Introduction

Malaria is a preventable and curable disease. However, in 2018, around 228 million cases were registered worldwide.¹ Timely prevention of malaria cases allows for the allocation of adequate resources for controlling the disease and planning its elimination.³ The initiative called ‘The Malaria Eradication Research Agenda’ analyzed different important aspects for the global elimination of malaria. Predictive modeling of cases was suggested as a tool for helping the Health Surveillance sector to plan infection control actions.⁴

Monthly time series of new cases can be statistically adjusted in mathematical functions, using software.⁶ Time series can be broken down according to three basic components: (i) the seasonal component, representing the cyclic pattern of the disease over time; (ii) the linear component, understood as an increasing or decreasing linear trend in the disease over time; and (iii) the stochastic component, regarding intervening factors affecting time series without a specific pattern.

Statistical time series models may be used for predicting future cases. Autoregressive integrated moving average (ARIMA) is a pioneer method for describing and predicting time series.⁹ The exponential smoothing model (ETS) represents an alternative to ARIMA.¹⁰ Exponential smoothing models for adjustment of complex seasonal patterns (TBATS and BATS) are methods deemed more efficient than ARIMA.¹¹ Another alternative can be the model dividing the seasonal component into subcomponents (STLM).¹²

The models described above may be deterministic and have statistical structure to break down and adjust seasonal and linear components in time series, but they cannot estimate the stochastic component. Therefore, machine learning computational approaches have been proposed to quantify the effect of the third component: the structural model (StructTS), the neural network model (NNETAR), and machine learning models (ELM, MLP) are examples of such approaches.⁹ Models may be compared with a null model, defined by the constant value of the last observation.¹⁷

The premise of this article is that health surveillance can use predictive models of time series to predict the impact of malaria on a given Brazilian state. Malaria case incidence in the state of Amapá between 2015 and 2018 was, on average, 17 cases per 1,000 inhabitants, one of the highest rates in the country in comparison with the rates found in the states of Acre (37/1,000 inhabitants), Amazonas (17/1,000 inhabitants) and Roraima (19/1,000 inhabitants) in the same period. It is important to identify future time series of malaria cases for planning control measures. This study aims to evaluate the predictive power of different malaria case time-series models in the state of Amapá.

Methods

A statistical and computational approach was applied to the Amapá state health service, complementary to malaria control activities in Brazil as a whole, in accordance with the terms set by the World Health Organization (WHO) for eliminating this disease.²

This was an ecological time series study, using the number of malaria cases recorded in the state of Amapá in the period from 1997 to 2016.

Amapá is one of Brazil’s most important endemic malaria regions (Figure 1). Its predominant climate, according to Köppen-Geiger's climatic classification, is tropical monsoon, that is, hot and very humid, with average rainfall indexes of 3,300mm annually. The largest portion of the state (73%; 97,000km²) is covered by native vegetation. In 2019, Amapá's population totaled 830,000 inhabitants, distributed over 16 municipalities.

Figure 1  – Average incidence (per 1,000 inhabitants) of confirmed malaria cases in Brazilian states 

Malaria data (autochthonous cases, identified through microscopy slides testing positive for Plasmodium – using the thick smear technique) were obtained from the following Health Ministry Health Surveillance Secretariat information systems: National Malaria Control Program Information System (SISMAL) for the period 1997-2003; and the Malaria Epidemiological Surveillance Information System (SIVEP-Malaria) for the period 2003-2016.

Three quantitative discrete variables were used with natural logarithm transformation:¹⁸

The statistical approach used was based on time series models. The first procedure involved the Dickey-Fuller test, increased to a 5% significance level, to test whether the time series was stationary or not. The stationary premise was assumed using statistical models, described later in this text.

The time series encompassed 240 months, from January 1997 to December 2016, divided into two periods: training period, from January 1997 to December 2015; and test period, from January to December 2016. The number of monthly cases of malaria in the training period was used to adjust each one of the statistical models and estimate time component parameters (seasonality; linear trend; stochastic effect). The number of monthly cases of malaria in the test period was used for comparison with the values estimated by the statistical models. The test was conducted with three prediction time horizons: 12 months in advance (January to December 2016); six months in advance (July to December 2016); and three months in advance (October to December 2016). The result of each test was used to assess the predictive power of the models.

