## Abstract

An effort is made to evaluate the multifractal properties of malaria cases in India from 1980 to 2014. The possible changes in these properties in a global warming scenario (during 2015–2049) are also quantified. The malaria cases for this purpose are derived from the VECTRI dynamical malaria model, which uses rainfall and temperature data of Coupled Model Intercomparison Project Phase 6 models for the historical and SSP5–8.5 projection scenario, respectively. It is found that the generalized Hurst exponent *h* (*q*) and multifractal spectrum width Δ*α* are strongly nonlinear decreasing functions of order *q*, thus confirming the multifractal nature (and heterogeneous distribution) of the malaria cases in India. The multifractal spectrum of the malaria cases exhibits right-tailed nature along with high inter-model variability, suggesting that the time series under consideration are fine scale and highly complex in nature. The multifractal spectrum width and change in singularity dimension are used to quantify the strength of multifractality for the historical and future projection scenario. It is demonstrated that the strength of multifractality of malaria cases is likely to decrease with an increase in greenhouse gas concentration, which may be happening due to a persistent increase in malaria cases in India as a result of global warming.

## HIGHLIGHTS

Effect of climate change on malaria cases in India is quantified with the help of a dynamic malaria model.

Multifractal analysis of malaria cases is carried out in a global warming scenario.

Malaria distribution in India is complex with right-tailed multifractal spectrum.

Strength of multifractality of malaria will decrease in a warming environment.

## INTRODUCTION

Global warming and associated climate change (e.g., variability of temperature and rainfall) are emerging as the foremost environmental problems in the 21st century, especially in developing countries (Ekwueme & Agunwamba 2021). Earth's temperature (and associated rainfall pattern) is continuously changing in a global warming scenario (Doi 2022), including in the Indian subcontinent region (Pandey *et al.* 2020; Chaturvedi & Dwivedi 2021). The climatic conditions are now considered a non-biomedical, natural resource for studying and addressing human health challenges (Droli *et al.* 2022). Even though the mortality rate due to vector-borne disease malaria has considerably decreased in India in the past 1–2 decades, it is still a significant public health challenge and one of the main diseases occurring in the subcontinent, especially in the summer monsoon and post-monsoon months in India (Lauderdale *et al.* 2014; Dhiman & Sarkar 2017). Recent dynamical modelling studies by Chaturvedi & Dwivedi (2020, 2021) have clearly shown that the transmission intensity of malaria disease in India is greatly influenced by the non-uniform distribution of climatic parameters such as temperature and rainfall of the region. It was also highlighted in these studies that the climate change is significantly impacting the malaria transmission intensity in India.

To study the past and future spatio-temporal variability, heterogeneity, scaling, and predictability properties of malaria transmission in India, it is imperative to have the long-term (e.g., 3–4 decades longer) continuous gridded data of malaria at most of the grid points in India. However, such long-term observations of malaria, even if available, are limited to only certain specific locations in India. The recourse is, therefore, taken by running the state-of-the-art dynamical malaria models to generate the high-resolution (in space and time) quality-controlled three-dimensional data of malaria transmission in India.

There are several dynamical malaria models which have been used for this purpose in different regions in the world. These models include Liverpool Malaria Model (Leedale *et al.* 2016), Mapping Malaria Risk in Africa (Craig *et al.* 1999), Modeling Framework for the Health Impact Assessment of Man-Induced Atmospheric Changes (MIASMA) (van Lieshout *et al.* 2004), and Vector-borne Disease Community Model of International Centre for Theoretical Physics Trieste (VECTRI) (Tompkins & Ermert 2013). Of all the malaria models described earlier, the VECTRI model is the only fully dynamical model which incorporates a representation of human migration for transporting the malaria parasites to those new regions which may become suitable for future transmission (Caminade *et al.* 2014). Unlike other dynamical models, the VECTRI model takes into account the impact of climate (e.g., temperature and rainfall variations), surface hydrology, and population density on the development cycles of the malaria vector and on malaria distribution (Caminade *et al.* 2014; Leedale *et al.* 2016; Chaturvedi & Dwivedi 2021). As such, in the present study, this model is employed to estimate the malaria transmission intensity in terms of a number of clinical cases (‘Cases’; Tompkins & Ermert 2013) using the historical and future projection scenario data of temperature and rainfall derived from the Coupled Model Intercomparison Project Phase 6 (CMIP6).

