## 3 Data, Modelling Methods and Analysis Tools
### Hurst Exponent and Entropy to Assess Memory Effects in Stochastic Series  
We use two different metrics to assess memory in series of malaria incidence, empirical and simulated, investigating whether long-range dependencies could occur close to phase transition near disease elimination when compared to more stable epidemic scenarios. The Hurst exponent is defined as  
\[\frac{R}{D}=kT^{H} \tag{1}\]  
where the first member represents the rescaled range as a dimensionless ratio between \(R\) (represents the maximal range of all observations) and \(D\) (represents the standard deviation of all observations). An explicit mathematical definition is given below in Equation (4).  
Figure 2: Six different simulations of malaria incidence using different scenarios of gametocytemia, specifically, (**a**) 63 days (\(w_{h}=0.420\)), (**b**) 68 days (\(w_{h}=0.453\)), (**c**) 70 days (\(w_{h}=0.467\)), (**d**) 75 days (\(w_{h}=0.500\)), (**e**) 90 days (\(w_{h}=0.600\)), and (**f**) 110 days of positive gametocytemia over 150 days of expected disease duration (i.e., \(w_{h}=110/150=0.733\)).  
stands for the time index (number of observations in the time series), \(k\) is some constant to be determined and \(H\) represents the Hurst exponent.  
With the Hurst exponent estimation it became possible to distinguish among three different regimes characterizing the time series: (i) the anti-persistent regime, characterized by \(0.0<H<0.5\), where, if the series increases (resp. decreases) in one period, it is very likely that it will decrease (resp. increase) in the next period; (ii) the persistent regime, characterized by \(0.5<H<1\), where, if the series decreases (resp. increases) in one period it is very likely that it will decrease (resp. increase) in the next period; and (iii) the memoryless regime, characterized by \(H=0.5\) when the process is uncorrelated in time.  
A Hurst exponent estimation close to 0.5 (random walk process) is found in empirical time series with heavier disease transmission. In malaria time series from our model simulations, transitions from prevalence to disease elimination are characterized by values of the Hurst exponent larger than 0.5 and close to 1, a footprint of a persistence time series.  
As for entropy, it is related with the complexity of the time series [29,30]. The complexity of stochastic processes may be calculated with the use of entropy-based measures. For that purpose, several functions may be employed. The significance of complexity, emergence phenomena and self-organization may provide us with useful information concerning continuous as well as discrete systems, in the form of time series results.  
Information entropy is supported by the equation:  
\[S=-\sum_{i=1}^{N_{k}}P(x_{i})\log P(x_{i})\, \tag{2}\]  
where \(P(x)\) is the probability density function of the observable \(x\), which, in its discretized form, is estimated by a histogram with a finite set of values \(P(x_{i})\) for the set of bin-points \(x_{i}\). The log function is base two. Information theory defines entropy in terms of information uncertainty in the evolution of time series results. Other entropy-related terms, such as mutual information, may be used as alternative methods for time series analysis in malaria.