Mosquito‐repellent controlled‐release formulations for fighting infectious diseases

Polymer microcapsules are systems that consist of a functional barrier between core material containing the active ingredient, e.g. repellent, as an internal phase, and a wall (natural or synthetic polymer) also known as a shell or a membrane (Fig. 2) [26]. In order to adapt to the several types of core and wall materials as well as to produce different particle sizes, shell thicknesses and permeabilities, hence, adjusting the release rate of active ingredient [27], several methods can be used to prepare polymer microcapsules. The common methods include spray drying [27‐29], interfacial polymerization [30], and coacervation [31‐34].

A nanomeulsion device comprises a boundary between the oil and water phases at which the surfactant is located. Those systems are typically formed only in a specific and narrow range of concentrations for a given surfactant-oil-water structure. The relationship between the phase behaviour of different mixtures and their composition is generally depicted using a pseudo-ternary phase diagram (Fig. 7), where a corner represents a binary mixture of surfactant-co-surfactant, water-drug and oil-repellent. Outside the nanoemulsion region, the amount of surfactant is too low to allow the formation of a single nanoemulsion phase, therefore leading to the existence of multiphase systems [51].

Based on the components, nanoemulsions are divided in: (i) oil-in-water (O-W) (the oil phase is dispersed into a continuous water phase); (ii) water-in-oil (W-O) (the water phase is dispersed in a continuous oil-phase), and, (iii) bi-continuous or multiple emulsions in which micro domains of phases of the oil and water are interdispersed [45].

The nanoemulsion morphology has been obtained by microscopy [52]. As an example, the surface morphology of nanoemulsions stabilized by semi-solid polymer interphases were investigated by transmision electron microscopy (TEM) as is presented in Fig. 8 [48]. The results showed that the type of oil affected the morphology of nanoemulsions. In addition, the optical micrographs of a nanoemulsion confirmed that the resulting droplet size was in the range of a few micrometres (Fig. 9) [48].

Due to problems caused by classic formulations, that is, solutions and lotions, available on the market including irritation and skin dryness when applied on the human skin, the nanoemulsions of repellents are promising systems for reducing mosquito-borne diseases [35]. This may be explained by their intrinsic physicochemical properties, such as, uniform and very small droplet sizes (20–200 nm), low viscosity and optical transparency [53]. Finally, several works have reported the activity of nanoemulsified repellents against mosquitoes, boosting in this way their use in the control of infectious diseases [53]. For example, a study done by Sugumar et al. [54] reported a good performance of eucalyptus oil nanoemulsion against the mosquito Culex quinquefasciatus, where the activity of essential oil nanoemulsions was related to the size of the oily droplets. Nuchuchua et al. [55] also reported that the small oily-droplet size displays an important role in the efficacy of oil nanoemsulsions. Specifically a study conducted by Anjali et al. [50] showed that the small size of the neem seed oil nanoemulsion was much more effective against C. quinquefasciatus when compared to oil nanoemulsions with a medium and large sizes. Sakulku et al. [56] showed the effect of citronella oil nanoemulsion against A. aegypti. The results of protection time was directly related to the release rate of citronella oil from the nanoemulsion as is shown in Fig. 10. Furthermore, the effect of emulsification on the release rate of citronella oil aiming to understand the relationship between droplet size and the release rate was also investigated by Agrawal et al. [57]. Release rate studies of samples with different droplet diameters of 65 nm and 72 nm but the same composition was carried out at 35 ^oC. It was found that the evaporation rate was high for a sample with small droplets compared to large droplets. Similar behaviour was observed by others [55, 56].

Due to the small size and low polydispersity of droplets in nanoemulsions, those have advantages over other repellents-carrier systems [58]. The advantages include: (i) better physical stability against creaming, flocculation, sedimentation, and Ostwald ripening than ordinary emulsions; (ii) efficient permeation ability and enhanced bioavailability because of high surface area/volume ratios and small droplet size, thereby enhancing the transfer of molecules through biological membranes; (iii) improved water solubility, tunable loading capability and enhanced chemical stability; (iv) slow release of bioactive compounds through encapsulation and solubilization, and (v) low dosage of emulsifiers, compared with microemulsions (the dosage of emulsifiers in microemulsions is roughly ≥ 20 % while it is between 2 and 10 % in nanoemulsions) [58‐60].

