Modelling the Impacts of Early Intervention on Desert Locust Outbreak Prevention

To preserve crop production losses, monitoring of desert locust attacks is a significant feature of agriculture. In this paper, a mathematical model was formulated and analyzed to protect crops against desert locust attack via early intervention tactics. We consider a triple intervention approach, namely, proaction, reaction, and outbreak prevention. The model integrates a stagestructured locust population, logistics-based crop biomass, and blended early intervention via pesticide spray. We assume that the amount of pesticide spray is proportional to the density of the locust population in the infested area. Conventional short residual pesticides within ultralow volume formulation and equipment control operations are considered. The trivial and locustfree equilibrium of the model is unstable, whereas the interior equilibrium is asymptotically stable. Numerical simulations validate the theoretical results of the model. In the absence of intervention measures, desert locust losses are approximately 71% of expected crop production. The model projection shows that effective proactive early intervention on hopper stage locust contained locust infestation and subdued public health and environmental hazards. Relevant and up-to-date combined early interventions control desert locust aggression and crop production losses.


Introduction
Locusts are divisions of the grasshopper species Acrididae, which includes most short-horned grasshoppers. The desert locust (Schistocerca gregaria Forskål, 1775) is the most devastating migratory member of all locust species [1]. In response to environmental stimuli, compact and extremely mobile adult desert locust swarms can form [2]. The nonflying hopper (or nymph) stage can form cohesive masses that are called hopper bands [3]. They are ravenous eaters who consume their weight per day, targeting food crops and forage. A single adult desert locust can consume two grams, and the hopper stage can consume half of the amount that an adult locust can consume per day. Just a single square kilometer of the swarm can contain up to 80 million adults, with the capacity to consume the same amount of food in one day as 35,000 people. The current desert locust upsurge could have posed an unprecedented consequence, potentially causing large-scale crop damage and threatening food security. The desert locust invasions, recognized for thousands of years, can follow one another at a high frequency if there are no intervening control measures. The recession periods are generally short, whereas the invasions can last for one decade or more [4]. Since the 1960s, desert locust outbreaks are now better controlled and are often shorter in duration and have reduced impact. However, upsurges continue to occur, most often as a result of a reduction in resources during recessions, or due to insecurity in key areas where preventive control should be undertaken. The role of climate change on such outbreaks remains a matter of debate.
Damaging desert locust infestation in decades is at the onset in East Africa since July 2019. The current upsurge of the desert locust is very devastating and reminiscent of situations experienced a very long time ago [5]. Swarms of desert locusts are threatening large areas of pastures and crops, overwhelming countries in the Horn of Africa, the Middle East, and South Asia. The UN Food and Agriculture Organization (FAO) says that these swarms represent the worst infestation in 25 years in Ethiopia and Somalia, 26 years in India, and the worst in 70 years in Kenya. The trauma has affected 23 countries to date, from Pakistan to Tanzania [2,6,7]. The Food Security and Nutrition Working Group (FSNWG) conducted a regional desert locust impact evaluation revealing that nearly 42%-69% of crop production has been lost in the fragile areas [8,9].
