The analysis performed is based on a compartmental, meta-population, and age-structured model representing the transmission dynamics of meningococcal infection. This model was specifically designed for assessing the impact of Hajj pilgrimage on disease transmission and calibrated with historical IMD data in Saudi Arabia.
Mathematical model
The world population, including both pilgrims and non-pilgrims people, is modeled in homogeneous groups called clusters. Each cluster shares the same representation of the infection and demographic processes. The Hajj-specific impact is mainly represented through interactions between the different clusters included in the model. A full mathematical description of the model is given in Additional file 1: Text 1.
Following Hethcote [16], we consider a population with a continuous flow of individuals. In keeping with several previous publications [17, 18], we simplified N. meningitidis transmission by aggregating all serogroups. This facilitated a stronger focus on the consequences of MGs. Regarding the infection process, an important specificity relates to the distinction between short-term and asymptomatic carriage. This enables a better representation of the time between infection and disease occurrence which is an important factor for time-limited events such as Hajj pilgrimage.
Hajj-related transmission is first represented by dividing the population of each cluster at the start of each Hajj period between pilgrims and non-pilgrims. The proportion of pilgrims is both cluster and age-dependent. We also account for Hajj vaccination that occurs in the model at the time pilgrims are transferred to the pilgrim-specific states. At the end of a given Hajj period, subjects in pilgrim-specific compartments are reintegrated in the non-pilgrim corresponding compartment. It is worth mentioning that this representation is a simplification compared to the actual pilgrimage process. We notably consider that pilgrims all begin and end their pilgrimage at the same time and did not account for differences in the propensity to become a pilgrim according to disease status. The demographic process is unchanged for pilgrim-specific compartments. Pilgrims can change age group or die during the Hajj. We however restricted births to non-pilgrims compartments. We also account for an increased risk of carriage acquisition for the local population of the Hajj site linked to contacts with pilgrims.
Data
This study is supported by various data sets including population demography, outbreak cases and incidence, disease prevalence, vaccine schedule, Hajj schedule, annual number of pilgrims and their characteristics (age, immunization, etc.). Population demography and carriage rates are used to calibrate endemic equilibrium and outbreak data are used to calibrate model parameters and validate the model predictions. Detailed information on data used are presented in Additional file 1: Text 1.
The Hajj pilgrims gather from more than 180 countries which include both low and high meningitis endemic regions (Table 1). A majority of pligrims originate from Asian countries (Additional file 1: Text 1, Table S1). In general, there is a low prevalence in Europe and American countries whereas African countries in which the so called ‘meningitis belt’ falls are highly endemic. Since it is impractical to consider each country separately and include their population in a model to study the disease transmission, we split the globe into several clusters. While there is no fine line that may be drawn to create the homogeneous zones, we divide the participant countries (except KSA) into three clusters (or zones) based on the prevalence of meningitis and infection risk labelled high, medium, and low transmission clusters [19]. In addition, Mecca and KSA outside of Mecca are considered as two separate clusters due to their strategic locations (subject to an increased risk during the Hajj of meningitis transmission for the population belonging to these areas). This gives us a total of 5 clusters shown in Table 1.
Table 1 Cluster information Carriage rates varies across clusters. For high, medium and low transmission clusters, carriage rates are based on published estimates [19] and derived from calibration results for the two KSA clusters. For the 1995–2001 period, the estimated carriage rate is respectively 4.2% [2.0; 17.8] for Mecca and 1.2% [0.5–3.2] for the rest of KSA. For these two clusters, the range of variation reflects year-to-year variation in meningitis transmission.
Calibration of transmission parameters
All transmission parameters were calibrated. The first calibration step allowed us to obtain cluster-specific transmission parameters while the second step focused on Hajj-specific transmission parameters.
Calibration of cluster-specific transmission parameters \((\beta _c)\), was based on available evidence on the proportion of carriers for each cluster given its level of endemicity (high, medium or low). We used the analytic solution for the endemic equilibrium in the absence of vaccination at the cluster level to adjust \((\beta _c)\) to these proportions. These endemic equilibria were further used for initializing the model. We considered a burn-in period of 95 years before the period used for calibrating Hajj-related parameters.
Three type of Hajj-related parameters were calibrated: Hajj density effect \(\beta _{H}\), impact of Hajj on local transmission \(\beta _{L}\) and year-to-year variation in meningitis transmission in KSA clusters \(\beta _y\). The calibration of Hajj-related parameters was based on a comparison between model outcomes and historical annual data on IMD in KSA in 3 groups: pilgrims, local population in Mecca and local population in the rest of KSA. The period with available data was divided in 2 periods: a calibration period directly used for fitting parameters values and a validation period used for assessing model accuracy. To get a range of possible values for Hajj-related parameters, we used 100 random samples assuming that each observed data point was Poisson distributed with a mean corresponding to the observed value. Hajj-related parameters were fitted for each of these random samples through the maximization of a quasi log-likelihood function.
Model simulations
For each scenario considered in the simulations, we generated at least 100 samples using for the transmission parameters the set of possible values obtained through calibration. For each of this scenario we calculated both the median values and 95% credible intervals for the number of IMD cases for specific clusters or specific population (pilgrims and non-pilgrims). We also calculated probability for the number of IMD cases to exceed a given threshold.