Bayesian SEIR model
The model used in this case is the one proposed in Hauser et al. (2020), but adapted to the estimation of mortality and hospital admissions in the Basque Country.
As indicated in the paper by Althaus et al. (2020) ( referred to the reader for more details), this is an SEIR (Susceptible- Exposed-Infected-Recovered) model stratified by age, with a distinction between symptomatic and asymptomatic infections.
The population is divided into nine age groups (0-9 years, 10- 19 years,. . . , 70-79 years, over 80 years). The population of each k age group is subdivided into five compartments: susceptible (Sk ), incubating or exposed (Ek ), infected with symptoms (Ik ), infected and asymptomatic (Ak ), and recovered and dead (Rk ) (see Figure 1).
The number of individuals in each compartment is scaled according to the total population of the Basque Country (2,188,017 inhabitants in 2019), so that the sum of all compartments is 1.
Figure 1: SEIR Model (Source: Hauser et al. 2020).
About the dynamics of the infection (see Figure above), the strength of the infection λk depends on the number of infectious individuals in each age group and the rate of contagion/infection by contact with an infectious individual β. Besides, the model includes a time-dependent function (f(t)) to account for the reduction in transmissibility following the introduction of control measures on 15 March 2020. The strength of the infection λk is, therefore, expressed as follows:
Concerning the other parameters, after the incubation period (1/τ ), a proportion of infected persons develop symptoms and become infectious, while the remaining persons remain asymptomatic and do not transmit the disease. After a period of 1/μ, individuals are identified and isolated, and therefore no longer infectious.
The dynamics of the infection is controlled by 4 parameters (see Figure 12): {β, τ, ψ, μ}. Note that the model assumes that susceptible individuals can be infected by contact with infected individuals with symptoms from any other age group. This means that it assumes that incubating as well as asymptomatic individuals are not infectious. For correcting this limitation, a high proportion of symptoms (80%, i.e. valor ψ ≈ 0.80 in Figure 12) has been considered to estimate the model. Additionally, a sensitivity analysis has been done considering: (1) various proportions of symptom carriers; and (2) that the symptom carriers transmit the disease. Results of the sensitivity analysis show that the model is robust against these specifications in estimating hospital admissions and deaths (our goal). For the remaining (fixed) parameters that control the dynamics of the infection, the results shown in this report have considered τ = 1/5.95 days (based on the literature) and μ = 1/5.4 days (based on the report of positive individuals in the Basque Country).
The estimation of the model is made from a Bayesian perspective, and the code provided by the original authors has been adapted to the situation and objectives in the Basque Country. Concerning the data used to estimate the model, in our case, these are hospital admissions and mortality per day and age group. In this regard, it should be noted that according to the indications for the diagnostic test for the detection of the virus described in the epidemiological surveillance protocol, in the first weeks of the pandemic the test was preferably performed on individuals with severe symptoms.
However, in recent weeks these indications have changed, and the scope of the test has been extended to include individuals with mild symptoms. Thus, the number of positive cases is not only an under-representation of the actual number of infected, but this under-representation has changed over time. For this reason, the plausibility of the model only incorporates information on hospital admissions and mortality per day and age.
Finally, it is worth noting that the hospital admissions are calculated on the incident cases in the group of symptomatic persons, and the mortality is calculated based on said hospital ad- missions. In both cases, the model takes into account that there may be a delay.