A paradox of epidemics between the state and parameter spaces

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.