Ultrametric model for covid-19 dynamics: an attempt to explain slow approaching herd immunity in Sweden

We present a mathematical model of infection dynamics that might explain slower approaching the herd immunity during the covid-19 epidemy in Sweden than it was predicted by a variety of other models; see graphs Fig. 2. The new model takes into account the hierarchic structure of social clusters in the human society. We apply the well developed theory of random walk on the energy landscapes represented mathematically with ultrametric spaces. This theory was created for applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, the virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchys levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider linearly increasing (with respect to hierarchys levels) barriers. This structure of barriers matches with a rather soft regulations imposed in Sweden in March 2020. In this model, the infection spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This models feature matches with the real situation during the covid-19 epidemy, with its cluster spreading structure. Clusters need not be determined solely geographically, they are based on a number of hierarchically ordered social coordinates. The model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. In particular, our model describes such a specialty of spread of covid-19 virus as the presence of "super-spreaders" who by performing a kind of random walk on a hierarchic landscape of social clusters spreads infection. In future, this model will be completed by adding the SIR-type counterpart. But, the latter is not a specialty of covid-19 spreading. O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=126 SRC="FIGDIR/small/20146209v1_fig2.gif" ALT="Figure 2"> View larger version (8K): org.highwire.dtl.DTLVardef@1b3b3fcorg.highwire.dtl.DTLVardef@ed8c38org.highwire.dtl.DTLVardef@190b208org.highwire.dtl.DTLVardef@9848be_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFigure 2:C_FLOATNO Asymptotic behavior of probability to become immune; increasing of the herd immunity (for fixed social temperature T, the upper graphs correspond to one-step barrier growth 10 and 100 times, respectively. C_FIG