The SEIR model, widely used for analyzing and predicting the spread of virus infection, is here extended to considering the movement of people across regions. In contrast to previous models considering the risk due to the travelers from/to other regions, here we consider two factors. First, we consider the movements of susceptible (S), exposed (E), and recovered (R) individuals who may get infected after movement , as well as infected (I)  . Second, the infectious contacts between people from different regions are dealt with separately from the first factor. By this, we can reflect the potential future influence on the infection spread due to the movements of S and E that are hard to detect by the infection test at the time of immigration . As a result , we show the results of simulation where the influence of traveling across regions, which mean prefectures here, and of the choice of regions to vaccinate with priority , on the risk of infection spread. We find the regions of large population or large effective regeneration number should be vaccinated with priority and that frequent travels tend to cause an influence to enhance the spread especially in the uptrend stage of infections and in regions of smaller populations. Moreover, we show vaccination at a slow pace in Japan will not suppress the infection spread sufficiently to extinguish the current 4th wave within a few months. However, a restricted quantity of vaccines can be used efficiently by maximizing a diverted index value? the conditional entropy ? of vaccination distribution. Furthermore, travel activity across regional borders that accelerated infection spread so far is found to rather sup-press it if vaccination is processed at a high pace .
The SEIRS model, an extension of the SEIR model for analyzing and predicting the spread of virus infection, was further extended to consider the movement of people across regions. In contrast to previous models that con-sider the risk of travelers from/to other regions, we consider two factors. First, we consider the movements of susceptible (S), exposed (E), and recovered (R) individuals who may get infected and infect others in the destination region , as well as infected (I) individuals . Second, people living in a region and moving from other regions are dealt as separate but interacting groups with respect to their states, S, E, R, or I. This enables us to consider the potential influence of movements before individuals become infected, difficult to detect by testing at the time of immigration , on the spread of infection. In this paper , we show the results of the simulation where individuals travel across regions, which means prefectures here, and the government chooses regions to vaccinate with priority . We found a general law that a quantity of vaccines can be used efficiently by maximizing an index value, the conditional entropy Hc, when we distribute vaccines to regions. The efficiency of this strategy, which maximizes Hc, was found to outperform that of vaccinating regions with a larger effective re-generation number. This law also explains the surprising result that travel activities across regional borders may suppress the spread if vaccination is processed at a sufficiently high pace, introducing the concept of social muddling .
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