Abstract
Malaria is a Plasmodium parasitic disease transmitted by infected female Anopheles mosquitoes. Climatic factors, such as temperature, humidity, rainfall, and wind, have significant effects on the incidence of most vector-borne diseases, including malaria. The mosquito behavior, life cycle, and overall fitness are affected by these climatic factors. This paper presents the results obtained from investigating the optimal control strategies for malaria in the presence of temperature variation using a temperature-dependent malaria model. The study further identified the temperature ranges in four different geographical regions of sub-Saharan Africa, suitable for mosquitoes. The optimal control strategies in the temperature suitable ranges suggest, on average, a high usage of both larvicides and adulticides followed by a moderate usage of personal protection such as bednet. The average optimal bednet usage mimics the solution profile of the mosquitoes as the mosquitoes respond to changes in temperature. Following the results from the optimal control, this study also investigates using a temperature-dependent model with insecticide-sensitive and insecticide-resistant mosquitoes the impact of insecticide-resistant mosquitoes on disease burden when temperature varies. The results obtained indicate that optimal bednet usage on average is higher when insecticide-resistant mosquitoes are present. Besides, the average bednet usage increases as temperature increases to the optimal temperature suitable for mosquitoes, and it decreases after that, a pattern similar to earlier results involving insecticide-sensitive mosquitoes. Thus, personal protection, particularly the use of bednets, should be encouraged not only at low temperatures but particularly at high temperatures when individuals avoid the use of bednets. Furthermore, control and reduction of malaria may be possible even when mosquitoes develop resistance to insecticides.
1. Introduction
Malaria is a Plasmodium parasitic infectious disease transmitted by infected female anopheles mosquitoes. Sub-Saharan Africa and many tropical and subtropical regions of the world experience a large number of malaria-related mortality and morbidity with most of the disease burden in sub-Saharan Africa [1]. Climatic factors, such as temperature, humidity, rainfall, wind, duration of daylight, and vapor pressure, significantly affect the incidence of many vector-borne diseases, including malaria [2, 3]. These climatic factors affect the host, vector, and parasite behavior. They also affect the duration of the vector and parasite/pathogen life cycles [4–7].
The average global surface temperatures since the late century have increased by about - [3, 8]; unfortunately, due to tropical warming, this trend is expected to continue [8]. Although there is significant spatial heterogeneity in the anticipated changes in the global average, the greater temperature increase is expected in sub-Saharan Africa [8].
Variability in weather and climate determines the geographical distribution, seasonality, yearly variability, and longer-term trends of malaria transmission [9, 10]. The distribution and duration of the disease are modified by periods of unusually high rainfall, altered humidity, or warmer temperatures leading to increased transmission [11]. Similarly, periods of long-term drought can reduce malaria transmission [9–11]. Variability in the El Nio cycle and other long-term meteorological cycles can increase the disease burden and periodic upsurges in cases resulting in malaria epidemics [9, 11].
The greatest impact due to climate change is likely to be observed in temperature ranges – and –. These temperature ranges have a significant impact on the transmission of many vector-borne diseases [12]. Places at the lower end of the temperature range (–) will see an increase or extension in the transmission periods [13]. For malaria transmission, many temperature-related effects have been documented and established. For example, the female adult anopheles mosquitoes tend to feed more frequently as temperature increases because blood is digested more quickly. However, the juvenile mosquitoes maturate more quickly in warmer waters as a result of the shortened incubation period (leading to a faster growth rate of the vector larvae) [12, 14]. The average lifespan of an adult female mosquito is about 21 days, and this is also affected by temperature; the mosquito survival rapidly decreases when temperature rises beyond [15]. Furthermore, temperature equally affects the malaria parasites inside the vector; at , it takes 19 days for these parasites to mature within the mosquito, but the maturation period decreases to eight days at [15, 16].
