Hysteresis loop of nonperiodic outbreaks of recurrent epidemics

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

Figure 1

Real data of epidemics. Part (a) represents the weekly consultation rates of influenza-like illness (per 1000 consultations) in Hong Kong from the general practitioners (GPs) [46], where the value of $C$ is from 0 to 150. Parts (b) and (c) represent the time series of reported cases of infective measles $I$ in New York and Baltimore, respectively, where the variable $I$ is from 0 to $3\times {10}^4$ in (b) and from 0 to $8\times {10}^3$ in (c) [47].

Figure 2

A typical hysteresis loop in real data on epidemics. (a) Amplification of the data around the peak between 1938 and 1939 in Fig. 1, where the dashed line indicates the location of the rescaled zero point. (b) The hysteresis loop of (a) in the rescaled framework, where the squares and circles denote the growing and recovering phases, respectively.

Figure 3

(a) and (b) The SIS model. (c) and (d) The SIRS model. (a) and (c) Hysteresis loops for the SIS and SIRS models, respectively, where the curves with squares, circles, up triangles, and down triangles represent the cases of $T=1$, 2, 5, and 1000, respectively. (b) and (d) $S$ and ${\beta }_m$ vs $T$ for the SIS and SIRS models, respectively. The results are an average of 1000 realizations.

Figure 4

Cases of the SIS model with ${\beta }_r<{\beta }_m$, where (a)–(d) represent the cases of $T=1$, 2, 5, and 10, respectively, and the curves with squares, circles, up triangles, and down triangles represent the cases of ${\beta }_r=0.3$, 0.2, 0.15, and 0.1, respectively. The results are an average of 1000 realizations.

Figure 5

Case of the SIRS model, where (a)–(d) represent the cases of $T=1$, 2, 3, and 10, respectively, and the curves with squares, circles, up triangles, and down triangles represent the cases of ${\beta }_r=0.08$, 0.12, 0.16, and 0.2, respectively. The results are an average of 1000 realizations.

Figure 6

Dependence of the hysteresis loop on the chosen $\mathrm{\Delta }\beta$ with $N=10000$, $\langle k\rangle =6$, $\mu =0.2$, and $T=1$, where the curves with squares, circles, and triangles represent the cases of $\mathrm{\Delta }\beta =0.01,0.001$, and 0.0001, respectively. Parts (a) and (b) represent the cases of the SIS and SIRS models, respectively, with $\delta =0.5$ in (b).

Figure 7

Comparison between numerical simulations and theoretical approaches of mean-field theory (MFT), the microscopic Markov-chain approach (MMCA), and effective degree theory (EDT) with $\mu =0.2$, $\langle k\rangle =6$, and $T=1$, where the squares, circles, up triangles, and down triangles represent the case of ${\beta }_r=0.4$ for the four approaches, respectively, and the left triangles, right triangles, diamonds, and pentagons represent the case of ${\beta }_r=0.15$ for the four approaches, respectively. The results for the simulations are an average of 1000 realizations.