As previously reported, the nanoemulsions present several advantages due to the small droplets size they contain, high optical clarity, good physical stability and droplet aggregation, and enhanced bioavailability of encapsulated substances, which make them suitable for final applications [59]. Depending on desired formulation, preparation method should be selected to optimize droplet size distribution since it strongly affects stability behaviour. Therefore, systems with droplets diameter smaller than 200 nm and a monomodal distribution usually have a homogeneous structure, that is, a structure with well distributed droplets that do not show flocs [59]. A structure with these characteristics remains unchanged for long time, up to six months, given the nanoemulsion enhanced stability compared to conventional emulsions with the same formulation [59]. Therefore, the development of systems with potential technological application also requires the existence of long-term stability. For example, Lucia et al. [61] evaluated the stability of the emulsions after an aging of 28 months. The results demonstrated that the essential oils based on nanoemulsions before and after aging showed that samples containing eugenol oil as main component did not present any significant change on their homogeneity after 28 months of aging, maintaining their monodisperse character and a constant droplets size. The findings on the stability of essential oils based on nanoemulsion could be considered a promising pre-requisite to the development plant-derived repellents formulations against mosquitos.

The mechanisms of repellent release are diffusion, degradation, and swelling followed by diffusion. Solid lipid nanoparticles may display any or all of these mechanisms of release of repellents. Diffusion can occur across the lipid matrix on a macroscopic scale due to pores therein or by passing through lipid chains on a microscopic level. The repellent diffusion rate through SLNs typically reduces with time since initially repellent diffuses from the surface of particles to release media followed by diffusion from interior layers. Consequently, repellent moves gradually slower and needs an extended diffusion time to release [62]. Wissing et al. [67] reported that the drug release through SLNs systems is influenced by modification of the lipid matrix, surfactant concentration and production parameters.

Repellents based on solid lipid nanoparticles have demonstrated a good potential for insect control. Studies reported various essentials oils into SLNs as repellent-based products against mosquitoes [68, 69]. For example, Iscan et al. [70] reported that DEET-loaded solid lipid nanoparticles controlled the release of repellent and its skin permeation. Other works also showed that the repellent-based solid lipid nanoparticles are suitable carriers for pest control. A study conducted by Lai et al. [71] demonstrated that essential oil incorporated into SLNs controlled the evaporation rate of oil over 48 h at 35 ^oC during storage. Furthermore, the performance of garlic essential oil loaded SLNs against the Red flour beetle Tribolium castaneum maintained effective repellence over five months, because the active ingredient continued to be released slowly through particles [72]. This suggests that it is possible to incorporate garlic essential oil into polyethylene glycol (PEG) coated nanoparticles to control pests that affect the stored products. Finally, those studies showed that SLNs systems show the capacity to reduce quick volatility and degradation of repellents, it also improves stability and keeps the minimum effective amount needed and facilitates their use.

Solid lipid nanoparticles (SLNs) have emerged as an alternative to other novel delivery approaches due to various advantages such as feasibility of incorporation of lipophilic and hydrophilic drugs and improved stability [73]. For example, Lai et al. [71] incorporated Artemisia arborescens essential oil into solid lipid nanoparticles by high-pressure homogenization method. The obtained Artemisia arborescens essential oil-loaded SLNs demonstrated high physical stability at various storage temperatures over 2 months, and they were able to reduce the rapid evaporation of essential oil during in vitro release experiments compared with reference emulsions. Furthermore, Tian et al. [74] evaluated the thermal stability and chemical stability of citral oil-loaded solid lipid nanoparticles. The results demonstrated that the thermal stability of citral improved when it was encapsulated in the citral-SLNs. In terms of chemical stability of citral in the citral-SLNs formulation remainaded stable during 12 days of storage at 37 °C. From the results, it can be said that the citral-SLNs could be used as a carrier to effectively protect citral from degradation in acidic environments.

The repellent loading can be done by chemical conjugation or physical entrapment [75]. In the chemically conjugated repellent, the release rate of repellent occurs by bulk degradation of the polymer matrix or surface erosion [75]. Physically, for the repellent entrapped into polymeric micelle, the repellent release is controlled by diffusion through the core of the polymeric micelle, stability of micelles and the copolymer degradation rate. If the polymeric micelle system is stable and the polymer degradation rate is slow, the repellent difussion rate is determined by the compatiblity between the drug and core forming block copolymer, the quantity of repellent entrapped, the repellent molecular volume, and length of the core forming block polymer [76].