To plan for an effective control strategy for malaria disease as well as for its better parameterization, modelling, and prediction by the dynamical models, it is important to understand the structure and dimensionality (e.g. scaling properties) of malaria transmission in India in the past, present, and future climates. In particular, it will be useful to know that similar to the various other processes of nature, whether it is possible to obtain self-similarity, i.e., fractal property in the malaria transmission in India. However, since the distribution of malaria transmission in India is spatio-temporal varying in a highly complex nonlinear manner (Chaturvedi & Dwivedi 2020, 2021), as such it may not be possible to obtain a single scaling exponent, e.g., monofractal dimension (Dwivedi 2012; Kumar *et al.* 2018) for it. This is in conformity with the previous findings which suggest that a single scaling exponent is not able to capture the underlying dynamical nature of natural systems, owing to the nonlinearity, entangled nature, and complexity of these systems (Ihlen & Vereijken 2010; Ihlen 2012). Several studies have demonstrated that real-world natural processes, especially the biometeorological processes, are multifractal in nature, instead of being monofractal (Suckling *et al.* 2008; Moreno *et al.* 2011; Ihlen 2012; Xue & Bogdan 2017; Hou *et al.* 2018; Racz *et al.* 2018; Philippopoulos *et al.* 2019; Maruyama 2020; Dwivedi & Pandey 2022).

There are several techniques which are used to estimate the multifractality of time series data. However, the multifractal formalisms based on the Fourier analysis and the partition function do not correctly estimate the multifractal behaviour of non-stationary signals or signals affected by trends (Bacry *et al.* 2001; Telesca *et al.* 2005). The wavelet transform modulus maxima (WTMM) technique (Muzy *et al.* 1993, 1994) and multifractal detrended fluctuation analysis (MFDFA) technique (Kantelhardt *et al.* 2002, 2006) are widely used methods for estimating the multifractality of non-stationary signals with or without trends. Oświe¸cimka *et al.* (2006) suggested that in the majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA is recommended in WTMM. They found that using WTMM may give biased outcomes with spurious multifractality. Moreover, it was suggested that while it is possible to apply MFDFA in a more automatic fashion, WTMM must be applied with care. The MFDFA also produces less overall variability than WTMM, particularly with smaller datasets (Kantelhardt *et al.* 2002; Oświe¸cimka *et al.* 2006). Oświe¸cimka *et al.* (2005) showed that MFDFA allows a global detection of multifractal behaviour, while the WTMM method is suited for the local characterization of the scaling properties of signals. Telesca *et al.* (2005) also argued that MFDFA is a more reliable tool for multifractal analysis compared to the WTMM. In several head-to-head comparisons of MFDFA with WTMM on the same datasets, it is found that MFDFA produces more consistent results (Kantelhardt *et al.* 2002; Galaska *et al.* 2007; Zorick & Mandelkern 2013) and requires less computational and programming effort as compared to the WTMM (Telesca *et al.* 2005; Kantelhardt 2009). On the basis of the aforementioned conclusions, the MFDFA technique is used in the present study for estimating the multifractality of malaria time series data.

The MFDFA technique is a generalization of the detrended fluctuation analysis technique (Kantelhardt *et al.* 2002, 2006; Ihlen & Vereijken 2010; Ihlen 2012; Vergotte *et al.* 2018) and helps to compute multiple scaling exponents (multiple fractal dimensions) instead of a single exponent. The technique has been applied in different kinds of studies including a variety of processes from different disciplines (Kantelhardt *et al.* 2003; Makowiec *et al.* 2006, 2011; Dutta 2010; Bogachev & Bunde 2011; Ihlen 2012; Yadav *et al.* 2012, 2013; Hou *et al.* 2018; Agbazo *et al.* 2019; Krzyszczak *et al.* 2019; Gos *et al.* 2021). These studies have shown that with the help of a multifractal spectrum, it is possible to identify a wide range of different scale-invariant structures of underlying mechanisms. The higher degree of multifractality of a time series indicates the greater complexity and tangled nature of the underlying mechanisms involved in explaining the processes/phenomena under consideration (Vergotte *et al.* 2018; Torre *et al.* 2019).

