Incorporating equity in infectious disease modeling: Case study of a distributional impact framework for measles transmission

Introduction Deterministic compartmental models of infectious diseases like measles typically reflect biological heterogeneities in the risk of infection and severity to characterize transmission dynamics. Given the known association of socioeconomic status and increased vulnerability to infection and mortality, it is also critical that such models further incorporate social heterogeneities. Methods Here, we aimed to explore the influence of integrating income-associated differences in parameters of traditional dynamic transmission models. We developed a measles SIR model, in which the Susceptible, Infected and Recovered classes were stratified by income quintile, with income-specific transmission rates, disease-induced mortality rates, and vaccination coverage levels. We further provided a stylized illustration with secondary data from Ethiopia, where we examined various scenarios demonstrating differences in transmission patterns by income and in distributional vaccination coverage, and quantified impacts on disparities in measles mortality. Results The income-stratified SIR model exhibited similar dynamics to that of the traditional SIR model, with amplified outbreak peaks and measles mortality among the poorest income group. All vaccination coverage strategies were found to substantially curb the overall number of measles deaths, yet most considerably for the poorest, with select strategies yielding clear reductions in measles mortality disparities. Discussion The incorporation of income-specific differences can reveal distinct outbreak patterns across income groups and important differences in the subsequent effects of preventative interventions like vaccination. Our case study highlights the need to extend traditional modeling frameworks (e.g. SIR models) to be stratified by socioeconomic factors like income and to consider ensuing income-associated differences in disease-related morbidity and mortality. In so doing, we build on existing tools and characterize ongoing challenges in achieving health equity.


Crude birth rates
Data on crude birth rates (CBRs) and total fertility rates (TFRs) by wealth quintile from the 2016 Ethiopian Demographic and Health Survey (DHS) were used to derive quintile-specific CBRs [1]. The total (population-level) CBR was assumed to be the average of the CBR values by quintile. The ratio of CBRs by quintile was assumed to equal the ratio of TFRs by quintile: the ratio of CBRs for each pair of quintiles was assumed to be proportional to the ratio of TFRs for each pair of quintiles. Under this assumption, the CBR for quintile j could be rewritten as the product of the CBR of the first quintile and the ratio of TFR for quintile j and the TFR for the first quintile (see equations S1-S3 below). The procedure is captured in the following equations:

Case-fatality ratios (CFRs)
We drew from estimated CFRs (year 2015) from Portnoy and colleagues [2]: specifically, we used the estimated median overall CFR value for the low-income country category, i.e. 1.6% (0.5 to 5.0%). As an approximation, this CFR was subsequently distributed across each income quintile, using the relative distribution (across wealth quintiles) in the percentage among children for whom treatment was sought from the Ethiopian DHS [1], as an attempt to capture differences in healthcare access across different socioeconomic groups in the Ethiopian population. The following proxied CFR estimates (from quintile 1 to 5) were then derived: 2.18%; 1.89%; 1.60%; 1.31%; and 1.02%.

Derivation of transmission rate matrices
Simulating the transmission matrices for scenarios 1, 2, and 3 Below we describe our methods for generating each simulated transmission matrix for scenarios 1, 2, and 3 described in the main text (Methods section). For simplicity, note that we assumed symmetry in the transmission matrices: >: = :> .
1. Draw 5 values of R0,i from a uniform distribution with defined minimum (10) and maximum (22) values that yields a mean value of 16 (see Table 1).
4. Sample 4 values for the R1j terms (for quintile 1, j¹1) from a uniform distribution with mean equal to the previously sampled sum of the 4 R1j terms (in step 3) divided by 4. Repeat this step n=100 times.
5. Extract the sums of each of the n combinations of R1j terms from step 4, and select those combinations that yield a sum that equals the previously sampled sum of R1j terms in step 3 (rounded to the nearest tenth digit).
We repeat steps 1-7 above for each of the n=1000 SIR model iterations and translate the simulated matrix into >: (in the SIR model equations (1) in the main text) terms, using the following relationship: where >,E represents the initial population size of quintile i ( >,E = E ) and is the recovery rate.

Scenario 4 using Mexican "contact" survey
We also derived contact matrices and corresponding transmission matrices from the Leo et al.
2016 study [3]. The ratio "number of links" (phone calls or texts) observed between individuals in all possible decile pairings and the number of such links expected in a social network with random mixing, or the L(si,sj) terms, were directly extracted from the study (Figure 3b    For all five scenarios (1, 2, 3, 4, and 5), the susceptible S and recovered R populations' size for quintiles 1 to 5 were initialized at the number of unvaccinated and successfully vaccinated individuals, respectively, among 15,000 individuals per quintile (reflective of the population sizes for each wealth quintile as reported in the DHS using quintile-specific vaccine coverage levels) [1]. We initialized infection by seeding one infected case in each quintile and assumed that no individuals were deceased at the start of the simulation.

Disparities and estimation of concentration indices
To estimate the concentration index (CI) resulting from each distributional vaccination case (see Methods section in the main text), we calculated the cumulative sum of each quintile's proportional population size (such that the cumulative sum at quintile 5 equals 1), and the cumulative sum of each quintile's proportional number of measles deaths. The ensuing concentration curve is then constructed as a plot of the cumulative share of number of deaths for each quintile against their cumulative share of the total population. Subsequently, the CI is computed as twice the area under the concentration curve above the "line of equality" [4]. Figure S1: Susceptible (blue), Infected (red), Recovered (green), and Deceased (brown) model dynamics for all quintiles (1=poorest; 5=richest) under transmission scenario 1. Solid lines indicate mean values while shaded areas indicate 95% uncertainty intervals for each compartment.