Temperatures greater than (extreme temperature) negatively affect the survival of malaria vectors, parasites, and, invariably, the transmission rate [17]. A shift in the distribution of vectors into nonendemic areas is expected as global temperature increases so that an increase in malaria incidence is expected in malaria-endemic areas with lower temperatures, while a decline in incidence is expected in regions at temperatures beyond the transmission peak (provided all other factors remain constant and do not change) [18, 19].
Furthermore, malaria transmission is enhanced by rainfall, which generally increases the availability and productivity (conditional on ambient temperatures) of mosquito breeding sites [20, 21]. On the contrary, excess rainfall can lead to the flushing out of Anopheles breeding sites [22] and a decrease in the mosquito population.
Agusto et al. in [2] assessed the role of temperature variations on malaria transmission and identified mean monthly temperature ranges where malaria-associated burden increases in West African cities as , in Central African cities as , in East African cities as , and in KwaZulu-Natal, South Africa, as . The study carried out uncertainty and sensitivity analysis using Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC) to identify parameters with the most impact on the total number of infectious individuals as output function; the parameters with the most impact on the output are mosquito carrying capacity , the mosquito biting rate , the transmission probability per contact for susceptible mosquitoes and susceptible humans , the maturity rate from larvae stage to adult mosquito , and the human recruitment rate .
The study also carried out uncertainty and sensitivity analysis by incrementing temperature by ; they found that the larvae maturation rate has a minimum PRCC value at and a maximum value at . Furthermore, the study showed that the mosquito biting rate has a maximum PRCC value at and a minimum PRCC value occurs at . The implication of this result on control policy is that when the mean monthly temperatures are in the range , individuals in the community should use both mosquito-reduction strategy and personal protection against mosquito bites; however, for higher mean monthly temperatures in the range , mosquito-reduction strategy should be emphasized in the community (or household) ahead of personal protection.
This study aims to determine if this should be the control protocol. In particular, should individuals use bednets when temperatures are low? Or should they desist from using bednets when temperatures are high? To address these questions, we apply optimal control theory using the results from the sensitivity analysis carried out in [2] to identify the time-dependent control variables to be implemented in our study.
The paper is organized as follows: the model formulation is given in Section 2, alongside details of the temperature-dependent parametrization adopted. The optimal control problem is given in Section 3. The section further includes the optimal control analysis and the characterization of the control variables. In Section 4, we give results of the numerical simulation of the control problem. In Section 5, we introduce a mathematical model with insecticide-resistant mosquitoes; this is due to the results obtained from applying the optimal controls in Section 4.1 involving insecticide-sensitive mosquitoes. The optimal control analysis and numerical simulation for insecticide-resistant mosquitoes are implemented in Section 5.1. Finally, the discussion and conclusions are stated in Sections 6 and 7, respectively.
2. Methods
2.1. Model Formulation
In this paper, we consider the simple malaria model used in the study of Agusto et al. [2] since the model with the property of gradual boosting of immunity against malaria due to repeated exposure has marginal (or no) effect on predictions of disease burden. The human population, of size , is split into mutually exclusive subpopulations of individuals who are susceptible , exposed , infectious , and recovered from the infection so that
Similarly, the total mosquito population size at time , denoted by , is subdivided into immature mosquitoes (eggs, larvae, and pupae) , susceptible mosquitoes , exposed mosquitoes , and infectious mosquitoes .
The simple model is given by the following deterministic, nonautonomous, system of nonlinear differential equations, where dot notation denotes the derivative with respect to time and and denote air and water temperature:
The flow diagram of malaria model (2) is shown in Figure 1; Table 1 describes the state variables and parameters.
