## 1 Introduction  
In Babylon, during the year 323 BC, [PERSON] fell ill after returning from Persia, and died shortly after at the age of 32. The reason of his death was most probably a disease now known to be caused by an infection with the parasite _Plasmodium falciparum_[1]. Malaria cases have declined globally from 238 million cases, in 2000, to 229 million, in 2019 [2]. Its mortality also decreased worldwide from around 736 thousand in 2000 to 409 thousand deaths in 2019 [2]. However, this reduction has stabilized since 2016, and several challenges and risks remain, namely, as concerns children under 5 years old. Due to their limited immune protection, they are a highly vulnerable group, accounting for 67% of global fatal malaria cases in 2019 [2]. Of particular importance in malaria transmission is the amount of gametocytes in the blood circulation. Gametocytes are the parasite forms which mediate transmission from humans to mosquitoes and, therefore, are also obvious targets in implementing preventive actions such as vaccine immunization or anti-malarial drugs. Submicroscopic gametocytemia detection is difficult and often not as precise as statistical data collection from e.g., malaria incidence varying heterogeneously from one community to another [3, 4]. In this paper we propose an approach to estimate effective _Plasmodium_ transmissibility levels, which reflect submicroscopic gametocytemia in individuals of a specific community, based on collected time series of malaria incidence in that same community. Time series models have been used as important tools and metrics, not only in epidemiology, but also in economics, geophysics, biology and ecology [5]. One of them, the so-called Hurst exponent, was introduced by [PERSON] in 1951 [6] in the context of hydrology planning, specifically, during the study of flooding levels in the river Nile. The Hurst exponent is a measure of memory in a series of values. Applying it to the series of Nile flooding levels, based upon Egyptian ancient hydrology data collected over 847 years, [PERSON] was able to estimate the rate at which the autocorrelation of that series decreased as the time interval between measurements increased. In this way, [PERSON] used the collected data to model the flooding levels of the Nile, and, with that, assessed optimum dam sizes to contain extreme rain events. Soon the applicability of the Hurst exponent extended to many other fields, particularly in the study of financial theory [7] and complex phenomena with evidence of fractal features [8; 9; 10]. However, while different time series models have been implemented in epidemic infections in general [11; 12; 13; 14; 15; 16; 17; 18], and in malaria incidence forecasting in particular [19; 20]--mostly auto-regressive and linear models--the application of the Hurst exponent in malarial spreading is still lacking. To relate gametocytemia levels with malaria incidence time series we introduce a methodology based upon the use of Hurst exponents. To the best of our knowledge, this is the first analysis of empirical data of malaria incidence series using Hurst exponent to derive an estimate of a hidden variable, namely, the level of transmissibility or easiness of spreading within the community from which the series of malaria incidence is collected. We show that the Hurst exponent is able to grasp long-range dependencies close to the phase transition between disease elimination and stable prevalence scenarios of malaria. In a more general scope, complexity in time series may also be related to the level of information entropy, which is commonly used to address emergence phenomena and self-organization. In this paper, we analyze the effects of memory in time series of malaria incidence. Two types of memory have been defined in random stochastic processes, long and short memory, where the transition between these regimes may be represented as a phase transition in the context of a stochastic random process.