Hopper bands and adult swarm stage of the desert locust can cause significant deterioration to vegetation and crops in the infested areas. Food security, industrial raw materials, and export earnings may also be severely threatened in affected areas. Consequently, it is not a surprise that extensive control efforts are mounted whenever hopper bands or swarms of the desert locust emerge in or invade a region. Various strategies are implemented to control the desert locust infestation and conquer crop production losses [10]. Reaction, proaction, and outbreak prevention are three broadly defined desert locust control strategies. However, this has remained a topic of considerable debate, and in recent years, there has been extensive discussion of early intervention strategies [11]. However, the strategy advocated for decades by the FAO of intervening as soon as possible (i.e., proactive or even preventive treatments) seems to have been effective and has largely contributed to containing invasions for over 60 years. Obviously, curative control operations are still essential when outbreaks cannot be stopped at an early stage. Besides, applying standard pesticides sprayed directly onto hopper bands and swarms should be the principal control [12]. Therefore, to prevent catastrophic swarms from maturing from hoppers, it is critical to strengthen ground and aerial surveillance efforts. One means of support is identifying potential breeding sites for timely and effective management of hopper bands. During the hopper stages, proactive ground controls are cost-effective. Once locusts reach the adult stage, aerial control operations will be employed [13]. However, they are very toxic and pose acute risks to human health and the environment [14]. To reduce contingent risks, balancing the control of desert locust populations and the amount of pesticides used is necessary [15,16]. Currently, the use of ultralow volume (ULV) spraying equipment and oil-based ULV formulations reduces droplet evaporation and allows the use of only 0:5 to 1:0 liter of pesticide per hectare, thus reducing environmental contamination and transportation, handling, and storage costs [17][18][19]. It is largely considered that ULV treatment technique is the most effective way of applying pesticides for locust control [20].
Mathematical modelling can be essential to better understand pest population dynamics and different control strategies to optimize the efficacy of desert locust management [21]. Mathematical models have been widely used to assess the impact and effectiveness of viral infection control strategies against crop pests [22][23][24]. Anguelov et al. [25] formulated and analyzed pest-insect control using mating disruption and trapping. Various authors have studied the role of integrated pest management (IPM) and farmer awareness campaigns on crop pest management [26][27][28][29][30]. Farmers use insecticides to kill insects to protect the crop. Excessive use is hazardous to human health and the environment. Therefore, it is assumed that the amount of insecticides used are proportional to the density of insects. The computational mathematical model assessed the effects of insects and insecticides on crop production [31,32]. Nevertheless, during our review of the literature on models, we did not find a dynamic model of interaction between crop damage and desert locust attacks. Moreover, the control strategies and attack aggressiveness of desert locust attacks are different from other crop pests. To address this gap, we aimed to study the dynamics of interactions between desert locusts, crop damage, and various management strategies. We formulated our model based on the well-known general pattern of invasions and the practices commonly used to control desert locust populations.
In this paper, we considered ongoing control intervention strategies, crop biomass, and stage-structured locust populations. We divided the locust population into two subgroups: wingless hoppers or nymphs and adult locusts. We admitted that hoppers and adult locusts do not have an equivalent attack rate. At all stages, pesticide spraying of hopper bands and adult swarms was considered the main method of desert locust control. The model is derived assuming that the amount of pesticide spray is proportional to the number of desert locusts in the infested area. In addition to locust pesticides, calibrating integration, ultralow volume (ULV) formulation, and equipment control operation leads minimizing pesticide-driven human health and environmental uncertainty. Mathematical analysis of system dynamics and parameter value identification are studied. Finally, we perform numerical simulations to illustrate the theoretical results and assess the impact of intervention strategies.

