Collateral Effects of Insecticide-Treated Nets on Human and Environmental Safety in an Epidemiological Model for Malaria with Human Risk Perception

Malaria remains a major health problem in many parts of the world, including Sub-Saharan Africa. Insecticide-treated nets, in combination with other control measures, have been effective in reducing malaria incidence over the past two decades. Nevertheless, there are concerns about improper handling and misuse of nets, producing possible health effects from intoxication and collateral environmental damage. The latter is caused, for instance, from artisanal fishing. We formulate a model of impulsive differential equations to describe the interplay between malaria dynamics, human intoxication, and ecosystem damage; affected by human awareness to these risks and levels of net usage. Our results show that an increase in mosquito net coverage reduces malaria prevalence and increases human intoxications. In addition, a high net coverage significantly reduces the risk perception to disease, naturally increases the awareness for intoxications from net handling, and scarcely increases the risk perception to collateral damage from net fishing. According to our model, campaigns aiming at reducing disease prevalence or intoxications are much more successful than those creating awareness to ecosystem damage. Furthermore, we can observe from our results that introducing closed fishing periods reduces environmental damage more significantly than strategies directed towards increasing the risk perception for net fishing.


Introduction
Malaria is a vector-borne disease caused by intracellular parasites of the genus Plasmodium. Five species are known to infect humans: P. falciparum; P. vivax; P. ovale; P. malariae; and P. knowlesi [1,2]. The disease is mainly transmitted into the human blood by the bites of female Anopheles mosquitoes, who feed for survival and the production of eggs; thereby, the infective sporozoites from their salivary glands are passed to the human circulation system and multiply inside its red blood cells [1,2]. Other mechanisms of malaria transmission are via blood transfusion and congenitally [3,4]. Children under 5 years of age are most vulnerable to malaria, as is also the case for other vector-borne diseases [1,5]. Additionally, malaria remains a major cause of perinatal mortality, maternal anemia, and low birth weight [3,6].
Malaria is a major health problem, mainly affecting tropical and subtropical counties, causing 241 million cases worldwide in 2020 and 627,000 deaths [1,7]. This disease is endemic in 85 countries and the WHO African region accounted for 95% of the global cases in 2020 [8]. Despite a significant effort made by the WHO and other international agencies in the fight against malaria, its control still remains a public health challenge [8]. malaria cases, in combination with other mitigation strategies [1,7,12,14,41]. This result has been supported and analyzed by mathematical models [35,[42][43][44][45][46].
Malaria dynamics and the implementation of mitigation measures occur at different timescales, which may be difficult to implement through classical models of ordinary differential equations. Impulsive differential equations, through their hybrid systems, are capable of modeling these dynamics with greater precision [47][48][49]. There are some studies that have used these models to describe malaria dynamics, for instance, to express impulsive releases of sterile mosquitoes [49] or to describe a pulse strategy to control mosquitoes through insecticide spraying [48]. Other articles have included pulse effects regarding intoxications and their effects on people's health [25,50]. The novelty of our work is to model, through this type of equations, the dynamics of malaria, pulse human intoxication, and pulse environmental effects due to the use and misuse of ITNs. Additionally, these dynamics are affected by different levels of risk perceptions, which determine the awareness of humans towards malaria disease, intoxication from ITNs, and environmental damage.
The structure of this paper is as follows. In Section 2, we present and explain the mathematical model of impulsive differential equations. In Section 3, we generate numerical simulations to observe the trajectories corresponding to malaria infection, human ITN intoxication, and ecosystem damage due to misuse of ITNs as fishing gear, and how these are impacted through risk perception and ITN coverage levels. Finally, in Section 4, we discuss our results and propose future directions.

