Investigations of a possible connection between population density and the propagation and magnitude of epidemics have so far led to mixed and unconvincing results. There are three reasons for that. (i) Previous studies did not focus on the appropriate density interval. (ii) For the density to be a meaningful variable the population must be distributed as uniformly as possible. If an area has towns and cities where a majority of the population is concentrated its average density is meaningless. (iii) In the propagation of an epidemic the initial proportion of susceptibles (that is to say persons who have not developed an immunity) is an essential, yet usually unknown, factor. The assumption that most of the population is susceptible holds only for new strain of diseases. It will be shown that when these requirements are taken care of, the size of epidemics is indeed closely connected with the population density. This empirical observation comes as a welcome confirmation of the classical KMK (Kermack-McKendrick 1927) model. Indeed, one of its key predictions is that the size of the epidemic increases strongly (and in a non linear way) with the initial density of susceptibles. An interesting consequence is that, contrary to common beliefs, in sparsely populated territories, like Alaska, Australia or the west coast of the United states the size of epidemics among native populations must have been limited by the low density even for diseases for which the natives had no immunity (i.e., were susceptibles).Provisional. Comments are welcome.
Establishing unambiguously the existence of speculative bubbles is an on-going controversy complicated by the need of defining a model of fundamental prices. Here, we present a novel empirical method which bypasses all the difficulties of the previous approaches by monitoring external indicators of an anomalously growing interest in the public at times of bubbles. From the definition of a bubble as a self-fulfilling reinforcing price change, we identify indicators of a possible self-reinforcing imitation between agents in the market. We show that during the build-up phase of a bubble, there is a growing interest in the public for the commodity in question, whether it consists in stocks, diamonds or coins. That interest can be estimated through different indicators: increase in the number of books published on the topic, increase in the subscriptions to specialized journals. Moreover, the well-known empirical rule according to which the volume of sales is growing during a bull market finds a natural interpretation in this framework: sales increases in fact reveal and pinpoint the progress of the bubble's diffusion throughout society. We also present a simple model of rational expectation which maps exactly onto the Ising model on a random graph. The indicators are then interpreted as "thermometers", measuring the balance between idiosyncratic information (noise temperature) and imitation (coupling) strength. In this context, bubbles are interpreted as low or critical temperature phases, where the imitation strength carries market prices up essentially independently of fundamentals. Contrary to the naive conception of a bubble and a crash as times of disorder, on the contrary, we show that bubbles and crashes are times where the concensus is too strong!
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