To the best of our knowledge, this kind of study which estimates the dynamical nature of malaria transmission in India in terms of multifractal properties has not been carried out yet. The main aim of this study is to use the three-dimensional dynamical malaria model output (corresponding to temperature and rainfall data of several CMIP6 models) for investigating the power-law behaviour (e.g., scaling and self-similarity) and multifractal properties of malaria transmission in India. Another aim is to estimate the effect of global warming on the multifractal nature of malaria transmission. The MFDFA technique is used to examine the multifractal properties of the historical data (1980–2014) of malaria transmission in India (in terms of VECTRI simulated malaria ‘Cases’ daily time series corresponding to five CMIP6 models). The change in the multifractal properties as a result of increased greenhouse gas (GHG) concentration is quantified by using the future projection dataset of the years 2015–2049 (35 years). This article is organized as follows. A brief description of the VECTRI model and methodology related to the MFDFA technique are given in Section 2. The results are summarized in Section 3. Conclusions are presented in Section 4.

## MODEL AND METHODS

### Dynamical malaria model VECTRI

For the estimation of malaria transmission intensity, the dynamical malaria model VECTRI (Tompkins & Ermert 2013) is used in this article. The VECTRI is a mathematical–biological model developed by the International Centre for Theoretical Physics (ICTP), Italy (Tompkins & Ermert 2013; Chaturvedi & Dwivedi 2020, 2021). The model estimates the three-dimensional (space-time varying) malaria transmission intensity with the help of spatio-temporally varying temperature, rainfall, and population density. It takes into account the key life cycle processes of the malaria mosquito; specifically the *Anopheles cruciferae* vector and the *Plasmodium falciparum* parasite (Tompkins & Caporaso 2016). Our model setup does not use multiple vector types and their dispersion. The VECTRI treats the host and host–vector interaction as a unified system, instead of treating them as two different entities (Tompkins & Ermert 2013). The impact of temperature in VECTRI is modelled using the concept of degree days that need to be completed in order for the *Anopheles* mosquito to proceed from one stage to the next in its developmental cycle. In VECTRI, the impact of rainfall is parameterized in terms of a simple, physically based model of surface hydrology in which mosquito larvae development increases during low rainfall rates (albeit above a threshold), whereas it decreases during very high rainfall rates as a result of flushing of breeding sites (Tompkins & Ermert 2013; Asare *et al.* 2016; Chaturvedi & Dwivedi 2021). The parameterization makes an estimate of available permanent and temporary water bodies that form after a rain episode, and also takes into consideration the necessary aquatic environment for oviposition and development of aquatic stages of the mosquito (Tompkins & Ermert 2013). The relative humidity is known to influence malaria transmission through the desiccation of malaria vectors (under very dry conditions). The evaporation and infiltration rate are set to 5 and 245 mm day^{−1}, respectively (Tompkins & Ermert 2013). The surface hydrology in our model setup does not take into account terrain slope, soil type, and land use characteristics. The biting rates of mosquitoes and their transmission probabilities get impacted by the population density (Tompkins & Caporaso 2016). However, sudden, large-scale population movements (e.g., due to conflict, environmental disasters, and forced resettlement schemes) are neglected in the model. The model also lacks explicit incorporation of interventions commonly employed for malaria eradication. Further details of the model setup were given by Chaturvedi & Dwivedi (2021).

In the present study, daily rainfall and temperature data of five CMIP6 models are taken to run the VECTRI model. The CMIP6 models used in this study are CESM2, GFDL-CM4, IPSL-CM6, MIROC6, and NorESM2. The daily rainfall and temperature data of these models are interpolated to a regular 0.25° model grid. The model rainfall and temperature datasets are validated against the corresponding India Meteorological Department (IMD) observed gridded rainfall dataset (Pai *et al.* 2014) and ERA5 reanalysis temperature dataset (Hersbach *et al.* 2020), respectively (see e.g., Chaturvedi & Dwivedi 2021). The gridded population data are taken from the Global Rural-Urban Mapping Project (version 1) (GRUMPv1), which are derived from the Gridded Population of the World (version 4) (GPWv4) dataset (Balk *et al.* 2005). The initial condition uncertainty is not taken into account since it plays a negligible role in long-term climate change investigations using VECTRI (Tompkins & Thomson 2018).

