Group-certain program off ODEs
In order to address the potential impact of the implementation and easing of lockdown measures, we expand the model structure to group-specific compartments. Specifically, for given groups a = 1, …, A and any time t, set Sa(t) as the number of susceptible people in group a at time t, Ea(t) as the number of exposed people in group a at time t, and so on. The group-specific compartment model is characterised by the ODE system (3) for all groups a = 1, …, A, which is a direct extension of the ODE system of the basic compartment model for the special case A = 1. We define (4) as the effective contact rate between groups a and b, where w is the secondary attack rate, mabdominal is the specific mitigation effect by lockdown measures with regard to contacts between groups a and b, r is a general factor that accounts for compliance to distance, isolation and quarantine orders, hb is the proportion of infectious people in group b in need of hospitalisation and ?ab is the basic contact rate between groups a and b when no lockdown measures are in place. As we are primarily interested in short-term prediction, we do not model biological aging, i.e. transitions between demographic groups. Therefore, for any time t, compartment-specific additivity is assumed, i.e. S(t) = ?a Sa(t), E(t) = ?a Ea(t), I(t) = ?a Ia(t), R(t) = ?a Ra(t) and D(t) = ?a Da(t) and N = S(t) + E(t) + I(t) + R(t) + D(t). The system is closed, meaning that the sum of all ODEs is 0 at each time t.
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In the absence of any lockdown measures, the general contact patterns are characterised by the basic contact rates ?ab, which represent how intensive/often group a has any contact with group b sufficient for potential virus transmission. In the POLYMOD study , 7,290 participants from 8 countries including Germany reported the number and extent of their social contacts during a randomly assigned 24 hour period, using a written diary. The age and gender of the contacted persons were recorded, among other information. Overall, the study contains information on 97, 904 contacts, distributed across the 8 participating countries. For Germany, the matrix of age-specific gender ratios of contact rates is shown in Fig 4. Squares in red color stand for higher contact rates among women (values below one), blue squares for higher contact rates among men (values above one). In 41 of 64 cells (8 ? 8 age groups), women have higher contact rates than men (Table 1). This is especially true for the youngest age group, 0–9 years, where contacts with women aged 20 and above are always far higher than contacts with men of these ages. Among adolescents (ages 10–19), there is still a surplus of female contacts, although men dominate contacts with persons aged 50 and older. This pattern is reinforced in the 20–29, 30–39, and 50–59 age groups. Female contacts again predominate in the 60–69 and 70–79 age groups, and this trend becomes stronger with age.
The behaviour of the epidemiological model is primarily governed by the effective contact rates ?ab which result from the basic contact rates ?ab by accounting for the secondary attack rate and lockdown measures. It is implicitly assumed here that hospitalised cases are effectively isolated from the remaining population and can not spread the disease. Note that the product (1 ? mab)(1 ? r)(1 ? hb) represents the proportion of potential virus transmissions that are not prevented.
