## 3 Results and Discussion
### Analytical Expressions for Limited Growth Based on the Verhulst Model  
If the relative growth rate is constant in time for a statistical homogeneous group, the risk \(1/\tau\) is constant, and the number of infections grows exponential:  
\[N(t)\equiv N\propto e^{t/\tau}. \tag{8}\]The Verhulst model is applied with a risk decreasing in time as:  
\[\frac{1}{\tau(t)}=\frac{1-\left(N(t)/M\right)^{1/2}}{\tau}, \tag{9}\]  
and \(M\) is, at most, the population number under investigation minus the number of vaccinated people and those living completely isolated. The equations are summarized in Table 1.  
The aim of the simulation is to enlighten some main ideas in a broad-brush picture. Therefore, the above flexible analytical phenomenological model \(V_{n2}\) (logistic with \(n=2\)) is chosen [4; 5]. The refined discrete models are at the expense of too many unknown parameters and a complex set of equations with less insight. Models based on deep learning and artificial intelligence use a huge amount of data and uncertain parameters, are less intuitive and are not considered here. The aim of the simulation is to use a simple and intuitive model that takes into account the statistical inhomogeneity of age groups and explains better the following crucial effects:  
1. The possibility of multiple peaks in the epidemic curve without changing rules;
2. That partitioning in subgroups with strong isolation is an efficient approach;
3. That isolation of high-risk elderly people from normal risk younger people is the best procedure in retirement houses if vaccination is not available;
4. That a high plateau value after a peak is caused by'spontaneous' outbreaks at random places and at a random time by a lack of rules or a lack of local compliance with the rules;
5. That health care pressure asks to lower the risk of contamination by imposing efficient rules, especially II) and III) and by a population complying with the rules;
6. That the presentation on a semi-logarithmic format has an advantage over a linear one: exponential growth is visible as a straight line, large and small countries fit in the same plot, and noisy data is better represented (except, e.g., for numbers switch ing between 1 and 0) and the apparent 'lag time' in linear formats for exponential growth is not discernable in a semi-logarithmic plot;
* The rule of thumb: A perfect vaccine applied to non-vaccinated people, e.g., reduces the number of non-vaccinated by a factor of four (from 100% to 25% or from 80% to 20%), and results in a reduction by a factor four in hospitalization (the parameter, M, in the model). An example of a more complicated model is given in by [PERSON] et al. [9]. The handling of noisy data in models (transfer functions) with noisy parameters is proposed by [PERSON] et al. [10]. There is a long history of deterministic epidemic models starting with fast exponential growth and, in the recovery phase, showing a slower exponential decay [11]. More recent references to the Verhulst model exist [12; 13]. In [12], the work investigates not only the time-dependent risk as in Equation (10) for \(Vn\left(1-\left(N(t)/M\right)^{1/n}\right)/\tau\) but, e.g., also the risk proportional to \(\left(1-N(t)/M\right)^{1/n}/\tau\) and time lag. Reference [13] is in support of the fact that the limiting value \(M/m\) for \(m\) equal subgroups, as applied in Section 3.3, is an overestimation. The parameter \(M/m\) should be reduced by taking into account the population density (surface concentration) in the subgroups. Here, the statistical inhomogeneous population is considered with not too many details, equations and unknown parameters. In all models, there is uncertainty about parameter values. For simplicity, the uncertain lag time between the curves for contamination, hospitalization and deaths and between the moment of imposing new behaviour rules and its effect is not taken into account.