## 6 Discussion and Conclusions  
The utility of time series models is still a long way from becoming standard practice in malaria prevention. Differences in climate and geographic factors between world regions act as confounding factors in the strictly mathematical time series approach, lowering malaria forecast precision. In recent years the Box-Jenkins theory has become a consistent development in malaria forecasting [5, 18, 35, 36]. However, little attention has been devoted to the Hurst theory, information entropy, short- and long-memory stochastic processes or long-range dependence. It is remarkable that the Hurst theory was initially implemented in the field of hydrology, as malaria surges are clearly correlated with rainfall, temperature and climate seasonality [6]. Malaria epidemic time series consistently present different memory patterns, depending on disease transmission intensity. By comparing time series from our model simulations to real-data malaria time series from different parts of the world, it was possible to obtain a better definition of epidemic stability, according to disease transmission efficiency, from field data time series results. In stationary time series, long-memory processes have been related to the presence of the long-range dependence (LRD) between present and past results [37]. At low H-to-M disease transmission intensity, time series patterns were consistent with the presence of LRD. However, at high disease transmission intensity, this pattern reverted to a low-memory process. By looking at the present model time series with changing \(w_{h}\), one could witness significant differences in stochastic memory patterns. Additionally, in the presented empirical time series, the Hurst exponent and entropy correlated reasonably well with different epidemic growth rates. A similar pattern was evident when looking at malaria incidence correlation with the Hurst exponent and Shannon entropy. As these parameters may be affected by the time series length, their use in a normalized setting should be considered as a reliable option. By using the standardized forms of the Hurst exponent (\(H_{36}\)) and entropy measurement (\(S_{36}\)) it was possible to define the type of memory of stochastic malaria incidence time series with greater precision. This fact may be of significant relevance as both parameters may become additional and useful tools in malaria forecasting. Figure 6: Repeating Figure 5 with the effective gametocytemia for each empirical series (see text). In this paper we used the 36-month standard time length for a specific analysis. We considered it a compromise between a shorter time length (24 months) with less information available to the Hurst exponent estimation, and a longer time length (48 months) with less data available for analysis in shorter empirical time series, such as the series of [PERSON] et al. [16] (60 months). Other alternative methods, such as the generalized Hurst exponent and adaptive fractal analysis [38; 39], could be applied, but, in our case, the results did not show significant improvements. See Appendix B. Being a standard approach in time series analysis, we also performed SARIMA models for the simulation and empirical cases. Figures 7 and 8 show two illustrative cases of each. Details of the SARIMA model are given in Appendix C. In the case of the seasonal component, the regressive component of the SARIMA term vanishes in the Okech model such that we have \((P,D,Q)=(0,1,1)\). On the contrary, the seasonal SARIMA model of Landoh is purely a regressive model with one-time differencing, where we have \((P,D,Q)=(1,1,0)\). The SARIMA forecast of both empirical series is presented in Figure 8, while its equations and coefficients are available in Table A4. Figure 8: (**Left**) Empirical time series (black) and SARIMA model forecasting (blue) from Okech, 2008 [27] empirical time series, along with declining malaria incidence. The SARIMA forecast predicts rapid disease elimination evolving towards negative values.