The parameter in the system (2) is the human recruitment rate, is the temperature-dependent force of infection of healthy susceptible humans (after contact with infectious mosquitoes), and is the natural death rate of humans. The infection rate, , is defined aswhere is the susceptible human transmission probability of malaria infection per bite from an infectious mosquito and is the mosquito temperature-dependent per capita biting rate; the conservation law of mosquito bites was used to derive the temperature-dependent force of infection [23–25]. The parameter, , represents exposed humans progression rate into the infectious class . Also, the parameter represents the rate at which individual in the class recovers (and moves to the recovered class). The parameter represents the rate at which the recovered individuals in loose their infection-acquired immunity and move to the class. The parameter is the rate at which individuals in the infectious human class suffer malaria-induced mortality.
After successfully obtaining blood meals from human hosts, the female mosquitoes will rest for a few days allowing the blood to digest and their eggs to develop. The process of mosquito egg laying is sensitive and dependent on temperature [26]; the laid eggs typically hatches within 2-3 days and may take up to 2-3 weeks in colder temperate climates [26, 27]. The temperature-dependent egg deposition rate is denoted by . We assume that the immature mosquito population (comprising of the larvae and pupae) is limited by the carrying capacity ; this parameter is related to the amount of available nutrients and space [28–30]. Thus, represents the logistic growth rate for the immature mosquitoes. The parameter denotes the temperature-dependent metamorphosis rate (i.e., maturation rate) of immature mosquitoes [31–34], and the temperature-dependent mortality rate of immature mosquitoes is give by the parameter (it is worth noting that, in this study, the dynamics of the three aquatic stages of the mosquito is modeled using the single compartment, , for mathematical convenience (see also [25, 29, 30])). Susceptible adult female mosquitoes become infected, following effective contact with an infectious human (from an infectious blood meal); this force of infection is also temperature-dependent, and it is given bywhere is the probability that a bite from a susceptible mosquito to a human with infectious gametocytes leads to the infection of the mosquito. The parameter is the temperature-dependent progression rate of exposed mosquitoes [35, 36]. The parameter represents the temperature-dependent death rate of adult mosquitoes; this parameter is dependent both on temperature and humidity and the ability of the mosquito to successfully obtain blood meals while avoiding host defenses [26]. We assume there is no disease-related death for the mosquitoes.
Following Agusto et al. [2], the temperature-dependent parameters of the model (2) are defined as follows: using the quadratic functions used in [37], the mosquito biting rate, , and egg deposition rate, , are given by
Similarly, the temperature-dependent parameters , and are defined in [37] as follows:
The temperature-dependent function represents the lifetime number of eggs laid by the female mosquito, the probability that the egg laid survives to become an adult mosquito is represented by , and the development time from egg to adult mosquito is represented by [37]. The temperature-dependent function, , is defined aswhere is the number of eggs laid per female per day. Hence, , the total number of eggs laid by a mosquito (in a lifetime), is equal to the number of eggs laid per day multiplied by the average adult lifespan [37]. These temperature-dependent parameters are expressed as [37]
Hence, substituting the temperature-dependent function (9)–(12) into (7), we have
Lastly, the parameter [38] is defined as
Numerical simulations of the model are carried out using the mean monthly temperatures from 2011 to 2013 for ten cities in sub-Saharan Africa (three cities each in West, Central, and East Africa and KwaZulu-Natal, South Africa) as were implemented in [2]. The cities in West Africa are NZerekore, Guinea; Bamako, Mali; and Niamey, Niger. The cities in Central Africa are Lubumbashi, DRC; Tchibanga, Gabon; and Ati Chad, while the cities in East Africa are Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya.
The temperature within the three cities of West, Central, and East Africa is chosen to represent the minimum, mean, and maximum mean monthly temperature, respectively, in the given region. As in [2], we used the temperature profile of KwaZulu-Natal as a representative sample of cities in southern Africa, since most cities in southern Africa have similar mean monthly temperatures [39].
The next section investigates the control strategies for curtailing malaria following the results obtained from sensitivity analysis in [2].
