## 3 Data, Modelling Methods and Analysis Tools
### Estimating the Hurst Exponent and Entropy in Series of Malaria Incidence
In the Okech time series, the behavior of \(H_{36}\) and \(S_{36}\) is also quite correlated with the behavior of \(I_{36}\) in time. As the Hurst exponent (\(H\)) and Shannon entropy (\(S\)) are influenced by the length of the time-series sample, we also considered a 36-month moving window estimate. In the case of the Hurst exponent, we compute the quantities \(R\) and \(D\) as  
\[R(T) = \max_{t=1,\ldots,T}\left(\sum_{k=1}^{t}(I(k)-I_{36}(T))\right)- \min_{t=1,\ldots,T}\left(\sum_{k=1}^{t}(I(k)-I_{36}(T))\right), \tag{4a}\] \[D(T) = \left(\frac{1}{36}\sum_{k=T-35}^{n}(I(k)-I_{36}(T))^{2}\right)^{1/2}. \tag{4b}\]  
Figure 3: (**a**) Illustration of how a 36-month moving average of malaria incidence, symbolized as \(I_{36}\), is obtained from an original series of monthly malaria incidence values, either empirical or simulated. Similar procedures are used to compute (**b**) \(H_{36}\) and (**c**) \(S_{36}\). See text for details. In this case, we used an agent-based simulation with a low human-to-mosquito transmission efficiency of \(w_{h}=0.420\). Having the series of 36 values \(R(T)/D(T)\) for each window, we then applied Equation (1) to find fitting values for \(k\) and for the Hurst exponent \(H\). As for Shannon entropy, an estimate of the probability density was first computed given by the set of values \(P(l_{i})\) for an assumed set of bin values \(l_{i}\), with \(i=1,\ldots,N_{b}\), of malaria incidence (\(I\)) and then Equation (2) was applied, running the sum only over the 36 incidence values observed within each time window. In this way, we have computed the average of malaria incidence \(I\), \(H\) and \(S\) for 36-month windows, independently of the size of the empirical series. In order to quantify \(I\), \(H\) and \(S\) for each empirical case, we then obtained the average of the corresponding time series and computed its standard deviations. Table 1 shows the empirical values of the three observables. Notice that, in these cases, higher moments seems to be not particularly relevant, which can be seen comparing the average in Table 1 with the median and quartiles in Table 2.  
Qualitative Analysis and Robustness Assessment of the Hurst Exponent and Entropy in Empirical Time Series Behavior  
While the eight empirical cases show a broad range of values for malaria incidence and the Hurst exponent, the entropy seems much more resistant to changes. The cases of Alhassan, Elipe and Okech form a separated group from the other series which form a second group. We will address these cases in more detail when building the model for effective human-to-mosquito transmissibility (gametocytemia). We have also looked at the way those indices behaved in a typical low-transmission empirical time series, such as that obtained from Okech, 2008 [27]. In this empirical time series, when the Hurst exponent was evaluated in 36-month partial intervals (\(H_{36}\)), it revealed a clear inverse correlation with malaria annual incidence. Futhermore, the Shannon entropy consistently decreased with progressive lower values of malaria incidence. This correlation pattern was similar to the one found in model simulations when comparing high- and low-transmission scenarios. By consistently searching for evidence of the presence of long-range dependence in malaria time series we looked into the time evolution of information (Shannon) entropy in scenarios with stronger disease transmission, and compared the results with those from other time series of lower disease transmission. In our model simulations, at low levels of disease burden near an eradication-prevalence transition, we consistently found malaria time series with lower information entropy. In Appendix A we describe, in more detail, the eight particular cases. Table 3 shows the correlation between the three properties, \(I\), \(H\) and \(S\), for the eight empirical data sets. One finds evidence of linear correlation between malaria incidence and