Materials and Methods
We present a mathematical model of impulsive differential equations that include malaria dynamics in humans and mosquitoes, two types of intoxication in humans produced by ITNs exposure, and two states representing the environment and its damage due to misuse of ITNs. Figure 1 depicts malaria dynamics in humans, dividing the total human population (N h ) into susceptible (S h ), exposed (E h ), infectious (I h ), and recovered (R h ). The superscript of each of these states (see Figure 1) represents the level of toxicity in humans: no superscript representing no toxicity, p representing acute intoxication, and w representing possible health effects after prolonged exposure (chronic intoxication). Malaria is transmitted to susceptible humans by infected mosquitoes, I r v and I v , insecticide-resistant and non-resistant mosquitoes, respectively, in continuous time (solid lines) according to a force of transmission β h,a (t) where the subscript h represents transmission from mosquitoes to humans and a, with a ∈ {1, 2, 3, 4} indicating four transmission levels that may differ annually due to seasonal mosquito presence and may be location-dependent. The transmission rate is defined as where β * h,a is the natural transmission rate, η is the ITN coverage, and K d (t) is a variable representing risk perception of humans to malaria disease, such that an increase in risk perception from a natural risk perception to malaria, K * d , results in a reduction of the transmission rate. The dynamics of this risk variable are determined by the following differential equation, as in [25,[51][52][53]: where λ d 1 represents resistance to change and λ d 2 represents the velocity of reaction to the presence of disease. We observe that individuals increase their risk perception to malaria when there is an increase in the number of infectious individuals. Humans transition from exposed to infectious at a rate δ h , recover at a rate γ h , and lose immunity at a rate ω h . Insecticide intoxication occurs in discrete time (segmented lines), considering that exposure to the toxic substance happens during handling or through direct contact with ITNs. When these events happen, a proportion µ K η := µ(K * p /K p (t))η of individuals become acutely intoxicated (superscript p), where K p (t) is a dynamic variable representing peoples' risk perception regarding health effects from exposure to insecticides against mosquitoes, in bed-nets. The higher the risk perception is, the less intoxication occurs. The dynamics of this risk perception variable are given by the following differential equation: which is similar to Equation (2) and where K * p represents the natural risk perception to health damage due to insecticide exposure in the population. We observe that individuals increase their risk perception when there is presence of intoxicated individuals. Acutely intoxicated individuals can present, over time, chronic intoxication with possible health effects, and hence can transition from the X p h states to the X w h states at a rate g, with X ∈ {S, E, I, R}. Detoxification from both intoxication states occurs at rates f and φ, respectively.
Schematics showing the flow between malaria disease states: Susceptible (S x h ), exposed (E x h ), infectious (I x h ), and recovered (R x h ); the flow between intoxication levels due to insecticide exposure, where the superscript x ∈ {{}, p, w} represents poison states in humans: no toxicity, acute intoxication, and toxicity with possible health effects (chronic intoxication), respectively. The dashed lines represent discrete moments in time when individuals are exposed to the toxicity, and the solid lines represent continuous rates. A description of the variables and parameters used can be found in Tables 1 and 2, respectively.
All individuals are born disease-free and intoxication-free at a rate Λ h . Mortality occurs at a rate d h in all states, and there is malaria-disease-induced mortality for infectious individuals occurring at a rate d h . Note that we did not include the mortality rates in Figure 1 in order to preserve the visual clarity of the schematics. Figure 2 shows the malaria dynamics of mosquitoes, dividing their population into susceptible, S r v and S v , and infectious, I r v and I v , representing insecticide-resistant and non-resistant mosquitoes, respectively. Mosquitoes become infected by infectious humans according to a force of transmission β v,a (t)( , with β * v,a being the natural transmission rate from humans to mosquitoes, K d (t) the risk perception to malaria of humans as in Equation (2), and a ∈ {1, 2, 3, 4} representing four transmission rates that may be seasonally and location-dependent, as mentioned before. Mosquitoes may die at a rate d v . We omitted incubation time, loss of immunity, and disease-induced death for mosquitoes due to their short lifespan. One of the novelties of our model is that it includes a pulse, modeling the exposure of the vectors to the ITNs (dash-dotted line), assuming two possibilities: (i) mosquitoes can acquire some level of resistance to the toxic substance, transitioning from the Y v states to the Y r v states, Y ∈ {S, I}, such that a fraction ση does so; (ii) the insecticide satisfies its purpose, increasing the mortality rate of mosquitoes by a proportion θ. Mosquitoes are recruited into the susceptible class S v without being insecticide-resistant, at a rate Λ v . Repurposing and misuse of ITNs, for instance, for fishing, is of increasing concern since it can harm aquatic ecosystems due to the capture of species that play an important ecological role, which may lead to overfishing and the loss of ecosystem functioning. Here, we call "collateral damage" or "environmental damage" the detrimental effect of ITN fishing on the aquatic ecosystem, leading to an increase in the mortality rate of its species. We model that damage considering two states for the aquatic ecosystem: susceptible to damage (S c ) and damaged (D c ), such that the damaged ecosystem has a higher mortality rate (d c > d c ). Figure 3 shows the schematics of the collateral damage flow. The moments (pulses) when a proportion Ω K η of susceptible species in the ecosystem are damaged correspond to instants when ITNs are being used for fishing (dashed-dotted-dotted line).