3. Optimal Control Problems
Following the results obtained from the sensitivity analysis in [2], we introduce into the transmission model (2) two time-dependent controls, namely, personal protection against mosquito bite and mosquito-reduction strategies. The mosquito-reduction strategy is further divided into two (larvicides and adulticides) according to the life stages of the mosquitoes. For simplicity, we assume that the use of bednets reduces the contact between mosquitoes and humans and that the mosquito population is reduced by the mosquito-reduction strategies (larvicides and adulticides), which we assume in this case chemical-based insecticides. However, there are other environmental friendly larvaciding methods that can be used to target the larvae stage of the mosquitoes, such as larvivorous fish which prey on the larvae [40–44] and bacterial larvicide which causes the mosquitoes to starve to death once they ingest the spores of the bacterial larvicide [45–48]. In a future study, we will incorporate the insecticide capacity of the bednets [49] and the actions and kinetics of these larvacide methods. Thus, the three time-dependent control variables are , and , representing personal protection, larvaciding, and adulticiding. Thus, model (2) becomes:where
The factor reduces the force of infection associated with humans and mosquitoes, respectively [50–52].
This study aims to minimize the objective function defined assubject to the differential equations (15), where is the final time. This performance specification involves minimizing the number of infected humans and the mosquito population, along with the cost of applying the controls . Often the cost of implementing a control would be nonlinear, and as such, a simple nonlinear quadratic objective functional is taken here. The costs can include funds needed for control implementation, cost of hospitalization, and the cost of manpower lost due to hospitalization. This assumption is based on the fact that there are no linear relationships between the effects of the intervention and the cost of the intervention on the infected populations; such quadratic costs have been frequently used [50–54]. The coefficients and balancing cost factors are .
We seek to find the optimal controls such that
The optimal control solution is given as
The control characterization is given in Appendix A.
In the next section, the results of the numerical solutions of the optimality system, the corresponding optimal control, and the interpretations of the various cases are considered.
4. Numerical Illustration
The forward-backward sweep algorithm, along with the fourth-order Runge–Kutta method, was used to obtain the numerical solutions of the optimal controls and state values. The algorithm requires that initial estimates for the control variables be made. Then using these estimates, the state variables are then solved forward in time using the dynamics (15). The results obtained for the state variables are used in computing the solution for the adjoint equations (A.4) together with given final conditions (A.7). These are solved backward in time, using the backward fourth-order Runge–Kutta method. Both the solution of the state and adjoint values are then used to update the control, and the process is repeated until there is sufficient convergence between the current state, adjoint, and control values [55]. The algorithm is implemented using MATLAB R2016A [56].
Thus, to implement the optimal control algorithm, the following initial condition values are used . The weight factors are chosen as . It should be pointed out that the values chosen for the initial conditions and weights used in the optimal control simulations are only of theoretical sense to demonstrate the control strategies proposed in this paper.
First, we consider the optimal control simulation for the maximum mean temperature of in KwaZulu-Natal, South Africa. This is the highest mean monthly temperature in the Province, and it occurs in November. The simulation is done for a period of 30 days representing the number of days in the month. In the next section, we consider the control using the mean monthly temperature over the twelve months in a year. The results of the optimal control simulations of model (15) are depicted in Figure 2. With optimal control, there are fewer infected humans in the community compared to the case with no control (Figure 2(a)). On the thirtieth day, there were a total of infected humans in the absence of control compared to approximately with the application of control. Similarly, for the infected mosquitoes, there are more infected mosquitoes in the absence of control, unlike when control is applied where the mosquitoes are significantly reduced (Figure (2(b)). In particular, there were infected mosquitoes in the absence of control on the thirtieth day compared to approximately 120 with the application of control.
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The corresponding time-dependent controls (, and ) are depicted in Figure 3. The time-dependent controls start at the upper bound of unity for the most part of the simulation period before reducing to the lower bound. Control starts to be low at 0.165 but quickly rises to the upper bound for most of the simulation period. The control starts at the upper bound for about eight days before gradually reducing to the lower bound.
