Assortative social mixing and sex disparities in tuberculosis burden

Globally, men have higher tuberculosis (TB) burden but the mechanisms underlying this sex disparity are not fully understood. Recent surveys of social mixing patterns have established moderate preferential within-sex mixing in many settings. This assortative mixing could amplify differences from other causes. We explored the impact of assortative mixing and factors differentially affecting disease progression and detection using a sex-stratified deterministic TB transmission model. We explored the influence of assortativity at disease-free and endemic equilibria, finding stronger effects during invasion and on increasing male:female prevalence (M:F) ratios than overall prevalence. Variance-based sensitivity analysis of endemic equilibria identified differential progression as the most important driver of M:F ratio uncertainty. We fitted our model to prevalence and notification data in exemplar settings within a fully Bayesian framework. For our high M:F setting, random mixing reduced equilibrium M:F ratios by 12% (95% CrI 0–30%). Equalizing male case detection there led to a 20% (95% CrI 11–31%) reduction in M:F ratio over 10 years—insufficient to eliminate sex disparities. However, this potentially achievable improvement was associated with a meaningful 8% (95% CrI 4–14%) reduction in total TB prevalence over this time frame.


Background
Overall, men bear higher burden of tuberculosis compared to women. Sex disparity in TB burden considerably varies between settings with the highest male :female ratio reported in Uganda and Viet Nam, while Ethiopia has the reported lowest. In this work, we provide an analysis of the role of assortativity, differential progression and detection on TB M:F ratio at the two extreme settings, Ethiopia (M:F=1.2) and Uganda M:F=4.5) (Figure 1).

Parameterisation
This section provides brief descriptions of selected parameters in the model. In this study, we explored three mechanisms that could contribute to sex disparities in TB prevalence. In particular, we modelled the impact of social mixing, differential disease progression and detection.

Social mixing
To understand the contribution of social mixing on TB M:F ratio, we used a single scaling parameter that takes spectrum of values from -1(mixing only with opposite sex) to 1 (mixing only with the same sex). This parameter is expressed in terms of the proportion of contacts that are with the same sex (m) as : m ∼ LN (−0.57, 0.085) [2] (2) Case detection rate The model is simulated using case detection proportion parameterised as an informative prior parameterised as Beta(5.6, 3.5) based on WHO report for all TB types [?]. case detection proportion sampled from this prior is converted into case detection rate to be used in a model: Case detection rates (δ) generated this way are average rates and sex-stratified case detection rates are generated as follows: where π-differential case detection, δ f -case detection rate in females and δ m -case detection rates in men.

TB progression parameters
TB progression parameters such as fast progression rate ( ), reactivation rate (ν) and relapse (ω) are not available in sex stratification in literature. Therefore, values from literature on these parameters are considered as average and sex stratified rates were generated using similar approaches used for case-detection rates above.
For example, sex stratified reactivation rates were generated in our model as follows where υ f and υ m are reactivation rates in men and women respectively, υ is the average reactivation rate, and α is the hazard ratio of progression in males compared to females.
TB mortality and self-recovery rates As we are not modelling by smear status in this study, we generated sex specific average mortality rates by taking the average of TB mortalities in smear positive and smear negative cases. Similarly, sex specific TB self-recovery rates were also generated by taking the average of smear positive and smear negative recovery rates in within each sex group. A log-normal prior was set around these parameters to use in our model [3].

Global and sex specific parameters
In this model, only five parameters are sex specific: infectious proportion, fast progression, reactivation, relapse and case detection rate. In addition,potential differential transmission risk is captured by the social mixing parameter. All other model parameters are identical for men and women.

Uncertainties in model parameters
All model model parameters were uncertain and specified via priors,within a Bayesian framework. However, fixed values were used for two parameters -natural mortality (µ) and treatment success (θ).

Condition for equilibria
As described in the main text, we solved for the equilibrium condition of the model ordinary differential equations, which yielded a quadratic equation for the equilibrium prevalence: where d = D/N , β is the effective contact rate and Ψ represents the parameters other than β. The coefficients in Equation 4 are

Impact of Intervention
The intervention considered in this study is narrowing the sex disparity in TB case detection rates.
To asses the impact of narrowing case detection gaps on TB prevalence and M:F ratio in TB prevalence, we simulated the intervention at different regions of joint posterior distributions of selected parameter combinations.

Sampling from different regions of posterior
We achieved sampling from three regions of joint posterior distributions by using ellipses with different radii and angle centred at different regions of the joint posterior. Data contained within each ellipse was extracted and used as input in a model used to simulate the impact of intervention. Data from different regions of parameter combinations were used to determine if the impact of an intervention is dependent on where the parameter is located in the parameter space ( Figure  2).

Compatibility of sampled regions with data likelihood
Before the impact of intervention was simulated using parameters from the sampled regions, we assessed if parameters in the sampled region were supported by the data likelihood. This is achieved by comparing the log-posterior at these locations with the log-posterior from the whole parameter spaces of all parameters included in the model.
The plots below show that log-posteriors from the sampled regions coincide with the log-posterior from the entire parameter space. Columns a, b and c represent different parameter combinations: Parameters along column a represent combinations of π (differential detection) and α (differential progresion). Similarly, while those along column b represent ρ (assortativity) and and α, those under column c are for joint posterior distributions of ρ and π (Figure 3). The impact of intervention was simulated at sampled regions of the selected joint distribution.

Disease dynamics under intervention
The below plot shows the dynamics of TB under intervention using 200 parameters sets sampled from the whole region (Figure 4). In a high M:F ratio setting, the intervention can reduce prevalence to some extent. In the low M:F ratio setting,however, the prevalence remains unaffected with intervention. Similarly, eliminating case detection gap can meaningfully narrow the M:F ratios in a high M:F ratio setting , although its impact on low M:F setting is limited.

Posterior estimates
The following table compares posterior estimates from the two exemplar settings. Our posterior estimates from the two exemplar settings were identical for most of the parameters involved. The estimates differed only with respect to the proportion of cases detected, differential detection and differential disease risk (Table 1).
NB: The π (differential detection) compares the hazard of males being detected compared to women. Similarly, α (differential risk of progression) measures disease progression in men as compared with women.