The statistical models used were:

Three criteria were considered for assessment of the predictive power of the statistical models:

A predictive power deemed acceptable was then defined as (i) lowest MAPE value, (ii) values greater than 2 for the null model MAPE scale, and (iii) values less than 1 for Theil’s U.

The analyses were conducted in the R computing environment, through a corrected and validated script that can be reproduced by Municipal and State Health Department surveillance staff in Brazil (Supplementary material).¹⁹

Results

The total number of malaria cases was 403,832, with an average of 1,771 cases per month and median of 1,518 cases/month, with 1,021 cases in the first quartile and 2,079 cases in the third quartile - standard deviation (SD) = 918 cases. The increased Dickey-Fuller value showed that the time series is stationary (test value = -5.352; p-value <0.01). The minimum number of cases was 487 (May 2014), while the maximum was 5,944 (October 2000), consistent with the seasonal nature of malaria in the state (Table 1).

Table 2 presents the performance of the models according to the criteria for assessing predictive power and the three selected prediction time horizons. In the 12 and six month time horizons, all deterministic models (ETS, ARIMA, STLM, BATS and TBATS) showed reliable predictive power. However, none of the stochastic models showed reliability in predicting future cases of malaria for those time horizons.

The three-month time horizon surprisingly showed itself to be challenging for all the models. Finally, no model was deemed reliable for predicting malaria cases three months in advance (Figure 2).

Figure 2 – Predictive power of the statistical models considering the comparison between the test variable and the estimated values for each model, according to three time horizons 

Discussion

Deterministic models showed themselves to be reliable for predicting the monthly number of cases of malaria in the next 12 or six months in the state of Amapá. This result may be interpreted as follows: the time series being studied has characteristics that enable better performance by deterministic models. This is because time series with (i) a strong seasonal component, (ii) relatively low linear trend (for example, temporal stationarity) and (iii) low or null stochastic effect are satisfactorily adjusted by deterministic models such as ARIMA. Theoretically, statistical models able to detect stochastic effects can present better performance, in comparison to deterministic models, in the case of time series with (i) lack of seasonality, (ii) high linear trend, and (iii) stochastic effects.²⁰ However, none of the assessed models presented reliable results for predicting the number of monthly cases of malaria in the three-month time horizon. Prediction of malaria in short future periods also proved to be unsatisfactory in another study, in which deterministic models were applied in districts of Sri Lanka between 1972 and 2005.¹⁸ That study discouraged using statistical models for predicting malaria cases in near future periods of one month in advance, notwithstanding the period of the study in Sri Lanka being shorter than the shortest period in our study.

A limitation of the present study is the impossibility of predicting malaria cases for short periods in advance, such as 3 months. Another limitation is the fact that, when using the state of Amapá as unit of analysis, local scale information is lost. The advantages of this approach, however, are the possibility of predicting malaria cases 12 or six months in advance and using deterministic statistical models. As these models are easier for health service managers to understand, this makes their implementation in health services more viable.²¹

The potential use of time series techniques in epidemiological studies, disease surveillance and malaria outbreak prediction has been explored in different studies.¹⁸ For example, a study with similar design, conducted in the north of Thailand from 1999 to 2004, found deterministic models enabling future prediction of malaria and dengue fever one to four months in advance, to the extent its authors suggest using such models for allocating resources in controlling and preventing these diseases.²² Another study with deterministic models, conducted in Sudan, between 2009 and 2013, showed that the predictive power of the models used varied according to each state (of that country).²³ Deterministic models used for predicting malaria in Afghanistan, between 2005 and 2015, led to reliable predictions for 12 to four months in advance.²⁴

Considering that, among deterministic models, ARIMA is the most widely applied in the literature and has predictive characteristics for diseases affected by seasonality,²² we recommend its implementation as a protocol for predicting monthly malaria cases for long-term horizons - 12 or six months – on a state-wide scale in the Brazilian Amazon state of Amapá.