Polymeric micelles can be used as mosquito repellent carrier-systems as well as for guided release due to their high encapsulation capacity [77]. A work done by Barradas et al. [77] reported that DEET placed into micellar formulations based on poly(ethylene oxide)–poly(propylene oxide)–poly(ethylene oxide) triblock copolymer exhibited slow release of DEET during up to seven hours. Therefore, there was a reduction of approximately 35 % in the amount of DEET absorbed through the skin after six hours. Balaji et al. [78] encapsulated an insect repellent, diethylphenylacetamide (DEPA) in polymeric nanomicelles and evaluated their performance against Culex tritaeniorhynchus. The median lethal concentration (48 h) for third instar C. tritaeniorhynchus larvae were found to be 0.416 mg/L for bulk DEPA and 0.052 mg/L for nano DEPA. The core-shell structure of the micelles was confirmed by SEM (Fig. 14a) and TEM (Fig. 14b).

Other studies reported the activity of polymeric micelle-based repellents for insect pest control. For example, Lucia et al. [79] evaluated essential oil components (EOCs) based polymeric micelles system against against Pediculus humanus capitits (Head lice), where the product was effective with a mortality above 60 %, being the most effective system containing linalool, 1,8-cineole, terpineol, thymol, eugenol, and geraniol. Therefore, the study suggested that essential oil-based polymeric micelles are alternative tools for insect control.

Stability studies ensuring the maintenance of product quality, safety and efficacy throughout the shelf life are considered as pre-requisite for the acceptance and approval of any pharmaceutical product. These studies are required to be conducted in a planned way following the guidelines issued by the WHO and or other agencies [87]. For example, Bezerra et al. [88] investigated the thermal stability of the complexes using the thermogravimetric analysis (TGA). The authors found that the citronella oil-ß-cyclodextrin inclusion complex presented higher thermal stability and lower presence of water. This fact has happened due to the encapsulation of citronella oil, which came to occupy the place previously taken by the molecules of water, as described. Finally, with the increase in temperature, the oil is released with the decomposition of the biopolymer that protects it. On the other hand, the thermal stability of the oil was improved.

Several studies have evaluated the of stability of liposomes formulations. Example, Valenti et al. [90] evaluated the stability of the liposome-incorporated Santolina insularis essential oil (EO). The kinetic study of the stability of liposomes encapsulating EOs was monitored in terms of liposome sizes, encapsulation efficiencies and zeta potential measurements. All prepared liposomal formulations showed a very good stability for more than one year, when stored at 4–5 °C. Therefore, the results obtained showed that at least 93–96 % of the incorporated oil was already associated with liposomes after one year. Dynamic light scattering measurements showed that average size and size distribution remained constant for at least one year. Moreover, the liposomes were able to prevent essential oil degradation compared to the neat oil. In summary, the liposome incorporation protected the essential oil from degradation and after one year its composition was still very close to that of the freshly prepared oil. A minimal loss of the more volatile compounds was observed and no identifiable degradation product was present even in traces. Lastly, it had been reported that Eucalyptus camaldulensis EO liposome had minor changes in the size and it remained stable after three (3) months of storage [92]. Additionally, more studies about the stability of liposomes encapsulating EOs were also investigated by authors [93‐96].

The phase behaviour of a linear low density polyethylene (LLDPE)/citronellal system is illustrated in Fig. 17. The system shows an upper critical solution temperature (UCST) displaying the stable single-phase region together with the metastable and unstable regions. The phase diagram indicates that the probability of forming a microporous matrix is high when the polymer is in the minority phase. In polymer-repellent mixtures, the loci of the phase boundaries is described by the Flory-Huggins theory [124]. At temperatures above the UCST, the system is fully miscible for all compositions. Below this temperature, phase separation can occur at a temperature that depends on the concentration of the system components [125]. The composition of the two phases in equilibrium at any temperature are defined by the binodal line. In the metastable region indicated in the phase diagram, the phase separation will occur via a nucleation and growth mechanism [126]. This is the usual scenario for liquid-liquid phase separation [126]. If the polymer represents the minority phase, it may initially lead to the undesirable formation of separate polymer particles that are suspended in the continuous liquid repellent phase.