where Ω is a constant and K c (t) is a dynamic variable that represents the risk perception of humans to the collateral damage produced by ITN fishing, such that the higher the risk perception, the less damage that occurs. The dynamics of this risk variable are given by the following differential equation: where K * c represents the natural risk perception to collateral damage of individuals and N c := S c + D c . From Equation (4) we can observe that individuals increase their risk perception to collateral damage as a reaction to the mortality of the damaged ecosystem. Damaged species may recover at a rate ψ, and we assume a constant nondamaged recruitment into the ecosystem at a rate Λ c .  Tables 1 and 2, respectively. Figures 1-3 is a model of coupled impulsive differential equations given in system (5). t i and t j correspond to instants at which humans and mosquitoes are exposed to the toxicity emitted by the mosquito net, respectively. t n corresponds to instants at which humans create damage to aquatic ecosystems due to the misuse of ITNs. At those time points, the malaria transmission rates β ∈ {β h,a , β v,a } increase their value by a factor α, since using ITNs for fishing entails not using them for malaria protection. A description of the variables and parameters used can be found in Tables 1 and 2, respectively.

Simulations
In this section, we first present, in Figure 4, the general dynamics of the model; second, in Figure 5, we analyze the effect of ITNs coverage (η) on disease dynamics and on chronic intoxication, and in Figure 6 how different levels of ITN coverage affect the risk perception of humans to disease, intoxication, and collateral damage; third, in Figure 7, we show how increasing risk perceptions affect disease dynamics, intoxication, and collateral damage over time; finally, in Figure 8, we study the effect of fishing restrictions on collateral damage to aquatic ecosystems. The general dynamics of the different compartments of the model are presented in Figure 4. Figure 4a shows the behavior over time of the epidemiological classes of malaria in humans, reaching an endemicity level. Figure 4b depicts the proportion of overall acute intoxication cases in humans at any particular time (blue) and the cumulative number of cases with chronic intoxication that may lead to possible health effects after exposure to ITNs (green). We assume that acute intoxications occur in discrete time (∆t i = 5 days); this situation is observed in the figure when looking at the respective jumps, while, on the contrary, the cumulative chronic intoxications are represented by a continuous curve. Figure 4c shows the general dynamics of the epidemiological classes of malaria for mosquitoes. We observe jumps, which explicitly represent the exposure of mosquitoes to the toxicity of ITNs in discrete time (∆t j = 5 days). It should also be noted that over time, the number of vectors that acquire resistance to the toxin increases (green and blue curves). Figure 4d shows the possible collateral damage produced by ITN fishing over time.
