Mathematical modeling of the SARS-CoV-2 epidemic in Qatar and its impact on the national response to COVID-19

Background Mathematical modeling constitutes an important tool for planning robust responses to epidemics. This study was conducted to guide the Qatari national response to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) epidemic. The study investigated the epidemic’s time-course, forecasted health care needs, predicted the impact of social and physical distancing restrictions, and rationalized and justified easing of restrictions. Methods An age-structured deterministic model was constructed to describe SARS-CoV-2 transmission dynamics and disease progression throughout the population. Results The enforced social and physical distancing interventions flattened the epidemic curve, reducing the peaks for incidence, prevalence, acute-care hospitalization, and intensive care unit (ICU) hospitalizations by 87%, 86%, 76%, and 78%, respectively. The daily number of new infections was predicted to peak at 12 750 on May 23, and active-infection prevalence was predicted to peak at 3.2% on May 25. Daily acute-care and ICU-care hospital admissions and occupancy were forecast accurately and precisely. By October 15, 2020, the basic reproduction number R0 had varied between 1.07-2.78, and 50.8% of the population were estimated to have been infected (1.43 million infections). The proportion of actual infections diagnosed was estimated at 11.6%. Applying the concept of Rt tuning, gradual easing of restrictions was rationalized and justified to start on June 15, 2020, when Rt declined to 0.7, to buffer the increased interpersonal contact with easing of restrictions and to minimize the risk of a second wave. No second wave has materialized as of October 15, 2020, five months after the epidemic peak. Conclusions Use of modeling and forecasting to guide the national response proved to be a successful strategy, reducing the toll of the epidemic to a manageable level for the health care system.


I. SARS-CoV-2 mathematical model structure
A deterministic age-structured meta-population compartmental model was developed to describe the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) transmission dynamics and disease progression in the population of Qatar, factoring subpopulation heterogeneity in exposure to the infection. The model stratified the population into compartments according to nationality subpopulation, age group (0-9, 10-19, 20-29,…, ≥80 years), infection status (infected, uninfected), infection stage (asymptomatic/mild, severe, critical), and disease stage (severe disease or critical disease). All Coronavirus Disease 2019 (COVID-19) mortality was assumed to occur in individuals that are in the critical disease stage. The model is based on extension and adaptation of our calibrated mathematical models developed to characterize SARS-CoV-2 transmission dynamics [1][2][3][4][5].
Epidemic dynamics were described using a system of coupled nonlinear differential equations for each age group and subpopulation (nationality) group. Each age group, a , denoted a ten-year age band apart from the last category which grouped together all individuals ≥80 years of age.
The population was divided into seven resident subpopulation groups i ( ) 1, 2,3, 4,5, 6, 7 i = representing the subpopulations of Indians, Bangladeshis, Nepalese, Qataris, Egyptians, Filipinos, and all other nationalities, respectively-these are the largest nationality subpopulation groups in Qatar. Qatar's population composition and subpopulations size and demographic structure were based on findings of "The Simplified Census of Population, Housing, and Establishments" conducted by Qatar's Planning and Statistics Authority [6]. Life expectancy was obtained from the United Nations World Population Prospects database [7]. Figure S1. Schematic diagram describing the basic structure of the SARS-CoV-2 mathematical model.
The model was expressed in terms of the following system of coupled nonlinear differential equations for each subpopulation group and age group: The definitions of population variables and symbols used in the equations are in Table S1.
Here β is the rate of infectious contacts, ( ) i ψ is the level of exposure profile in each subpopulation group i , and ( ) a σ is the susceptibility profile to the infection in each age group a .
To account for temporal variation in the basic reproduction number ( 0 R ), we incorporated temporal changes in the rate of infectious contacts. We parameterized the temporal variation (time dependence of β ) through the following combined function of the Woods-Saxon and logistic functions.
This function was mathematically designed to describe and characterize the time evolution of the level of risk of exposure before and after easing of restrictions. It was informed by our knowledge of SARS-CoV-2 epidemiology in Qatar [4], and it provided a robust fit to the data.
The mixing among the different age groups and subpopulation groups is dictated by the mixing matrices , ', ' a a i H (for age group mixing) and , i i Z ′ (for subpopulation group mixing). These matrices provide the likelihood of mixing and are given by the following expressions:    [4,8,9], and 11) nationality subpopulation distribution of antibody positivity [4,8,9].
Model input parameters were based on best available empirical data for SARS-CoV-2 natural history and epidemiology. Model parameter values are listed in Table 2. The following parameters were derived by fitting the model to data:  [12] and reported transmission before onset of symptoms [13].
Duration of infectiousness

days
Based on existing estimate [10] and based on observed time to recovery among persons with mild infection [10,14] and observed viral load in infected persons [12,13,15].  Proportion of infections that will progress to be infections that require hospitalization in acutecare beds ( ) S f a The distribution and age dependence of asymptomatic/mild, severe, or critical infections was based on the modeled SARS-CoV-2 epidemic in France [16].  Model-estimated relative risk of severe infection based on the SARS-CoV-2 epidemic in France [16].
Model-estimated relative risk of severe infection based on the SARS-CoV-2 epidemic in France [16].
Proportion of infections that will progress to be infections that require hospitalization in intensive care unit beds ( ) C f a The distribution and age dependence of asymptomatic/mild, severe, or critical infections was based on the modeled SARS-CoV-2 epidemic in France [16].    Figure S3. Model fits to A) daily hospital admissions in acute-care beds, B) daily hospital admissions in ICU-care beds, C) hospital occupancy of COVID-19 patients (number of beds occupied at any given time) in acute-care beds, and D) hospital occupancy of COVID-19 patients in ICU-care beds. v Figure S4. Evolution of the basic reproduction number R0 (A) and effective reproduction number Rt (B) in Qatar. Figure S5. Impact of the social and physical distancing interventions on A) cumulative number of infections, B) cumulative number of deaths, C) cumulative number of hospital admissions in acute-care beds, and D) cumulative number of hospital admissions in ICU-care beds. Figure S6. Uncertainty analysis. Mean and 95% uncertainty interval (UI) for the evolution of SARS-CoV-2 A) incidence (number of daily new infections), B) cumulative number of infections, C) active-infection prevalence (those latently infected or infectious), and D) attack rate (proportion ever infected), in the total population of Qatar. Figure S7. Uncertainty analysis. Mean and 95% uncertainty interval (UI) for the evolution of COVID-19 A) daily hospital admissions in acute-care beds, B) daily hospital admissions in ICUcare beds, C) cumulative number of hospitalizations in acute-care beds, D) cumulative number of hospitalizations in ICU-care beds, E) hospital occupancy of COVID-19 patients (number of beds occupied at any given time) in acute-care beds, and F) hospital occupancy of COVID-19 patients in ICU-care beds.