The VECTRI run is carried out at a resolution of 0.25° using the historical temperature and rainfall data of CMIP6 models for a period of 1980–2014 (35 years). To estimate the effect of extreme climate change, the VECTRI run is carried out with the help of future projection data of Shared Socioeconomic Pathway 8.5 (SSP5-8.5) for the period 2015–2049 (35 years). The malaria transmission intensity in India is estimated in terms of the VECTRI simulated number of clinical cases (‘Cases’; Tompkins & Ermert 2013). The model output for the historical Entomological Inoculation Rate data and Cases data is validated against the observed malaria transmission data as shown in the study by Chaturvedi & Dwivedi (2020, 2021). It is found that VECTRI realistically simulates the observed spatio-temporal distribution of malaria transmission intensity in India (not shown for brevity). The studies have shown that the malaria transmission intensity in India remains significant only from June to November (Chaturvedi & Dwivedi 2020, 2021). In other words, the malaria transmission intensity in India remains negligibly small during December to March. A time series of VECTRI simulated cases is created by taking the average of area in India [8N-36N; 68E-95E] for the months June to November. The time series of Cases created this way contains a total of 6,405 data points corresponding to the historical and SSP5-8.5 scenario of each CMIP6 model run. The multi-model mean (MMM) time series of Cases is also created by taking the mean of Cases corresponding to 5 CMIP6 models for historical and SSP5-8.5 runs.

### Multifractal analysis using MFDFA

The multifractal analysis of a time series helps us to determine the space-time variations in scale-invariant structure of that time series. This is in contrast to the monofractal analysis which assumes that scale invariance is independent of space and time (Ihlen 2012). The multifractal (e.g., MFDFA) analysis of VECTRI simulated Cases time series is carried out corresponding to the historical and SSP5-8.5 datasets of each CMIP6 model. The multifractal analysis is also carried out for the MMM time series.

*N*, the profile (or the cumulative sum of departure of from its mean value) is constructed as for

*i*= 1, … ,

*N*. These profiles are divided into

*N*= int(

_{s}*N*/

*s*) non-overlapping segments of equal length

*s*. Here,

*N*may not be an exact multiple of

*s*, and as such some part of the profile may remain unaccounted for. To include this part, the same procedure is repeated from the end of the profile and returning to the beginning, thereby creating 2

*N*segments altogether. The variance or root-mean-square error within each of these

_{s}*v*= 1, … ,2

*N*segments is then computed as follows:

_{s}Equation (2) is obtained by taking the mean over the *q*-order variances . Here, *q* can take any real value other than 0. In the MFDFA analysis, the variation of with scale *s* is examined for different values of *q*. The follows power-law behaviour with *s* as follows: , where is the generalized Hurst exponent. In general, *h*(*q*) is obtained using the log-log plot of versus *s* for different *q* values. In other words, the exponent is obtained as the slope of the linear portion of plot between log and log *s*. For a stationary time series, the value of at *q* = 2 is identical to the classical Hurst exponent *H*, whereas for a non-stationary time series (e.g., representing Brownian motion), *h*(*q* = 2) satisfies *h*(2) = *H* + 1 (Feder 1988; Movahed & Hermanis 2008). The exponent *H* = 0.5 for a time series suggests that the datasets under consideration are uncorrelated and random. The value of *H* in the range of 0.5 < *H* < 1 indicates long-term persistence, i.e., long-range correlation, whereas 0 < *H* < 0.5 suggests anti-persistence, i.e., short-range correlation between time series values (Carbone *et al.* 2004; Zhang *et al.* 2019). For a monofractal time series, remains independent of *q* and is the same as the constant Hurst exponent *H* (Kantelhardt *et al.* 2002). On the other hand, if varies with *q*, then the time series data indicate the presence of multifractality. For positive *q*, describes the scaling behaviour of segments with large fluctuations (significant multifractality), whereas, for negative *q*, denotes scaling behaviour of segments with small fluctuations (weak multifractality) (Tanna & Pathak 2014; Zhang *et al.* 2019).

If mass exponent (also referred to as multifractal scaling exponent) varies linearly with *q*, then the time series data exhibits monofractal nature. However, for a multifractal time series, varies nonlinearly with *q*. The stronger the nonlinear relationship between and *q*, the greater will be the strength of multifractality.