Inside the two-phase region there is another set of a phase envelope, the spinodal curves. In this region of the phase diagram, a homogeneous mixture is thermodynamically completely unstable. In contrast to the metastable bimodal region, the solution will spontaneously split into two phases through spinodal decomposition, into a polymer-rich phase and a solvent-rich phase. Phase separation through this mechanism leads to a finely dispersed microstructure through diffusion processes that amplify intrinsic thermodynamic spatial composition fluctuations. Ultimately this co-continuous structure may be fixed by either the subsequent crystallization of the polymer, or by vitrification of the polymer-rich phase. This means that the majority liquid phase is trapped inside a solid polymer-rich phase (which still may contain a minor amount of repellent) with a porous structure. In practice, such microporous microstructures are often achieved by rapid quenching of a homogeneous melt in a cold-water bath [14].

In particular the preparation of phase-separated polymer/repellent systems, in which the repellent is entrapped in a porous polymer matrix and slowly released to the environment, is considered as an option to obtain a controlled drug release device/tool. Figure 18 shows the schematic of the cylinder-shaped microporous membrane-strand, which serves as a model of the repellent-release characteristics [14]. The cross-section is assumed to be circular, and the structure of the inner polymer section is assumed to be microporous. Conceptually it corresponds to an open-cell polymer foam, which is initially completely filled with the liquid repellent. As the repellent is gradually released into the atmosphere, it is assumed that the outer pores are progressively emptied and the lost liquid is replaced by air and repellent vapour. Further, it is assumed that the location of the liquid-vapour boundary is concentric with the outer wall [14]. In order for the active compound to be released from the polymer strand, a portion of the liquid evaporates and diffuses through the porous matrix towards the outer membrane. The matrix polymer forms both the microporous structure and the outer membrane. The permeability of the repellent through this membrane is defined by the product of its solubility in the membrane and the diffusion coefficient inside the membrane. The implication is that the active ingredient is also dissolved in the rest of the microporous polymer structure [14].

The geometric features of this model were analysed by SEM, presented in Fig. 19, and shows the microporous structure (Fig. 19a) of LLDPE initially containing 30 wt-% DEET covered by membrane (Fig. 19b) [14]. It is important to note that the microporous polymer structures act as reservoirs to trap large amounts of repellent. Most of the studies [13, 14, 117] used microporous polyolefins as carriers for repellents, because they are widely available and cost effective.

Although polyolefins have a detrimental impact on the environment in the event that used products are discarded or littered after use, polyolefins are the most extensively used group of thermoplastics due to acceptable strength, light weight, low cost, easy processability and good water barrier properties. This would make the total cost of the repellent-based product affordable. For this reason, the possibility was explored to develop repellent-based socks against outdoor mosquito bites [13]. Polyolefins are obtained through polymerization of olefins such as ethylene, propylene, butene, isoprene, and pentene, including their copolymers [127]. A fundamental common characteristic of all polyolefins is a nonpolar, nonporous, low-energy surface that is not receptive to inks and lacquers without special oxidative pre-treatment [128‐130].

Since the first production of polyolefins following the development of Ziegler-Natta catalysts, commercial exploitation has been very rapid because of their attractive characteristics. However, polyolefins are notch sensitive and brittle when exposed to low temperature and high rate of impact [130, 131]. In order to increase the application of polyolefins, fillers are incorporated into polyolefins to increase the stability, heat distortion, stiffness, strength and impact resistance without sacrificing their processability and barrier property [131‐133]. In the case of malaria control, clay was added in some polymer-repellents formulations. Xu et al. [134] reported that intercalated or incomplete exfoliated structures and dispersed tactoids with several layers can effectively enhance the barrier properties of the polymer matrix. The aim of having exfoliated clay present is to control the release rate of the active ingredient, i.e., the volatile repellent, through the polymer membrane-like structure to the surfaces of the open-cell polymer-repellent system (Fig. 20). Therefore, when impermeable nanoparticles are added to a polymer, the permeating molecules are forced to wiggle around them in a random fashion, and consequently diffuse through a tortuous pathway [135].