In particular, the damaged ecosystem, D c , is represented by the two blue dotted curves, picturing two damage levels that alternate at different instants, simultaneously altering the susceptible ecosystem level, S c , represented by the black curves. More specifically, the instants at which ITN fishing happens are visualized through the jumps in the diagram (∆t n = 5 days, pulses happening between the pulses at t i and t j ), such that at every moment when fishing occurs, the level of damage, D c , of the ecosystem increases (higher pulses in the blue curve), while S c decreases (lower pulses in the black curve). Figure 4e shows the curves of risk perception towards damage to health due to the toxic exposure to ITNs (K p , black), to disease (K d , blue), and to collateral damage due to the misuse of ITNs for fishing (K c , red), with the risk perception to disease being the highest among the three, followed by the risk perception to health effects due to ITN exposure. Figure 5 depicts different levels of ITN coverage (η) in malaria-endemic regions, affecting disease dynamics and intoxication dynamics. More specifically, Figure 5a shows the proportion of infectious humans for levels of coverage between 0% (η = 0) and 100% (η = 1), which decreases with increasing coverage; while Figure 5b depicts the cumulative intoxication cases over time for the same ITN coverage levels, which increase with increasing coverage. Figure 6 shows how ITN coverage levels between 0% (η = 0) and 100% (η = 1) affect risk perception of humans to (a) toxicity from ITN use, (b) malaria disease, and (c) collateral damage due to misuse of ITNs. Specifically, Figure 6a shows a directly proportional behavior between ITN coverage and level of risk perception to ITN exposure, K p , i.e., the risk perception increases with increasing coverage; the same can be observed in Figure 6c for the risk perception to collateral damage, K c . On the contrary, the risk perception towards malaria disease, K d , behaves indirectly proportional to ITN coverage, as pictured in Figure 6b. We observe that η = 0 represents the case of no ITNs availability in the population, and as soon as η > 0, the pulses in the model are activated, which indirectly has a pulse effect on the risk perceptions.     Figure 7 shows the effect of different risk perception levels on (a) the cumulative cases of chronic intoxications after exposure to ITNs, (b) the proportion of malaria-infected individuals, and on (c) the collateral damage (D c ) on the ecosystem from misuse of ITNs as fishing gear. The risk perception was changed, making a decimal variation to the resistance to change parameter (λ 1 ) and the parameter (λ 2 ) that represents the reaction speed of people against the threat in each risk case. It is observed that after decreasing the resistance to change (λ p 1 ) and increasing the reaction speed (λ p 2 ), there is a significant decrease in the cumulative cases of chronic intoxications after exposure to ITNs (see Figure 7a). A significant decrease can also be observed regarding the number of malaria-infectious humans when applying the same decimal variation in the behavioral parameters of the risk perception towards malaria (λ d 1 and λ d 2 ) (see Figure 7b). Finally, when applying the variation to the λs of the risk perception to collateral damage (λ c 1 and λ c 2 ), a less pronounced reduction of collateral damage can be observed (see Figure 7c). Specifically, in Figure 7c, it can be seen that after the decimal variation in the behavioral parameters regarding ecosystem damage, the respective baselines-which is when ITNs are not used for fishing-coincide for the λ c 1 and λ c 2 values used (lower dotted line), representing low levels of ecosystem damage. On the contrary, when fishing with ITNs, pulses are activated (jumps from the baseline, every ∆t n = 5 days between t i and t j ), and high levels of damage are observed (upper dotted lines). However, there is no significant difference in the collateral damage caused by ITN fishing under different levels of risk perception, since the difference between the upper lines is small.  Finally, Figure 8 shows the effect of imposing fishing restrictions, in particular limiting ITN fishing. We observe that increasing the duration of closed fishing periods, of length ∆t n , produces, on average, lower levels of damage to the aquatic ecosystem (average between lower lines and upper lines for each color), and also a spacing out of the dashes of the higher curves, representing time intervals with large collateral damage in each case. Hence, the longer the closed fishing periods, the lower the average collateral damage level and the shorter the timeframes producing large damage to the ecosystem.

Discussion
Malaria is a vector-borne disease transmitted by Anopheles mosquitoes and caused by parasites of the genus Plasmodium. It represents a major health problem, with Sub-Saharan Africa being most seriously affected. Insecticide-treated nets (ITNs) were massdelivered to endemic countries and have without doubt-in combination with insecticide residual spraying-proven instrumental in the successful reduction of malaria incidence over the past two decades [1]. ITNs, if properly handled, are, in general, safe to use, and potential health risks are likely below an acceptable threshold [19][20][21]. Nevertheless, due to the poisonous nature of insecticides and the lack of more studies, there have been concerns about the potential for acute and chronic intoxication from ITN exposure [27][28][29][30]. Additionally, there have been recent discussions about the damaging effect of ITN fishing on fish stocks and aquatic biodiversity, driven often by the necessity of communities to ensure food security [13][14][15][16][17][18]. We present a compartmental model of impulsive differential equations that studies the dynamics of malaria disease, acute and chronic intoxications in humans due to pulse ITN exposure, and collateral effects on ecosystems due to pulse ITN fishing. Each of these dynamics depends on a dynamic variable representing, respectively, risk perception to disease, toxicity, and collateral damage.