*q*(e.g., slope of mass exponent curve). The smaller (larger) value of suggests the higher (lower) degree of singularity. The value of

*α*= 1 indicates the uniform distribution of time series (Zhang

*et al.*2019). The singularity dimension represents the dimension of time series segments corresponding to singularity strength . It is related to mass exponent via a Legendre transform (Yadav

*et al.*2012):

The variation of with results in a multifractal spectrum (also known as the singularity spectrum). The multifractal spectrum width characterizes the strength of multifractality in time series data. It is computed as , where and are the maximum and minimum values, respectively, of in a multifractal spectrum. The corresponding difference between maximum and minimum values of the singularity dimension (i.e., at and , respectively) is denoted as . The value of describes the proportion of the number of elements at the minimum and maximum values in the segments, indicating the proportion of the small and large peak values of the fluctuation time series data (Zhang *et al.* 2019). With an increase in the , the length of the range of fractal exponents in a time series becomes higher, thus resulting in more developed multifractality (Krzyszczak *et al.* 2019). The wider the range of possible fractal exponents, the richer the process in structure (Telesca *et al.* 2005). The higher the multifractal spectrum width, the stronger will be the multifractality (i.e., the time series will have a ‘fine’ structure). The strength of multifractality reduces with a decrease in multifractal spectrum width. For example, for a monofractal time series, the multifractal spectrum reduces to a point with (Philippopoulos *et al.* 2019).

## RESULTS

*F*(

_{q}*s*) with segment size (i.e., scale

*s*) is represented on a log-log scale in Figure 3 for the malaria cases time series of the historical period of all the CMIP6 models as well as its MMM time series. The power-law behaviour is seen in the plots for

*q*-order statistical moments ranging between

*q*= −5 to 5 at an interval of 0.1. A similar plot is obtained for the case's time series of SSP5-8.5 scenario (not shown for brevity). The value of

*q*is chosen in the range of −5 to 5 to avoid divergence of moments in the fat tails of the fluctuation distribution by the freezing phenomenon (Lashermes

*et al.*2004; Kantelhardt

*et al.*2006; Ihlen 2012; Wang

*et al.*2013). The

*q*-orders should contain both positive and negative values to properly weigh the periods with large and small variations in a time series. In this context, largely negative and positive

*q*values are avoided because they inflict larger numerical errors in the tails of the multifractal spectrum (Ihlen 2012). The

*F*(

_{q}*s*) values get closer to each other with an increase in scale

*s*. In other words, the

*F*(

_{q}*s*) differences for different

*q*(between −5 to 5) become indistinguishable from each other with an increase in segment size (

*s*). The local periods with large and small fluctuations (e.g., positive and negative

*q*) are clearly distinguishable for small segment size

*s*. In contrast to this, since the large segments cross several local periods with both small and large fluctuations, the average of their differences in magnitude is obtained (Ihlen 2012). For large segment sizes, the

*F*(

_{q}*s*) values of the multifractal time series are similar to that of the monofractal time series.

*F*(

_{q}*s*) and

*s*on a log-log scale helps us to determine the

*q*-order generalized Hurst exponent . The slope of the linear (i.e., straight line) portion of variation of

*F*(

_{q}*s*) vs.

*s*on a log-log scale gives the value of corresponding to each

*q*for the malaria cases time series of each CMIP6 model and its MMM. For a multifractal time series, the varies nonlinearly with

*q*. On the other hand, for a monofractal time series, the remains constant with

*q*. The as a function of

*q*is shown in Figure 4. It is noticed that for all the CMIP6 models and its MMM, the is a strong nonlinear decreasing function of

*q*(as

*q*varies from −5 to 5). This helps us to establish that the malaria cases time series exhibits multifractality. A higher variability in the of CMIP6 models is noticed, especially for negative values of

*q*(i.e., −5 <

*q*< 0). In Figure 4, we also show the variability of with

*q*for the time series corresponding to the SSP5-8.5 CMIP6 dataset. It is found that the obtained from the SSP5-8.5 dataset shows the same nature as seen in the historical dataset; however, the curve remains lower than the historical dataset for each model as well as its MMM. The result suggests that due to an increase in global warming (e.g., GHG concentration), even though the malaria cases time series will remain multifractal, it is expected that the strength of multifractality of malaria cases in India will become less pronounced.

*q*-order mass exponent with the help of and

*q*. The vs.

*q*curve for the malaria cases time series corresponding to historical data of all the CMIP6 models and its MMM is shown in Figure 5. It is found that is a nonlinear increasing function of

*q*. This suggests that the malaria transmission variability in India shows a multifractal nature. The obtained using the SSP5-8.5 data is also shown in the figure. It is found that the difference between historical and SSP5-8.5 curves is higher for negative

*q*values (of higher magnitude, e.g., −5, −4). The difference between the two curves is the smallest for

*q*= 0. For negative (positive)

*q*, the of malaria cases obtained using SSP5-8.5 data is higher (lower) compared to the historical data.