In the next section, we consider the control using the mean monthly temperature over the twelve months in a year and determine the temperature range suitable for mosquitoes. The total number of infected humans and mosquitoes per mean monthly temperature is used as metric for mean optimal controls , and . We take only a single value for the total number of infected humans and mosquitoes and the mean optimal control level for each control , and since plotting the complete solution profiles will involve 36 solution trajectories resulting in a lot of figures to read and interpret. Therefore, using the total infected and the mean level of control, we can see clearly from the figures and draw interpretation quickly from the figures.
4.1. Optimal Control with Mean Monthly Temperatures
4.1.1. Mean Monthly Temperature: South Africa Profiles
The results of the optimal control simulations of the malaria model (2) in KwaZulu-Natal are depicted in Figures 4(a) and 4(b). First, the uncontrol and control cases for the mean monthly air temperatures in KwaZulu-Natal are superimposed with the total number of cases in each month generated using the model (2) with the baseline parameter values in Table 2. As previously obtained in [2], this figure show that, in general, malaria burden increases with increasing mean air temperature in the range . Figure 4(a) shows that there are fewer infected individuals with the application of the control measures.
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Furthermore, the results of the optimal control simulations of the malaria model (2) measured in terms of the average control efforts are depicted in Figure 4(c). This shows that the time-dependent controls and on average are close to the upper bound throughout the year for all temperature values, while the time-dependent control starts relatively low for low temperature and is relatively high for high temperature. It should be noted that the solution profile for mimics the trajectory of the infected mosquitoes as the mosquitoes respond to variations in temperature (Figures 4(b) and 4(c)).
4.1.2. Mean Monthly Temperature: Other Cities Infected Human Profiles
Similar behavior is observed in the nine cities, used in [2], and three cities each were chosen from West, Central, and East African regions (Table 3). In particular, Figure 5 depicts the mean monthly air temperatures in the three cities in West Africa superimposed on the mean total number of new cases generated each month using model (2). The figures for Central and East African cities are depicted in Figures 6 and 7.
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Figures 5(a)–5(e) show that in West Africa, malaria burden increases for temperatures in the range , while in Central Africa (see the infected humans in Figure 6), malaria burden increases for temperatures in the range . Lastly, the range in East Africa is (Figure 7 for the infected humans). These simulations clearly show the variability in the suitable temperature ranges for malaria transmission in the various regions of sub-Saharan Africa, as previously obtained in [2]. The solution profile for infected mosquitoes is given Figures 8–10. It also shows that the solution profile for control mimics the trajectory of the infected mosquitoes as the mosquitoes respond to variations in temperature (Figures 8–10).
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Specifically, in West Africa, Figures 5(a), 5(c), and 5(e) show that there are fewer infected individuals with the application of the control measures. The average time-dependent controls and are close to the upper bound for all temperature values in the three cities of West Africa (Figures 5(b), 5(d), and 5(f)). The time-dependent control starts relatively low for low temperatures and is relatively high for high temperatures. It should be noted that the solution profile for varies from about 0.25 to 0.7 as the temperature gradient varies across West Africa. The control is low at about 0.25 for low temperatures and about 0.7 for high temperatures.
In Central Africa, the application of the control measures leads to fewer infected individuals (see the infected humans in Figure 6). The average time-dependent controls and are all close to the upper bound for all temperature values in the three cities of Central Africa, see Figures 6(b), 6(d), and 6(f). The time-dependent control varies from about 0.3 to 0.8 as the temperature gradient varies across Central Africa. Similar to what was observed in West Africa, the control is low at 0.3 for low temperatures and is relatively high at 0.8 for high temperatures.
Also, in East Africa, there are fewer infected individuals with the application of the control measures (see the infected humans in Figure 7). The average time-dependent control varies from about 0.3 to 0.7, while the controls and are close to the upper bound as the temperature gradient varies across the three cities of East Africa, see Figures 7(b), 7(d), and 7(f). The time-dependent control starts relatively low at 0.3 for low temperature and is relatively high at 0.7 for high temperature.
