Phys Rev E. 2016 Dec;94(6-1):062318. doi: 10.1103/PhysRevE.94.062318.

**Hysteresis loop of nonperiodic outbreaks of recurrent epidemics.**

Hengcong Liu,^{1} Muhua Zheng,^{1} Dayu Wu,^{1} Zhenhua Wang,^{1} Jinming Liu,^{2} and Zonghua Liu^{1, 2}

*1 **Department of Physics, East China Normal University, Shanghai, 200062, P. R. China*

*2 **State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China*

(Dated: July 24, 2017)

**Abstract**

Most of epidemic studies so far are focused on the growing phase such as how epidemic spreads and what are the conditions for epidemic to outbreak in varieties of cases. However, we discover from real data that in large scale, epidemic spreading is in fact a recurrent event with distinctive growing and recovering phases, i.e. a hysteresis loop. We here show that the hysteresis loop can be reproduced in epidemic models, provided that the infectious rate is adiabatically increased/decreased before the system reaches its stationary state. Two ways to the hysteresis loop are revealed, which is helpful in understanding the mechanics of infectious in real evolution. Moreover, a theoretical analysis is presented to explain the mechanism of hysteresis loop.

**SUPPLEMENT**** **

So far, most of epidemic studies are mainly focused on its growing process, while little attention has been paid to its recovering process. In these studies, a common point is that their infectious rate β keeps constant during an evolutionary process. However, realistic situations are much complicated where β changes with time and the recovering process cannot be simplified only as an inverse process of its growing process. For example, in large scale, epidemic spreading is in fact a recurrent event where there is a recovering phase with gradually decreasing of β, except the growing phase with gradually increasing of β[1]. Previous studies in this line are mainly focused on the recurrence of seasonal influenza where serious epidemics occur in winter or spring followed by fade-out periods during the warmer months. For reflecting this influence, Ref. [1] assumed that β depends sinusoidally on time t.

Recently, Zheng *et al *have collected the recurrent epidemic data of Hong Kong, which comes from 64 General Out-Patient Clinics (GOPC) and 50 General Practitioners (GP) [2]. They surprisingly found that the data shows non-periodic outbreaks, i.e. non-seasonal or non-annual behaviors [3]. Then, they checked a large number of real data from other cities such as the Baltimore and New York etc [4], and also found that besides the seasonal periodic outbreaks of influenza, there are non-periodic outbreaks, indicating that the irregular outbreak is generic. To understand how the non-periodicity shows up, Zheng *et al *presented a network model of SIRS epidemic with both time-dependent infection rate and a small possibility of persistent epidemic seeds, representing the influences from the larger annual variation of environment and the infection generated spontaneously in nature, respectively. By this model, it is shown that the recurrent outbreaks of epidemic depend not only on the infection rate but also on the density of susceptible agents.

Further, Zheng *et al *noticed that there is a correlation of epidemic outbreaks between the system of GOPC and that of GP in Hong Kong, indicating that the two systems are coupled [5]. Then, they rechecked other coupled regions or cities and found the same phenomenon such as the coupled regions of California and Nevada, coupled Arizona and California, coupled cities of Boston and Fall River, and the coupled cities of Los Angeles and Sacramento. Very interesting, these coupled time series of recurrent epidemics can show either synchronized outbreak pattern where outbreaks occur simultaneously in both networks or mixed outbreak pattern where outbreaks occur only in one network but do not in another one. Therefore, the study of nonperiodic outbreaks is extended to a two-layered network model [5].

An important point in all these studies is that the infectious rate β changes with time, but an unclear question is how to reflect this change. Liu *et al *recently pointed out that β is adiabatically changed [6], where the initial conditions of infected seeds at each updated β are inherited from the final state of system with the last β. This way of adiabatical changing β is completely different from the previous studies where β is allowed to be updated only when system reaches its stationary state and the initial infected seeds for each updated β are always reset randomly. A consequence of this adiabatical change of β is that the growing and recovering processes become asymmetric and thus form a phenomenon of hysteresis loop, confirmed by all the recurrent epidemic data. According to the best of our knowledge, this is the first time to assume the adiabatical changing of β, it thus opens a new window to study the epidemic spreading form the angle of adiabatical changing of β. We believe that the caused phenomenon of hysteresis loop deserves definitely more studies and will highlight our understanding on the real epidemic data. For details, such as how the hysteresis loop is formed and what factors influence the size of hysteresis loop, the interesting readers are suggested to go to Ref. [6].

**References:**

[1] Stone L, Olinky R, and Huppert A. Seasonal dynamics of recurrent epidemics. Nature **446**, 533-536 (2007).

[2] Department of Health, HK. http://www.chp.gov.hk/en/sentinel/26/44/292.html. Date of access: 15/06/2014.

[3] Zheng M,Wang C, Zhou J, Zhao M, Guan S, Zou Y, and Liu Z. Non-periodic outbreaks of recurrent epidemics and its network modelling. Sci. Rep. **5**, 16010 (2015).

[4] B. Bolker, http://ms.mcmaster.ca/bolker/measdata.html. Date of access: 26/12/2014.

[5] Zheng M, Zhao M, Min B, and Liu Z. Synchronized and mixed outbreaks of coupled recurrent epidemics. Sci. Rep. **7**, 2424 (2017).

[6] Liu H, Zheng M, Wu D, Wang Z, Liu J, and Liu Z. Hysteresis loop of nonperiodic outbreaks of recurrent epidemics. Phys. Rev. E **94**, 062318 (2016).