*q*-order singularity strength is shown in Figure 6. The strongly nonlinear variation of with

*q*further supports the fact that the malaria transmission intensity in India is multifractal in nature. Further, the highest is found in the case of the NorESM2 model, whereas the lowest value of is obtained for the MIROC6. It is also noticed from the curve that the values of decrease sharply for negative

*q*(as

*q*increases from −5 to 0) compared to the positive

*q*(i.e., between

*q*= 0 and 5). For negative (positive) values of

*q*, the variation of

*h*(

*q*) and, therefore, describes the scaling behaviour of the segments with small (large) fluctuations. The levelling tendency shown in these curves for positive

*q*values reflects that the

*q*-order fluctuations are becoming insensitive to the magnitude of local fluctuations with large magnitudes (Ihlen 2012; Tanna & Pathak 2014). This kind of levelling results in a right-skewed (left-truncated) spectrum (shown later). Further, the curve corresponding to SSP5-8.5 dataset (shown in red colour) remains lower than the curve obtained using the historical data (blue colour). This suggests that global warming tends to reduce the multifractal singularity strength of malaria transmission cases.

*q*-order singularity dimension is computed using Equation (4). The multifractal spectrum of a time series is plotted with the help of a curve showing as a function of . The multifractal spectrum of the VECTRI simulated malaria cases time series corresponding to the CMIP6 models used in the study as well as their MMM is shown in Figure 7. The malaria cases time series corresponding to each CMIP6 model exhibits multifractal spectrum with distinct shape and width. The multifractal spectrum displays an asymmetric arc around the maximum value of . It is found that the multifractal spectrum of malaria cases time series is right tailed for all the CMIP6 models (and its MMM) used in the study. In other words, the vs. multifractal spectrum is left truncated (right-skewed), suggesting greater complexity and fine structure of the malaria cases time series (Shimizu

*et al.*2002; Krzyszczak

*et al.*2019; Agbazo

*et al.*2020). The truncation of curves happens due to the levelling of for negative/positive

*q*'s (Ihlen 2012). When the malaria cases time series of a model represents a multifractal spectrum, which becomes insensitive to the local fluctuations with small (large) magnitudes, then the spectrum will have a long left (right) tail. A right-skewed spectrum is related to relatively strongly weighted high fractal exponents, whereas a left-skewed spectrum is indicative of low fractal exponents (a more ‘regular’ time series) (Shimizu

*et al.*2002; Agbazo

*et al.*2020).

The multifractal spectrum is used to estimate singularity strength corresponding to the least and most singular values obtained in the multifractal analysis of the malaria cases time series. This is achieved by obtaining minimum () and maximum () values of in a multifractal spectrum. The multifractal spectrum width is computed by taking the difference of and . The corresponding difference between maximum and minimum values of the singularity dimension (i.e., at and , respectively) is denoted as . With the increase in the , the length of the range of fractal exponents in a time series becomes higher, thus resulting in more developed multifractality (Krzyszczak *et al.* 2019). The values of and are quantified for understanding the strength of the multifractal nature of malaria cases time series used in the study. These values are summarized in Table 1 for the malaria cases time series corresponding to all the CMIP6 models and their MMM. From Table 1 and Figure 7, we notice that, in general, for the models in which the peak value of the occurs for higher , the value of is higher. In other words, with the shifting of the value of towards the right-hand side in a multifractal spectrum, the width of the spectrum and, therefore, the strength of multifractality increase. Shimizu *et al.* (2002) suggested that a time series with high and a right-skewed spectrum is considered to be more ‘complex’ than a time series with the opposite characteristics. In other words, the malaria cases time series with higher and right-tailed multifractal spectrum will exhibit strong complexity and greater multifractality (e.g., with fine structure) (Krzyszczak *et al.* 2019). The wider the range of possible fractal exponents, the ‘richer’ is the process in structure (Telesca *et al.* 2003). The right-skewed spectrum obtained here suggests an unsmooth (uneven) fine structure of malaria cases in India. This indicates the presence of relatively strongly weighted high fractal exponents of malaria cases. The result suggests that the values in the malaria cases time series are less correlated (more complex) due to the irregular and heterogeneous distribution of malaria in India. Table 1 shows that the highest values (1.1706 and 1.1090) of for the malaria cases time series are obtained corresponding to the historical data of NorESM and CESM2 models, whereas the lowest value (0.9434) of is found for the GFDL-CM4 model. The MMM time series shows intermediate value of 1.0053. The values of the right-skewed multifractal spectrum shown in Figure 7 suggest that out of all the models used in the present study, the NorESM and CESM2 models exhibit the maximum strength of multifractality (e.g., the greater width of the spectrum) of malaria cases time series, whereas the minimum strength of multifractality is obtained for the GFDL-CM4 model.