Development of Model System
We have proposed a model using a stage-structured desert locust population via hopper/nymphs (locusts without wings) (H) and adult locust (A), crop biomass (C), and amount of pesticides (P). Before presenting our mathematical model, we make the following assumptions.
A1. In the absence of locusts, the production of crops grows in a logistic manner with a carrying capacity of KðK ∈ R + Þ and a growth rate constant such that rðr ∈ R + Þ. This is expressed as follows: A2. Considering the stage structure of a locust population, the total locust population L is divided into two classes: one is the juvenile locust or hopper (H) and the other is the adult locust (A), i.e., L = H + A. The nymphs become adults at a rate of σ (1/σ is the development or maturity period of hopper stage locusts).
A3. It is assumed that only an adult locust is capable of reproducing.
A4. We consider that locusts attack the crop at the rate β, which reduces crop production. Due to this consumption of crops by locusts, the density of the locust population increases at a rate ξβ, where ξ represents the conversion 2 Journal of Applied Mathematics efficiency of locusts. It is reduced by the rate of μ due to natural mortality.
A5. In a real-world natural ecological system, it is clear that the attack rate of nymphs is lower than that of adult locusts. So, we introduce a modification parameter η, which is 0 < η < 1.
A6. The intervention measure is implemented on both the hopper and adult locust populations at a rate of φγ 1 and φ γ 2 , respectively. The parameter γ 1 represents the uptake rate of hoppers, γ 2 represents the uptake rate of adults, and φ is the proportionality constant.
A7. The amount of pesticide spray is proportional to the locust population (L). Its growth rate is θ, which is depleted at a rate of θ 0 .
Based on the above assumptions, our model is formulated as follows: and the initial conditions are given as follows:  [3,33]. A desert locust adult can consume roughly its weight in fresh food per day, which is about two grams every day [3,34]. We set the consumption rate of a desert locust at β = 0:002 kg per day. However, the consumption rate of hoppers is relatively half that of adult locusts, that is, the modification parameter value η = 0:5 [35]. There can be at least 40 million and sometimes as many as 80 million locust adults in each square kilometer of a swarm [34]. We consider 1000 milliliters of pesticide per hectare for effective desert locust control. Therefore, we estimate the spray rate (θ) to be θ = ð1000 mL/hectareÞ/ð 400000 locust/hectareÞ = 0:0025 mL pesticide per locust. Following this, we set 10% pesticide depletion, and we estimate the depletion rate (θ 0 ) to be θ 0 = θ × 0:1 = 0:00025 mL per locust. We consider that total sprayed pesticides are equivalent to the sums of uptakes and depletions. Therefore, the uptake rate of pesticides per locust (γ 1 and γ 2 ) is estimated as γ 1 = γ 2 ≈ θ − θ 0 = 0:0025 − 0:00025 = 0:00225 mL per locust. Spraying a straightforward ULV formulation chemical pesticide on hopper bands and a settled swarm is considered to cause 80% mortality on desert locusts [36]. Therefore, the numbers of deceased desert locust (φ) due to a milliliter of pesticide is computed as follows: ϕ = 400000 locust/hectare 1000 mL/hectare x0:8 = 320locust per milliliter:
Proof. The Jacobian matrix at E 1 = ðK, 0, 0, 0Þ is Following some algebraic calculations, we obtain the following eigenvalues: Therefore, the eigenvalue λ 3 is positive, thus implying that the equilibrium point E 1 is unstable. Theorem 3. The interior equilibrium E * of the model is always locally asymptotically stable.
Proof. The Jacobian matrix at E * ðC * , H * , A * , P * Þ is where So, the corresponding characteristic polynomial of the Jacobian matrix is where Hence, by Routh-Hurwitz's criterion, equation (13) has all the negative roots or roots have a negative real part if and only if B 1 > 0, B 3 > 0, B 4 > 0, and B 1 B 2 B 3 > B 2 3 + B 2 1 B 4 ; otherwise, the system will be unstable around the interior equilibrium point. ☐ ☐

