow) to high (red), as indicated in the legend. Amplitude is relative to the mean of the weekly influenza time series in each province. (Right) Importance of semi-annual periodicities, measured by the ratio of the amplitude of the semi-annual periodicity to the sum of the amplitudes of annual and semi-annual periodicities. Yellow indicates strongly annual influenza epidemics, while red indicates marked semi-annual activity. See also <a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1001552#pmed-1001552-g004" target="_blank">Figure 4</a>.</p

Latitudinal gradients in seasonality of influenza A and B epidemics in China.

<p>Plots represent estimates of seasonal parameters as a function of latitude for influenza A (left panels) and influenza B (right panels). Top panels: Amplitude of annual periodicity, relative to the mean standardized influenza time series. Middle panels: Peak timing in weeks. Bottom panels: Contribution of the semi-annual cycle, measured by the ratio of the amplitude of the semi-annual cycle to the sum of the amplitudes of annual and semi-annual cycles. Open circles represent point estimates from seasonal regression models and horizontal dashed lines represent 95% confidence intervals based on 1,000 block-bootstrap samples. Symbol size is proportional to the average number of influenza virus isolates sampled each season, while colors represent different climatic zones (black, cold-temperate; blue, mid-temperate; green, warm-temperate; orange, subtropical; red, tropical). Purple lines represent linear regression of seasonal parameters against latitude (dashed line, unweighted regression; solid line, regression weighted by the inverse of the variance of province-specific seasonal estimates); R² and <i>p</i>-values are indicated on the graphs.</p

Influenza epidemiological regions and climate predictors.

<p>(A) Definition of three influenza epidemiological regions based on hierarchical clustering (colored rectangles; Ward's method, Euclidian distance between weekly standardized influenza time series). Province labels are color-coded by climatic region (black, cold-temperate; blue, mid-temperate; green, warm temperate; orange, subtropical; red, tropical). (B) Map of the three epidemiological regions identified in (A). (C) Climate predictors of the two main regions identified in (A) (subtropical regions 1 and 2 versus temperate region 3), based on stepwise discriminant analysis. (D) Climate predictors of subtropical regions 1 and 2, based on stepwise discriminant analysis.</p

Latitudinal gradient in the dominance of influenza B in China,

<p>measured by the proportion of influenza B among all influenza positive isolates each season. Median proportion of influenza B over seven seasons is displayed for each province, with grey horizontal bars indicating ±2 standard deviations. Symbol size is proportional to the average number of influenza virus isolates sampled each season. Colors represent different climatic zones (black, cold-temperate; blue, mid-temperate; green, warm-temperate; orange, subtropical; red, tropical). Purple lines represent linear regression of influenza B proportion against latitude (dashed line, unweighted regression; solid line, regression weighted by sample size).</p

Map of Chinese provinces conducting influenza surveillance (<i>n</i> = 30).

<p>Dots indicate the location of the capital city in each province. A total of 193 hospitals participate in disease surveillance, representing 88 cities. Colors illustrate different climatic domains (black, cold-temperate; blue, mid-temperate; green, warm-temperate; orange, subtropical; red, tropical). Different symbols indicate the type of surveillance scheme (circles, year-round surveillance; triangles, Oct through Mar surveillance).</p

Background characteristics of the 30 provinces involved in influenza surveillance and information on influenza sampling intensity, 2005–2011, China.

a<p>Number of cities and hospitals participating in surveillance.</p>b<p>Indicates year-round influenza surveillance before 2009 (all provinces switched to year-round surveillance in the post-2009 pandemic period).</p><p>C, cold temperate; GRP, gross regional product; MT, mid-temperate; ST, subtropical; T, tropical; WT, warm temperate;</p