Models . | Historical . | SSP5-8.5 . | ||
---|---|---|---|---|

. | . | . | . | |

CESM2 | 1.1090 | 0.7067 | 0.9763 | 0.5938 |

GFDL-CM4 | 0.9434 | 0.4122 | 0.8440 | 0.2971 |

IPSL-CM6 | 1.0303 | 0.8796 | 1.0429 | 0.5681 |

MIROC6 | 1.0365 | 0.3871 | 0.9673 | 0.3125 |

NorESM2 | 1.1706 | 0.8710 | 1.0776 | 0.9420 |

MMM | 1.0053 | 0.8293 | 0.9930 | 0.7492 |

Models . | Historical . | SSP5-8.5 . | ||
---|---|---|---|---|

. | . | . | . | |

CESM2 | 1.1090 | 0.7067 | 0.9763 | 0.5938 |

GFDL-CM4 | 0.9434 | 0.4122 | 0.8440 | 0.2971 |

IPSL-CM6 | 1.0303 | 0.8796 | 1.0429 | 0.5681 |

MIROC6 | 1.0365 | 0.3871 | 0.9673 | 0.3125 |

NorESM2 | 1.1706 | 0.8710 | 1.0776 | 0.9420 |

MMM | 1.0053 | 0.8293 | 0.9930 | 0.7492 |

The effect of global warming on the multifractal nature of the malaria cases time series in India is also quantified. The results of the multifractal analysis corresponding to the SSP5-8.5 dataset are shown by red curves in Figure 7. It is clear from the figure that the width of the multifractal spectrum decreases with an increase in global warming. In other words, the multifractal spectrum of malaria cases time series corresponding to CMIP6 models as well as their MMM becomes less wide in a warming environment. Thus, the strength of multifractality of the malaria cases time series will decrease with an increase in the GHG concentration. These findings are further substantiated by a similar decrease in values on using the SSP5-8.5 dataset as compared to the historical dataset (Table 1). For example, the MMM decreases from its value of 0.8293 for historical data to 0.7492 for the SSP5-8.5 dataset. To summarize, the strength of multifractality of the malaria cases time series is likely to decrease in the future climate under the effect of global warming. This may be due to the fact that with an increase in the GHG concentration (e.g., under the SSP5-8.5 scenario), a uniform increase in malaria cases is seen in most parts of India (compared to Figure 2). This indicates that the distribution of malaria in India tends to be increasingly homogeneous and regular in a warming environment, leading to a decrease in multifractal width and, therefore, multifractality, as a result of relatively smaller variations in the distribution of high and low fluctuations.

## DISCUSSION

There is no doubt about the fact that as a result of the implementation of useful control and mitigation strategies in the last several decades for eradication of malaria disease from the Indian subcontinent, the number of deaths due to vector-borne malaria disease has considerably reduced in the densely populated Indian states. However, it is still one of the prominent health hazards in India. The distribution of malaria in different regions of the country is highly irregular and non-uniform, which makes it difficult to model its dynamics and propose better prediction strategies for it. Therefore, it becomes important to estimate and understand the scale-invariant multifractal structure of malaria transmission in different regions of India. This article attempts to do this by carrying out a multifractal analysis of long time series of malaria cases in India. Apart from the study by Sequeira *et al.* (2022) who used the fractal (Hurst exponent) and entropy measures to predict effective transmissibility in empirical series of malaria incidence in different parts of Africa, there is no other study in which either fractal or multifractal characterization of malaria cases have been carried out. To the best of our knowledge, this is the first study in which the multifractal characterization of the long-time series of malaria cases in India is carried out for the present and future climate using the high-resolution dynamical malaria model output. The results (e.g., nonlinear variation of generalized Hurst exponent, mass exponent, and singularity strength) clearly demonstrate that malaria distribution in India cannot be described by a single scaling exponent. The malaria cases time series are multifractal in nature, exhibiting fine-scale complex structure of the disease. With an increase in global warming, the distribution of malaria in India becomes less irregular, which leads to a reduction in the strength of multifractality (e.g., less wide multifractal spectrum). The results suggest that malaria distribution is likely to become less complex in future climates under extreme global warming scenarios which will also lead to its increased predictability.

