## 3 Data, Modelling Methods and Analysis Tools
### Estimating the Hurst Exponent and Entropy in Series of Malaria Incidence  
Both the Hurst exponent and Shannon entropy are influenced by the length of the time series sample. Therefore, we introduced a standardization procedure, which is independent of the length of the time series. The procedure is illustrated in Figure 3 and is as follows. We define a 36-month moving average malaria incidence, symbolized as \(I_{36}(m)\), at month \(m\), as the average malaria incidence in the previous 36 months:  
\[I_{36}(m)=\frac{1}{36}\sum_{n=m-35}^{m}I(n)\, \tag{3}\]  
where \(I(n)\) is the monthly malaria incidence measured or simulated, composing the series of values. In the Alhassan malaria time series (Kasena Nankana municipality in Ghana, 2017) [21] one can see a declining trend in malaria incidence in the form of a decreasing moving average of 36 months (\(I_{36}\)), from \(\sim\)11 to \(\sim\)4 cases per 100 inhabitants, per year [_phy_], despite the small final outbreak in the last 6 months. Consistent this declining trend in \(I_{36}\), we may also witness a decreasing trend in \(S_{36}\), as well as a rising trend in \(H_{36}\) after month 55. In this case, the behavior of both the Hurst exponent and entropy in the preceding 36 months is well correlated with the malaria incidence trend. The Gomez-Elipe time series (Karuzi in Burundi, 2007) [24] is quite different from the remaining empirical examples. It shows a consistent stable pattern of malaria incidence below 110 cases _phy_, until month 45, peaking at \(\sim\)500 cases _phy_, around month 48, with a fast downward trend to \(\sim\)60 cases _phy_ at month 56, and a slower decline thereafter to a final value of \(\sim\)20 cases _phy_--see Figure 1. The 36-months malaria incidence (\(I_{36}\)) reveals a consistent upward trend peaking at month 64, with a downward pattern thereafter. In this case, we find a persistent oscillatory behavior in the value of \(H_{36}\) during the entire time series, usually above 0.9. Yet, in parallel with the outbreak in malaria incidence, we see a sudden fall in \(H_{36}\) to values close to 0.6 (closer to random noise) at around month 48. Entropy in the form of a 36-months moving average (\(S_{36}\)) reveals a similar behavior in relation to \(I_{36}\) from month 48 onwards. The Landoh malaria time series (Est Mono district in Togo, 2012) [25] reveals a consistent upward trend in malaria incidence in the form of a mild increase in the 36-months moving average (\(I_{36}\)) during the whole time series, ranging from the initial \(\sim\)18 cases _phy_, to a peak at \(\sim\)30 cases _phy_ in month 72. Consistent with the steady upward trend in \(I_{36}\) during the entire time series, we find a declining trend in the 36-months moving average of the Hurst exponent (\(H_{36}\)), from a peak of \(\sim\)0.85 at month 47, to an all-time-low value of \(\sim\)0.55. Along with this downward trend in \(H_{36}\) there is a consistent wave-like increase in entropy in the form of a 36-months moving average (\(S_{36}\)), during the entire time series. In the Landoh time series, the behavior of \(H_{36}\) and \(S_{36}\) is reasonably well correlated with the behavior of \(I_{36}\) in time. Of a similar form, but with an opposite trend to the Landoh time series, the Okech time series (Kenya, 2008) [27] also reveals a steady decreasing trend. In the Okech time series, malaria incidence consistently decreases from an initial peak of \(\sim\)400 cases _phy_, to a final value of close to 50 cases _phy_. Consistent with the steady downward trend in \(I_{36}\) during the entire time series, we find an upward trend in the 36-months moving average of the Hurst exponent (\(H_{36}\)) from an all-time-low of \(\sim\)0.6 at month 42, to a peak of \(\sim\)1.2 at month 82. Along with this upward trend in \(H_{36}\), there is also a consistent decrease in entropy in the form of a 36-months moving average (\(S_{36}\)), during the whole time series.