Our results show that, as expected, malaria prevalence decreases with increasing ITN coverage. In particular, an 80% coverage or more produces a significant decrease in malaria prevalence (see Figure 5a) and significantly lowers the risk perception to disease (see Figure 6b). On the contrary, the number of cumulative chronic intoxication cases increases with net coverage, almost doubling the cases with twice the coverage (see Figure 5b), representing a possible health risk from ITN exposure in populations with high ITN coverage and incorrect handling practices of nets. Nevertheless, we observe from Figure 6a that, as coverage of ITNs increases and as time passes, the risk perception towards intoxication naturally increases, slightly fluctuating due to the indirect effect of pulse exposure to toxicity. This result describes a population that, with time, naturally becomes more aware of the importance of proper handling of nets and the dangers of toxicity, while the level of awareness is proportional to the level of ITN coverage. The importance of the level of risk perception towards intoxication can, in fact, be seen in Figure 7a; to exemplify, we can observe from the figure that under a scenario of a net coverage of 50%, a higher risk perception towards intoxication produces a significant decrease in cumulative chronic intoxications in the population. The latter highlights the importance of educational campaigns that teach responsible handling of ITNs (reducing the urge to return to a low risk perception; see λ p 1 in Equation (3)) and informs about potential health effects of intoxications (increasing the reaction velocity to intoxication case count; see λ p 2 in Equation (3)), when ITNs are mass-distributed.
We can also observe from Figure 7b, and at a net coverage of 50%, that disease awareness campaigns-which aim to reduce the urge of individuals to return to a low level of awareness and to increase the reaction velocity to malaria point prevalence (see λ d 1 and λ d 2 in Equation (2))-significantly decrease the point prevalence of malaria. If those campaigns are kept, a sudden increase in malaria cases could be controlled more efficiently, even with a lower ITN coverage. For disease-preparedness purposes, it may be beneficial to maintain disease awareness campaigns, even at a high ITN coverage level of more than 80%, since under such coverage, the risk perception of individuals to disease is naturally reduced significantly (see Figure 6b). The latter is related to the fact that if net coverage increases, the number of those infected with malaria decreases, so the human population "relaxes" (see Figure 5b and Equation (2)).
Regarding the collateral effect produced by ITN fishing on the aquatic ecosystem, we can observe that the collateral damage decreases with time (see Figure 4d). This is due to the decrease in the overall number of species, since the mortality rate is higher in the damaged ecosystem, resulting in fewer species to harm. The increase in the susceptible ecosystem (not damaged) is associated with the fact that the collateral damage is not permanent, and only occurs when ITNs are used for fishing. The risk perception towards collateral damage slightly increases with increasing ITN coverage (see Figure 6c), but to a much lower extent than the risk perceptions towards disease or intoxications (compared with Figure 6a,b). In fact, due to the large jumps in the risk perception to collateral damage observed in Figure 6c), overlapping between cases of coverage, and due to the small scale, there is almost no variability in the average risk perception to collateral damage for different ITN coverage levels. Hence, a mass distribution of ITNs does not have a significant impact on the risk perception of individuals to ecosystem harm, despite its dynamics being dependent on the mortality of the damaged ecosystem (see Equation (4)), which becomes damaged through pulse ITN fishing.