## CONCLUSIONS

For effective control, prediction, and mitigation strategy of malaria transmission in India in a warming environment, it is desirable that its scaling properties are correctly estimated in the different parts of the country. However, since the malaria cases in the country are irregularly and non-uniformly distributed, as such, its scaling exponents (e.g., those obtained from power-law behaviour of fluctuation function) will also be different for different regions of the country. In other words, different regions of the country will have different fractal dimensions, which vary according to the distribution of malaria cases. For this purpose, first, the long time series of malaria cases are obtained from the high-resolution (0.25°) VECTRI dynamical malaria model, and then those time series are analysed using the MFDFA technique for quantifying the multifractal properties of malaria transmission in India in the present and future climate. The high-resolution malaria cases time series are obtained corresponding to five CMIP6 models (CESM2, GFDL-CM4, IPSL-CM6, MIROC6, and NorESM2) and their MMM for the historical and future projection (SSP5-8.5) scenario of the periods 1980–2014 and 2015–2049, respectively. The results show that malaria cases will increase in a high global warming scenario from June to November. The multifractal properties are estimated for the historical time series of malaria cases, and changes in these properties are quantified with an increase in the GHG concentration. It is found that the generalized Hurst exponent and multifractal spectrum width of the malaria cases time series (corresponding to all the CMIP6 models and their MMM in India) are strongly nonlinear decreasing functions of *q*. Moreover, the mass exponent shows a nonlinear increase with *q*. These results confirm the multifractal nature of malaria cases in India. The multifractal spectrum (i.e., versus curve) of the malaria cases time series shows right-tailed nature corresponding to all the CMIP6 models as well as the MMM time series, with inter-model variability. The right-tailed multifractal spectrum indicates the complexity and the presence of fine-scale structure of the malaria cases time series. The strength of the multifractality of the malaria cases time series is quantified for the historical and future projection scenario dataset in terms of the multifractal spectrum width and change in singularity dimension . It is found that these values will become smaller in an extreme global warming scenario. For example, and values obtained using the SSP5-8.5 data become smaller compared to the corresponding values obtained using the historical data. Our results clearly demonstrate that the strength of multifractality of the malaria cases is likely to decrease with an increase in GHG concentration. It is argued that the strength of multifractality of malaria cases will be decreasing in the next 2–3 decades as a result of persistent increase in malaria cases in India in a warming environment (leading to increased homogenization of malaria cases across India). Future research will focus on quantifying multifractal properties of malaria cases under different future projection scenarios of the CMIP6, as the radiative forcing levels are gradually increased from SSP1-1.9 to SSP5-8.5. Such an analysis will help us understand the systematic change in the multifractal properties of malaria cases in India with an increase in the GHG concentration. Moreover, a greater number of CMIP6 models will also be involved. Efforts will also be made to check the robustness of the multifractal nature of the malaria cases in India by employing other multifractal techniques such as the multifractal detrended moving average analysis and wavelet transformation-based analysis. It is hoped that the multifractal analysis carried out in this article will help the malaria modelling community towards the development of improved malaria prediction models.

## ACKNOWLEDGEMENTS

S. D. thanks infrastructural support provided to the KBCAOS under the DST-FIST scheme of DST, GoI. S. C. is thankful to the DST, Govt. of India for providing the INSPIRE Fellowship. The authors are thankful to the CMIP6 modelling community for making their data freely available. These datasets are obtained from https://esgf-node.llnl.gov/search/cmip6/. Thanks are also due to Dr Adrian Tompkins, ICTP, Italy, for providing VECRI related training to S. C. from time to time.

## FUNDING

No funding was received for conducting this study.

## AUTHOR'S CONTRIBUTION

Conceptualization, validation, and formal analysis were done by S. D. S. C. and S. D. contributed in developing methodology, investigation, and original draft writing. S. D. carried out the overall supervision. Both the authors have read and agreed to the present version of the manuscript.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.