A number of studies reported the development of intercalated or exfoliated nanocomposite structures with polyolefin/montmorillonite (MMT) [133, 136‐142]. These studies revealed that the incorporation procedure into a polymer matrix is important in order to obtain complete nanoclay dispersion. The first work done by Mapossa et al. [14] reported about the influence of organoclay in the barrier property of polyolefin nanocomposite matrices against volatile mosquito repellents. Results demonstrated that the organoclay known as Dellite 43B added in the polymer lead to the expected lowering of film permeability as well as in reduction of release rate of repellents from microporous strands.

For long-lasting protection against mosquitoes, the concentration of active ingredient in the repellent should have a constant rate of release during a sufficiently prolonged period. Work done by Mapossa et al. [14] showed that the mosquito repellent release from microporous straps can persist for several months at a constant rate. This is demonstrated by the constant slopes of the release curves in Fig. 21, suggesting that they can be developed into cost-effective long-life insect repellent systems. Furthermore, Brade and Davis [143] investigated the release of chemicals from a porous polymer. The authors used methyl nonyl ketone (KNK), dimethyl phthalate (DMP) and DEET as repellents. The porous polymer used was made from polypropylene. The results exhibited a slow release rate of DEET from porous polypropylene powder, being constant for 90 days.

A repellent dissolved in a polymer matrix may be released via a blooming mechanism. A disadvantage of this approach, however, would be an exponential decay of the release rate over time, initially being high but later, when still needed, being insufficient. Further disadvantages of this approach are a limited solubility of the repellent in the polymer device, and swelling of the polymer if the repellent is dissolved in the matrix. The latter means that the polymer will progressively shrink as the active substance is released, affecting the dimensional stability of the product. However, studies showed that the controlled-release technology based on polymer matrices are used largely due their low cost and versatility [144]. The mechanisms involved in slow release require polymers with a range of specific physicochemical properties [145] such as structure, chain length, molecular weight, and solubility [146].

Previous studies by [12‐14, 123] show that microporous polymers have been used as carriers for the controlled release of volatile repellents. For example, a study conducted by Mapossa et al. [14] demonstrated that the microporous polymers namely poly(ethylene-co-vinyl acetate) (EVA) and linear low-density polyethylene (LLDPE) containing repellents DEET and Icaridin, provided a long protection time of up to 12 weeks against An. arabiensis. Chattopadhyay et al. [147] investigated the repellence activity of an essential oil (from the bark of cinnamon, leaves of lemongrass and leaves of eucalyptus)-based ethylcellulose and polyvinylpyrrolidone polymers patch against Aedes albopictus. The product, however, provided only up to three hours protection. In addition, Islam et al. [148] investigated the stability of ethyl-anthranilate-based polymer matrix for prophylaxis against vector-borne diseases.

The results showed that the obtained product was successful and the ethyl anthranilate-based product stayed stable for six months under the conditions studied without significant changes. These studies demonstrated the suitability of different ways of incorporating mosquito repellent into polymer matrices and proved their effectiveness against insects. The polyolefins used for designing different systems not only enhanced the physical-chemical stability but also the safety by entrapping the volatile compounds internally and releasing them at a desired controlled rate [148]. Additionally, previous studies about the efficacy of mosquito repellents, repellent-based controlled release systems and repellent-based products available on the market against mosquitoes are presented in Table 1.

From Table 1, it can be seen that some of the repellent-based products available on the market continue to have problems related to the short time of protection. The studies showed that the time of protection for topical formulations of natural and synthetic repellents against several mosquitoes range from one to 14 h. However, DEET-filled bicomponent fibres knitted into socks provided effectiveness against An. arabiensis for up to 20 weeks [13]. Furthermore, DEET remains the most studied, efficient and effective mosquito repellent. Due to some cases of toxicity related to the use of DEET reported in literature [149, 150], both IR3535 and Icaridin may be considered as alternative mosquito repellents. This is confirmed with the results obtained by Mapossa et al. [14] that demonstrated that polymer (LLDPE and EVA) strands with 20 % and 30 % of Icaridin provided a good repellency activity against An. arabiensis for up 12 weeks.

