## 2 Concept of Combined Background Immunity and Specific Immunization  
The following approach was based on the idea that immunity cannot be regarded as a strict non-susceptible state, in general, but as a state of lowered risk of infection only, in many cases. Herd immunity is composed of individuals with different degrees of moderate immunization and individuals with a \"safe immunization state\". However, this \"safe state\" can also be quickly reduced to a less safe state by decreasing antibody concentration and by the emergence of infective mutants.  
In addition, it has to be considered that several mechanisms contribute to the overall immunity of each single individual, among them is a non-specific general immunity background; a low-specific immunization, which is caused by occasional responses to related proteins or viruses; and the high specific immunization caused by the infective agent or related vaccines.  
Finally, it is assumed that vaccination strategies have an impact on an individual's immunity state and its degradation, which is caused by new mutants, reduction in antibody production and decay of antibody concentration in the blood and in the concerned tissues. Thus, the infective agent itself provokes a molecular immune response against, more or less, all relevant bio macromolecules, which are foreign to the target organism. In the following, the very complex behavior of the immune system and the immune response are reduced to a set of very simple equations. This set is neither suited for reflecting the true kinetic network of biomolecular reactions nor to supplying a real quantitative description of immunization and infection processes. However, it illustrates the quantitative effects in these processes and can give an idea about the importance of individual responsivities and occasional exposition events in the risk of primary infection or re-infection in the case of previous immunization.  
The following parameters, for iterative simulation in time steps (1..100), are defined in the example of a viral infection:  
* \(\rm k_{vir}\) = replication rate of a virus,
* \(\rm k_{r}\) = reaction rate between virus particles and antibodies,
* \(\rm k_{st}\) = rate of stimulation of antibody production by a virus,
* \(\rm st_{oo}\) = maximum for stimulation,
* \(\rm v_{oo}\) = maximum concentration of a virus,
* \(\rm c_{ABv0}\) = initial concentration of antibodies,
* \(\rm c_{BGim}\) = non-specific and low-specific background immunity (counted as antibody equivalent),
* \(\rm z\) = random noise of \(\rm c_{BGim}\),
* \(\rm D_{AB}\) = decay rate of antibodies (specific protection),
* \(\rm D_{Vacc}\) = decay rate of primary applied vaccine.  
The following variables are used in the simulation:  
* \(\rm c_{vir}\) = concentration of virus particles,
* \(\rm c_{ABvir}\) = concentration of antibodies formed due to the stimulation of virus particles,
* \(\rm c_{ABvac}\) = concentration of antibodies formed due to stimulation by a vaccine,
* \(\rm c_{Vacc}\) = concentration of primary applied vaccine,
* \(\rm c_{ABtotal}\) = total concentration of antibodies, including an antibody equivalent for \(\rm c_{BGim}\).  
The following operations were applied in iterative steps:  
\[\rm c_{ABtotal}=\rm c_{ABvac}+\rm c_{ABvir}+\rm c_{BGim} \tag{1}\]  
\[\rm c_{vac}\ (t)=\rm c_{vac}\ (t-1)*(1-\rm D_{Vacc}) \tag{2}\]  
\[\rm c_{ABvac}\ (t)=\rm c_{ABvac}\ (t-1)*(1-\rm D_{AB})+\rm k_{st}*\rm c_{vac} \tag{3}\]  
\(\rm c_{vir}\ (t)=\rm roundoff3[1+(k_{vir}/c_{ABtotal})*(1-\rm c_{vir}(t-1)/v_{oo})]*k_ {r}*\rm c_{vir}(t-1)*\rm c_{ABtotal}\) (4)  
\[\rm c_{ABvir}=\rm c_{Avir}(t-1)*\rm sqrt[k_{st}*\rm c_{ABvir}(t-1)*\rm c_{vir} (t-1)*(1-\rm k_{st}*\rm c_{vir}(t-1)/\rm st_{oo})] \tag{5}\]  
\[*(1-\rm D_{AB})*\rm c_{ABvir}(t-1)*\rm c_{vir}(t-1)\]  
For simulation, a simple iterative method was sufficient. It could be run by an Excel table, for example. This allowed an easy change of parameters and start conditions. An example is given in the Table S1 (supplementary material).