4.2. Control in Temperature Ranges Suitable for Mosquitoes
4.2.1. Suitable Temperature Ranges: Infected Human Profiles
Next, we explore closely the optimal control strategies in the temperature range suitable for mosquitoes identified in the Mean Monthly Temperature: Other Cities Infected Human Profiles section [2]. In West Africa, the temperature range is , and in Central Africa, the temperature is . The temperature range in East Africa is , and in South Africa, the range is .
We observed, in Figure 11(a) in the absence of control, an increase in the total number of infected humans in West Africa until the midtemperature of before decreasing at a higher temperature. However, in Central, East, and South Africa, the total number of infected humans increases with increasing temperature (Figures 11(c), 11(e), and 11(g)). The application of the control strategies reduced the number of infected humans. Similar dynamics are observed for mosquitoes, and the plots are given in Figure 12.
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We observed in Figures 11(b), 11(d), 11(f), and 11(h) that the average time-dependent controls and are close to the upper bound for all temperature values in all the four regions. In West Africa, the average control increases until the midtemperature of before decreasing at a higher temperature, while in East, Central, and South Africa, the time-dependent control increases with increasing temperatures.
The optimal control strategies with temperature suggest that very high levels of larvicides and adulticides should be used. These high levels of insecticide, particularly from the scale-up use of insecticide-treated bednets, and indoor spraying have been observed to lead to the development of resistance mosquito [59, 60]. This widespread use of insecticide-treated bednets and indoor spraying has been largely successful in reducing the incidence of malaria across sub-Sharan Africa [61]. However, the widespread use of these has increased the pressure on mosquitoes to evolve resistance to these commonly used pyrethroid insecticides [59, 62].
Using sequencing techniques and genetic analyses, Barnes et al. [59] carried out a continent-wide population structure study of Anopheles funestus, a major African malaria mosquito. They located a gene region that has allowed mosquitoes to evolutionarily adapt to insecticides by enabling them to break down insecticides [59]. This resistance form of this gene has now swept through mosquito populations in southern Africa to become almost universal [60].
Coupling our result from the optimal control of insecticide mosquitoes and the findings of Barnes et al. [59], we develop a model to study the control of malaria in the presence of temperature variation and insecticide-resistant mosquitoes.
5. Malaria Model with Insecticide-Resistant Mosquitoes
Following the results obtained in the Optimal Control with Mean Monthly Temperatures section involving the high levels of adulticides and larvacides controls, we extend the simple malaria model used in Agusto et al. [2] to include both insecticide-sensitive and insecticide-resistant mosquitoes. The human population, of size , is split into mutually exclusive subpopulations of individuals who are susceptible , exposed , infectious , and recovered from the infection so that
Similarly, the total mosquito population size at time , denoted by , is subdivided into immature mosquitoes (eggs, larvae, and pupae) , susceptible insecticide-sensitive and insecticide-resistant mosquitoes , exposed insecticide-sensitive and insecticide-resistant mosquitoes , and infectious insecticide-sensitive and insecticide-resistant mosquitoes . We assume for simplicity that the mosquitoes differentiate into insecticide-sensitive and insecticide-resistant mosquitoes only at the adult stage.
Thus, the simple model is given by the following deterministic, nonautonomous, system of nonlinear differential equations (where the dot notation, as previously stated, denotes differentiation with respect to time and and denote air and water temperature, respectively):where
Furthermore, we assume that and . The parameters , and are modification parameters indicating the difference between the insecticide-sensitive and insecticide-resistant-related functions. We have assumed that ; indicating that the mosquitoes’ insecticide-resistant gene has no impact on mosquito biting rate and transmission probability, nor does it affect the parasite developmental rate. However, there are indications on the contrary [63–65]. We make these assumptions for simplicity, and without loss of generality, we believe that there will be no wide deviation from the conclusions of this study if we had incorporated these features. Furthermore, we suppose the resistance mosquitoes are twice likely to live longer than the sensitive mosquitoes; hence, we set . However, infected insecticide-resistant mosquitoes are more susceptible to insecticide [64, 66]. Note, and are given by the same functions as those in equations (6) and (8), respectively.
