## 3 Results: Illustration of Immunity Response after Vaccination and Infection  
### Immunity Decay after Vaccination  
The immune reaction of the human body protects against a new infection after a survived infection or after vaccination, for a certain time. The reliability of protection is dependent on the following main factors:* General immune status,
* Intensity of activation of the specific immune defense by the survived disease or vaccination,
* Intensity of a new virus exposition,
* Decay of specific immunity over time.  
Therefore, the seriousness of a second infection is dependent on the time span between the first and second immunization. This simple relation is illustrated in Figure 1. Only a small effect can be expected in the case of an early small virus impact (Figure 1a). In this case, the immune response is slightly enforced, but it is assumed that there are not strong symptoms of disease. Even in an early phase after immunization, a stronger reaction of the protected body might take place if it is confronted by a massive viral exposition. In this early phase the body is able to suppress the infection quickly (Figure 1b). After a longer time, a massive viral impact can result in the propagation of infection. In this case, the body can overcome the infection by means of the residual protection from the original immunization (e.g., vaccination) in combination with the virus-induced re-activation of the immune system (Figure 1c). The disease is obvious as there are significant symptoms, but the course of disease is less serious than without the early immunization. However, after a longer time span between vaccination and a massive viral impact, the common power of residual specific immunity, general immunity and instant immune reaction might be too small for overcoming the infection, and the virus wins the battle (Figure 1d).  
Besides the time since vaccination, the reaction rate between the virus and the immune system \(\mathrm{k_{r}}\) is important for the estimation of the seriousness of a massive viral exposition. The reaction rate is probably dependent on the individual immune system on the one hand and the properties of the antigen (virus) on the other hand. The simulation shows a low reaction in the case of a high infection rate (Figure 2a). Even, the re-activation of the immune system is low under these conditions. Much higher risks of illness and heavier symptoms have to be expected in the case of lower reaction rates (Figure 2b,c). However, such a high reaction is connected with stronger activation of the immune system, resulting in higher protection after the disease has passed by.  
A very important aspect is the viral replication rate \(\mathrm{k_{vir}}\). This parameter is a specific property of a virus with a certain molecular furnishing. It is dependent on variations in the virus's genome and can vary between different mutants. It is easy for a virus-exposed body to overcome the virus if its replication rate is low (Figure 3a). A slightly enhanced virus replication rate might cause the appearance of symptoms. A fast breakthrough and, therefore, a heavy infection with serious risks has to be expected in case of fast viral replication (Figure 3c).  
From the side of a concerned body, the ability to react to a virus impact by activation of a specific immune response is another important factor for estimating the risk of a breakthrough in a certain temporal distance after vaccination. At a higher stimulation rate \(\mathrm{k_{st}}\), the attack caused by a massive viral exposition can be fended off, even in the case where the protective effect of a vaccination is reduced by antibody decay after a certain time (Figure 4a). In contrast, the virus can easily breakthrough if the stimulation rate is too low and the specific re-activation of the immune response is slower than the virus replication (Figure 4b).