On the other hand, from Figure 7c we can see that a change in risk perception-by, for instance, increasing the awareness of fish mortality through ITN fishing-does not reduce the collateral damage significantly. In particular, we can even observe that the baseline for collateral damage (time intervals where ITN fishing is not performed) coincides for different levels of risk perception to damage. Hence, ecosystem damage awareness campaigns might not have a large effect on the reduction of collateral damage. This is contrary to what we have observed for malaria awareness and intoxication awareness campaigns and their respective effects on disease and intoxication case reduction. However, introducing management measures, such as implementing closed periods for fishing, does further reduce overall collateral damage and may help to ensure a sustainable use of fisheries. We observe especially that the longer the closed periods are, the longer the damage is kept at a low level (see longer dashes on lower lines in Figure 8). We point out that by applying such fishing restrictions, the overall collateral damage would be reduced, even with a high ITN coverage and, hence, a high availability of ITNs that makes the occurrence of ITN fishing more likely. However, the implementation of closed periods has proven to be challenging, since it leads to decreased fish trading activities, declining opportunities for small-scale fisheries, etc., affecting coastal livelihoods [69], and hence need to be analyzed considering socioeconomic impacts. Finally, we observe that strictly overseeing that fishing nets comply with the allowed mesh size-not affected through fishing the reproductive cycle of fish-would have an equivalent effect to increasing the length of closed periods, and hence may reduce ecosystem damage.
The main limitations of our work are the following: (i) there is little information available in the literature regarding the damage to aquatic ecosystems caused by ITN fishing, which is needed to model ecosystem damage more accurately; (ii) more information and studies are required to better understand the potentiality of health risks associated with ITN intoxication levels; (iii) additionally, more studies are needed to include in the model the loss of toxicity and efficacy of mosquito nets after being used for fishing; and (iv) the risk perception variables of our model do not consider explicitly psychological emotional factors, which may affect its dynamics. However, the generality of our model is capable of providing a framework and important qualitative results that in a novel way consider collateral effects of ITNs on human and environmental safety, considering human risk perception; hence, it is a contribution to the literature and may help to generate guidelines for decision-making in public and environmental health.
This work and our model leave a range of possibilities for future interdisciplinary work, such as (i) studying the impact of ITN fishing on the health of species inhabiting an aquatic ecosystem, in particular by analyzing the potential leaking of insecticides [14]; (ii) helping to analyze specific fishing management measures and their associated socioeconomic impacts and biological outcomes [69]; (iii) studying potential health effects in humans from the consumption of poisonous fish due to ITN fishing, and how this may affect human risk perception to intoxication; (iv) including a spatial variable in the model, etc. Finally, it is also worth mentioning that including people's risk perception or behavior within disease modeling is a dynamic aspect to consider in future work. This behavioral aspect alters the dynamics of different diseases [51][52][53]70,71], as well as the effects on health after exposure to pesticides [25], allowing us to study through mathematical models the impact of prevention and mitigation campaigns considering individuals' awareness.

Conclusions
Malaria disease represents a major health problem that has led to the implementation of several control measures, of which insecticide-treated nets (ITNs) have proven to be successful in reducing malaria in the last decades. However, the toxicity of the insecticides in the nets and misusing ITNs, for instance, for fishing, have been of concern for their potential capacity to harm human and ecosystem health. We present a compartmental model of impulsive differential equations to study the joint dynamics of malaria disease, acute and chronic intoxications in humans due to pulse ITN exposure, and collateral effects on ecosystems due to pulse ITN fishing, in combination with human risk perception to disease, intoxication, and ecosystem damage. Our results show that the point prevalence of malaria decreases significantly when increasing ITN coverage. However, this implies an increase in the cumulative number of intoxications, which is significant if the risk perception towards poisoning is low. Specifically, regarding risk perception levels, we can observe that the risk perception towards pesticides increases with increasing net coverage, contrary to the risk perception towards malaria disease, which decreases with increasing net coverage, and the risk perception towards collateral damage, which is much less impacted by coverage. Awareness campaigns on malaria and intoxications have an important impact on the reduction of malaria disease point prevalence and the cumulative number of intoxications. However, the same level of impact cannot be observed through awareness campaigns concerning the damage to the ecosystem due to the misuse of ITNs for fishing, i.e., they may not significantly reduce collateral damage. Therefore, we highlight the effect of restrictive measures taken to monitor the misuse of ITNs, and observe a greater reduction in collateral damage to the ecosystem when measures such as introducing closed fishing periods are implemented.