A study conducted in our research group evaluated the thermal-oxidative stability of repellents at elevated temperatures using oven ageing [151]. The work aimed to determine whether the repellents could withstand short-time exposure to such high temperatures because during extrusion of the microporous strands, the mosquito repellents were exposed to typical polymer processing temperatures, i.e. >180 ^∘C. Furthermore, the stability of relepellents during four months aged at 50 ^∘C was investigated. Various repellents such as ethyl anthranilate, DEET, Icaridin, IR3535, dimethyl phthalate, decanoic acid and Citrodiol^®. The results concluded that the repellents investigated were able to withstand typical polymer processing temperatures for short periods. That they also stayed essentially intact for several months at 50 ^∘C suggesting that they may retain repellent activity for comparable lengths of time. In a study done by Sibanda et al. [13] showed that DEET is physically and chemically stable at melt fibre spinning conditions. Furthermore, a work done by Annandarajah et al. [152] demonstrated that pyrethrum and DEET were sufficiently thermally stable to be extrusion compounded with PLA.

This occurs through friction with clothing and other objects. This can also occur through other physical activities, which allows the repellent to be lost [153, 154, 157].

The evaporation rate can be considered as one of the crucial physical properties of repellents which may affect their efficiency against mosquito bites, and this can be an indicator of the release rate of the repellent from polymer systems (i.e. microporous polymers matrices). This depends on the vapour pressure at ambient temperature and is associated to the boiling points of the repellents. Compounds that have a lower boiling point may allow better vapour repellence, but they may dispel faster. Compounds with higher boiling points have a low vapour pressure and would be ineffective in repelling at a pre-defined distance. This may allow mosquitoes to land but not bite. Generally, most repellents are effective up to a distance of about 4 cm of the skin [153, 154]. Mapossa et al. [151] evaluated the volatility of the different repellents using payne cups and thermogravimetric analysis. This study concluded that, if the repellent has a low volatility, it may provide effective protection. The repellents such as Icaridin and IR3535 were found as promising candidates for long time of protection against mosquito bites. Figure 22 shows the volatility of repellents conducted in convection oven at 50 °C. The repellents volatility was determined using the Eq. 1 that describes diffusion-controlled evaporation through a stagnant gas [158].

where \({{d{m_A}} {\left/ {\vphantom {{d{m_A}} {dt}}} \right. \kern-0pt} {dt}}\) represents TGA-measured rate of weight loss in (kg⋅s⁻¹); \(P_{A}^{{sat}}\) represents vapour pressure in (Pa) of sample at absolute temperature T in (K); R indicates the gas constant (8.3145 J⋅mol⁻¹ K⁻¹); D_A shows confficient of diffusion in (m² s⁻¹) of the repellent in air; M_A indicates the molar mass in (g⋅mol⁻¹) of the repellent, A corresponds the cross-sectional surface area in (m²) of the cup, and z is the depth in (m) of the gas filled part of the sample cup.

Previous studies considered the link between the evaporation rate of repellents in relation to the protection period achieved against mosquitoes [154, 156, 159]. These results demonstrated that the time of protection was inversely proportional to the rate of evaporation of repellent.

This goes hand-in-hand with evaporation and concerns the effect of temperature on the evaporative loss of the repellent [153]. Khan et al. [160] investigated the influence of temperature on protection time of DEET and other repellents. They found that the protection time was halved with every 10 °C rise above ambient temperature, and that more repetitive application of the repellent was needed at temperatures over 26 °C. Further, a study conducted by Chung et al. [161] showed that the temperature affected the release rate of oil from microcapsules. The authors set the storage temperatures at 4, 25, and 60 °C and conducted the release property measurement of microcapsules. From the results they observed that the amount of released oil at 25 °C was slightly higher than that at 4 ^oC. However, the release rate of oil was fast at 60 °C. At 60 °C, the mobility of molecules of the thyme oil became more active and thus volatile thyme oil rapidly diffused to the outside from the inside of microcapsules [161]. Similar behaviour was observed by Mapossa [162], when evaluating the release rate of DEET and Icaridin from LLDPE strands at 30 and 50 °C. As expected, the repellents DEET and Icaridin were released at a higher rate at a higher temperature of 50 than 30 °C [162]. The results obtained by the authors suggested that at an elevated temperature, this parameter affects evaporation of repellent as well as the protection duration of active ingredient. Additionally, other secondary factors such as wind velocity, loss from water wash-off and sweating also affected protection time.

