Abstract
It is recently revealed from amounts of real data of recurrent epidemics that there is a phenomenon of hysteresis loop in the state space. To understand it, an indirect investigation from the parameter space has been given to qualitatively explain its mechanism but a more convincing study to quantitatively explain the phenomenon directly from the state space is still missing. We here study this phenomenon directly from the state space and find that there is a positive correlation between the size of outbreak and the size of hysteresis loop, implying that the hysteresis is a nature feature of epidemic outbreak in real case. Moreover, we surprisingly find a paradox on the dependence of the size of hysteresis loop on the two parameters of the infectious rate increment and the transient time, i.e. contradictory behaviors between the two spaces, when the evolutionary time of epidemics is long enough. That is, with the increase of the infectious rate increment, the size of hysteresis loop will decrease in the state space but increase in the parameter space. While with the increase of the transient time, the size of hysteresis loop will increase in the state space but decrease in the parameter space. Furthermore, we find that this paradox will disappear when the evolutionary time of epidemics is limited in a fixed period. Some theoretical analysis are presented to both the paradox and other numerical results.
Highlights
Epidemic spreading has been well studied in the last two decades and its main attention has been focused on the influence of network topologies[1,2,3]
Considering that the collected epidemic data are from the state space but not from the parameter space, it will be definitely more convincing if we can directly explain the data from the state space, in contrast to explain them indirectly from the parameter space in ref.[42]
Correspondence and requests for materials should be addressed to Z.L. www.nature.com/scientificreports/. We address these questions directly from the state space, in contrast to the hysteresis loop indirectly observed in parameter space in ref.[42]
Summary
It is recently revealed from amounts of real data of recurrent epidemics that there is a phenomenon of hysteresis loop in the state space. We surprisingly find a paradox on the dependence of the size of hysteresis loop on the two parameters of the infectious rate increment and the transient time, i.e. contradictory behaviors between the two spaces, when the evolutionary time of epidemics is long enough. With the increase of the infectious rate increment, the size of hysteresis loop will decrease in the state space but increase in the parameter space. We compare the results between the state and parameter spaces and interestingly find a contradictory dependence of the size of hysteresis loop on the parameters Δβ and T in the two spaces when the evolutionary time of epidemics is long enough, indicating that there is a paradox between the two spaces. A theoretical analysis is presented to explain the numerical results
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