## 3 Results and Discussion
### How Behaviour Rules Can Lower the Risk 1/\(\tau\) and \(M\)  
The two crucial parameters in the Verhulst model and in our analysis are: 1/\(\tau\), the risk for infection or hospitalization or death per day, \(d\), and \(M\), a dimensionless number, which represents the number of non-vaccinated people in a group or subgroup. The characteristic time, \(\tau\), is an effective value and depends on the intrinsic effectiveness of the virus transmission, the risk to become ill and its incubation time. In addition, \(\tau\) depends on extrinsic factors, such as contact frequency, exposure time, distancing, quality of ventilation and contact tracing [3], intermingling [4], differences in culture (handshaking and kissing or polite greeting), the health care system (stocks of basic protective gear, face protection and testing kits), sanitation, food habits and immune status (BMI-range) [8]. Increasing contact time and frequency will increase the risk, 1/\(\tau\), and reduce the \(\tau\)-value. The \(\tau\)-value is intrinsically lower for some virus variants, and it has an outspoken effect if the risk by the external conditions is already kept low. The physical background of social distancing, mouth-nose masks, face shields and ventilation is the Brownian motion of water droplets, diffusion and the trapping model, as explained in [3].  
In the absence of vaccination, the risk reduction during the epidemic is achieved by imposing behaviour rules, such as: e.g., increasing social distancing, hand washing, mouth-nose mask, face shield and good ventilation with moderate airflow. Ultimately, the limiting value \(M\) represents the number of non-vaccinated in a group. Vaccination is the most effective way to lower the peak height of the pandemic curve. In the absence of vaccines, \(M\) can be lowered by, e.g., making people's bubbles smaller. Other behavioural epidemiology rules to increase \(\tau\) and decrease \(M\) span from imposing a curfew, testing, home isolation, reducing contact frequency, controlling the size of contact bubbles and tracing'super spreaders.' The trends between the parameter values: \(M\), \(\tau\) and rules are clear, but the precise values connected to rules are not so clear. That makes predictions uncertain. A weakening or lifting of behaviour rules increases risk and can trigger a second wave in a pandemic.  
Behaviour rules may be classified, e.g., in different stages of increasing'survival mode': stage I: social distancing, no fun shopping, or only within a time limit, only click and collect to reduce contact time, imposing teleworking where possible; stage II: in addition to stage I, avoiding social gathering, shutting down universities, secondary schools and sport, culture and religious accommodation; stage III: stage II and closure of all other schools, reducing contact bubble size, closing non-crucial shops and services and closure of airports; stage IV: like stage III, with travel stop and installing a curfew. Unfortunately, it is difficult to quantize the effect of an imposed stage in a precise value for the reduction in \(M\). A change in the pallet of rules for decreasing \(M\) will henceforth be denoted as 'imposing rules' or (after vaccination) 'lifting rules.' Yet, subdividing the population into a fast subgroup, with a high-risk group and a more robust low-risk subgroup, explains the observed pandemic curves better.