How many incoming travelers [I0 at time 0, equivalent to the “founders” in evolutionary genetics] infected with SARS-CoV-2 who visit or return to a region could have started the epidemic of that region? I0 would be informative about the initiation and progression of the epidemics. To obtain I0, we analyze the genetic divergence among viral populations of different regions. By applying the “individual-output” model of genetic drift to the SARS-CoV-2 diversities, we obtain I0 < 10, which could have been achieved by one infected traveler in a long-distance flight. The conclusion is robust regardless of the source population, the continuation of inputs (It for t > 0) or the fitness of the variants. With such a tiny trickle of human movement igniting many outbreaks, the crucial stage of repressing an epidemic in any region should, therefore, be the very first sign of local contagion when positive cases first become identifiable. The implications of the highly “portable” epidemics, including their early evolution prior to any outbreak, are explored in the companion study (Ruan et al. 2020).

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Exact solutions for spin-orbit (SO) coupled cold atomic systems are very important and rare in physics. In this paper, we propose a simple method of combined modulations to generate the exactly analytic solutions for a single SO-coupled ultracold atom held in a driven double well. For the cases of synchronous combined modulations and the spin-conserving tunneling, we obtain the generally accurate solutions of this system respectively. For the case of spin-flipping tunneling under asynchronous combined modulations, we get the specially exact solutions in simple form when the driving parameters are appropriately chosen. Based on these obtained exact solutions, we reveal some intriguing quantum spin dynamical phenomena, for instance, the arbitrary coherent population transfer (ACPT) with and/or without spin-flipping, the controlled coherent population conservation (CPC), and the controlled coherent population inversion (CPI). The results may provide a possibility for generating the accurate quantum entangled states and the exact control of spin dynamics for a SO-coupled ultracold atomic system.

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