Numerical Results and Discussions
In this section, we present the numerical results of model (2) using ode45 solver, via MATLAB software. We have to use baseline parameter values, which are list in Table 1 (unless otherwise stated). From this set of parameter values, we have to get the numerical value of equilibrium points of model (2), as follows: the model equilibrium point stability is justified here by using a set of parameter values in Table 1 The time series evaluation of system (2) with varying initial values are shown in Figure 1. All solution trajectories eventually become steady to the interior equilibrium point E * . Our result in Figure 1 shows that all eigenvalues are negatives; this implies that E * is stable. This numerical result is consistent with Theorem 3.

Impact of Intervention Measures.
Here, we observe the impact of attacking rate (β) in the absence of intervention measures (θ = θ 0 = γ 1 = γ 1 = 0), and the other parameters in Table 1 with various values of β. Figure 2 depicts simulation results that show a projected 4000 kg per hectare crop production gain from the agricultural land in the absence of a desert locust attack. However, crop production declines if the locust attack increases. Our model simulation in Figure 2 shows that we obtain 1145 kg out of 4000 kg crop production per hectare at the baseline value β = 0:002. The model simulation result noted that in the absence of relevant intervention strategies, a more than 71.375% crop production loss is expected. Therefore, the necessity of introducing relevant and appropriate intervention practices is undeniable.
A desert locust outbreak can cause significant and widespread agricultural crop losses. Food security, industrial raw materials, and export earnings may also be seriously threatened in affected areas. Consequently, it is not a surprise that extensive control efforts are mounted whenever hopper bands or swarms of the desert locust emerge in or invade a region. Reaction, proaction, and outbreak prevention are three broadly defined desert locust control strategies. The effect of integrated early intervention is evaluated by simulating model (2) in the absence and presence of intervention.
In Figure 3, we show the impact of integrated early intervention over the crop biomass. The numerical simulation shows that effective early intervention against desert locust outbreaks could protect against crop losses caused by locust invasion. The model projection shows that almost 71.375% expected crop losses are anticipated by baseline value integrated intervention strategies. Figure 4 depicts results showing how integrated intervention has a significant impact in overcoming the massive growth of the desert locust population. In particular, at the baseline value of the intervention rate (θ = 0:0025, γ 1 = γ 2 = 0:00225), the locust population was eliminated from the   Table 1 and varying the initial values. The interior equilibrium (3719.43, 12.0255, 1.0014, 130.269) of system (2) is asymptotically stable. 5 Journal of Applied Mathematics infested area. However, in the absence of integrated intervention (θ = θ 0 = γ 1 = γ 2 = 0), the locust population reaches its peak and finally persists in the region.
Reaction and outbreak prevention control strategies protect crops against swarms, but it is costly and disruptive. Proaction involves early intervention during outbreaks to avert further development to plague status; it is in current use because it is effective, relatively inexpensive, and it is the best available option for now. In desert locust control, proaction entails intervention against localized outbreaks to prevent them from reaching plague status. Proaction relies on early detection of bands and swarms, preferably in breeding areas, and prepositioning of campaign resources. Here, we assess the impact of early intervention (only on hoppers) on desert locust control (γ 1 ). Figure 5 depicts results showing how early intervention on hopper stage locusts protects crop production against desert locust attack. In particular, the simulation shows that in the absence of intervention, the agricultural land provides 1145 kg per hectare. Furthermore, at the baseline value of the intervention (γ 1 = 0:00225) only on hopper stage locusts, the  Journal of Applied Mathematics crop losses are rapidly recovered, and the optimal expected crop production is finally reached. Figure 6 depicts results that show the impact of early intervention on the total locust population. In particular, the simulation shows that in the absence of intervention desert locust persists in the infested area. However, effective proaction can contain locust populations. Early detection and strengthening survey and control capabilities at the hopper stage might be correlated with reduced pesticide application, economic costs, environmental risks, and duration and extent of the locust threat.

Conclusion
The desert locust is considered one of the world's most dangerous migratory pests. A typical swarm can have up to 80 million locusts per km 2 and consume the equivalent of crops that could feed 35,000 people for a day. We have evaluated the impact of early intervention to control outbreaks of this insect. We developed a mathematical model integrating crop biomass, locust population (structured by stages), and early interventions.
From our theoretical analysis model, we got three nonnegative equilibrium points. The trivial and locust-free equilibriums are unstable, whereas the interior equilibrium is asymptotically stable. The overall dynamic numerical results supported and verified the theoretical analysis of the model. Our numerical simulation has shown that the dynamical stability of interior equilibrium using times series evaluation. If antilocust measures are not implemented, numerical results have shown that we would get less than 28:8% of the expected agricultural production from infested areas. This will allow the emergence of a socioeconomic crisis and altered food security. Thus, developing and implementing an appropriate and timely control intervention is extremely important. Our study shows that the implementation of an integrated early intervention protects more than 70% of the total agricultural production. The simulation result showed that in the absence of pesticide treatment, this expected agricultural production would be lost due to desert locust attacks. In addition, early proaction intervention on hopper bands, with a standard implementation rate could protect against almost all agricultural crop losses.
In general, our study shows that crop production losses due to desert locust attacks should be manageable using efficient prevention and control strategies. In particular, proactive early intervention using ULV insecticides is costeffective, and has low risks to human health and the environment. However, the proposed model cannot address some of the concerns of integrated desert locust management. The main limitations of our study are as follows: (1) values of some parameters are not established from survey data; (2) the effects of weather and climate changes on locust breeding and movement are not considered; and (3) technologies related to geopositioning, spatial analysis, remote sensing, and early warning are not incorporated. Such limitations should be addressed in our future work.

Data Availability
The data used in this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.