The Higuchi equation is often used to analyse experimental release profiles, which may lead to erroneous conclusions about release mechanisms. The combination of different phenomena such as swelling, glassy/rubbery transitions, dissolution, and concentration-dependent diffusion may result in release data that display a square root dependence upon time [164]. Then, the release rate is given by Eq. 2.where, M_t and M_∞ are the cumulative amounts of drug released at time t and infinite time, K is the Higuchi constant. Usually, this model describes repellent release as a diffusion process based on Fick’s law when the correlation coefficients (R²) is nearly 1.0. As an example, Fig. 23 represents the plot of slow release rate of citronella oil predicted using the Higuchi equation [29].

The release of volatile actives from swellable polymers belongs to the category of diffusion problems known as moving-boundary or Stefan-Neumann problems [165]. Closed-form solutions are not generally available and it has become common practice to employ semi-empirical models to fit experimental data. The most widely applied models are due to Peppas and co-workers [165‐168]. They realized that the behaviour of such systems are determined by two competing release mechanisms, i.e. Fickian diffusion and a Case II relaxational process. On this basis they developed two important semi-empirical release models on heuristic grounds [167]. They observed that, regardless of the geometric shape of the release device, the first 60 % of a release curve is adequately described by the so-called Korsmeyer-Peppas power law model represented by Eq. 3 [168].

From Eq. 4, it can be represented as: where M_t/M_∞ is a fraction of core material released as the function of time t, k corresponds to the release rate constant and n is the diffusional exponent for active release. The diffusion mechanisms are represented as n = 0.5 indicating Fickian diffusion (Case I), n = 1.0 indicates relaxational transport (Case II), 0.5 < n < 1.0 corresponds non-Fickian diffusion (anomalous diffusion), and n > 1 corresponds Super Case II [169, 170]. The release rate of encapsulated repellent N,N-diethyl-3-methylbenzamide (DEET) is shown in Fig. 24. The Korsmeyer-Peppas model was used for data prediction.

The Weibull model provides another globally valid expression for modelling release rates represented by Eq. 5:

This expression is highly flexible and capable of correlating the behaviour of complex diffusion systems [171] including diffusion in fractal and disordered substrates [172]. The dimensionless exponent n is a shape parameter that determines the nature of the release curve. Interestingly, Weibull’s model also constitutes an extension of the Korsmeyer-Peppas model. This is because the exponent n is linearly associated to the exponent n of the power law derived from the analysis of the first 60 % of a curve of release [172]. Furthermore, the value that the exponent n indicates the transport mechanism of the active from the matrices of the polymer [172]. Values of n ≤ 0.75 represents Fickian diffusion mechanism [172]. Values in the range of 0.75 < n < 1 represents Fickian diffusion and Case II transport mechanisms. When n = 1, Weibull’s model reduces to classical first-order kinetics [173]. Complex release mechanisms are indicated when the n value exceeds unity [174, 175]. Figure 25 shows an experimental and predicted release rate profile of DEET from MICs by using Weibull’s model. The experimental and prection data are in close agreement, with R² equal 0.9914.

This model is defined by Eq. (non-polar, can be modelled with) that assumes quasi-steady state diffusion and is based on the assumptions of a dimensionally stable and inert solid scaffold. This means that it will break down if the polymer absorbs and swells in the presence of the repellent. In this case, polyethylene being non-polar, can be modelled with Eq. (6) [14, 176]: where X = m(t)/m(t→∞) represents the repellent mass (m) released after elapsed time (t) normalized with respect to the maximum that can be released; the characteristic time τ and the shape parameter β both depend on geometric features and the physical properties of the polymer device and mosquito repellent. Mapossa et al. [151] evaluated the release rate of different repellents, such as DEET, IR3535, Icaridin and Ethyl anthranilate from LLDPE strands covered by a thin mebrane barrier. The strands obtained from LLDPE approximated to this situation. The time dependent release from such strands followed the predictions of a simple implicit mechanistic model represented by Eq. 6. Therefore, the predicted data of release rate of repellents obtained by this equation was fitted well with the experimental data as is shown in Fig. 26.

Therefore, kinetic modelling of controlled release systems is necessary to predict the volatile compound release and protection time against insects. If the model is consistent, the behaviour of different combinations of active ingredients of compound and polymeric materials can be simulated at a reduced cost to achieve the desired performance [144]. Mathematical modelling of controlled release systems reduces the time and resources necessary for experimental work in product and process development (Additional file 1: Figure 1).