The flow diagram of the model is depicted in Figure 13, the associated state variables are described in Table 4, and parameters are described in Table 1.
5.1. Optimal Control Problem with Resistance Mosquitoes
The goal in this section is to minimize the objective function defined asthat is, we seek to find the optimal controls , such that
The optimal control solutions can be characterized by following the approach in the Optimal Control Problems section.
Our aim in this section is to apply optimal control theory to minimize the number of infected humans and mosquitoes (insecticide-sensitive and insecticide-resistant) in the temperature range suitable for mosquitoes in West, Central, East, and South Africa. As in the Control in Temperature Ranges Suitable for Mosquitoes section, we will use the mean total number of infected humans and mosquitoes to measure the impact of the mean level of controls in these regions of sub-Sahara Africa. We will also investigate the impact of monocontrol strategies where the time-dependent controls are applied one at a time.
5.2. Resistance Mosquitoes: Temperature Ranges Suitable for Mosquitoes
5.2.1. Control in Suitable Temperature Ranges: Infected Human Profiles
Next, we also explore the optimal control strategies in the temperature range suitable for mosquitoes identified in the Mean Monthly Temperature: Other Cities Infected Human Profiles section. In West Africa, the temperature range is , and in Central Africa, the temperature is . The temperature range in East Africa is , and in South Africa, the range is .
In Central, East, and South Africa, the mean total number of infected humans increases with increasing temperature (Figures 14(c), 14(e), and 14(g)). The application of the control strategies, depicted, in Figures 14(d), 14(f), and 14(h), reduced the mean number of infected humans, with the average level of the controls and close to the upper bound and the average control increases with the mean monthly temperature. Similar dynamics are observed in the mosquito population, and the plots are given in Figure 15.
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5.3. Monocontrol Strategies
Next, we investigate the effect of monocontrol strategies on malaria transmission in sub-Saharan Africa using malaria insecticide-sensitive model (2) and insecticide-resistant model (21).
We considered the following two monocontrol strategies:(i)Strategy A: controls —only while setting (ii)Strategy B: control —only while setting
We compare the solution profiles obtained from these strategies to the profile obtained from the control strategy when all the controls , and are used (Figure 11).
5.3.1. Monocontrol Strategies for Sensitive Mosquitoes: Infected Human Profiles
Comparing the mean total number of infected humans in Figures 11 and 16, we observed a higher number of infected humans in the absence of control (Figure 11) and lower number with the application of the controls regardless of the control strategies (Figure 16). As in the previous sections, we calculated the average level for all the controls , and . We observed that Strategy B was relatively better than Strategy A in reducing the number of infected humans when we have insecticide-sensitive mosquitoes only.
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We observed in Figure 11(a) increase in the mean total number of infected humans in West Africa in the absence of control until the midtemperature of before decreasing at the highest temperature of . The mean total number of infected humans was considerably reduced in the presence of control. Furthermore, in Figure 11(b), we observed that the average level for the time-dependent controls and is close to the upper bound for all temperature values in West Africa. The profile for control is low at low temperatures and is high at high temperatures.
This is not the case with infected mosquitoes. Strategy A led to fewer infected mosquitoes compare to Strategy B (see the profile for infected mosquitoes in Figure 17). These strategies perform relatively poorer than when the controls , and are all used.
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Thus, personal protection against mosquito bite reduces malaria burden in humans better than mosquito-reduction strategies. Mosquito-reduction strategies, on the contrary, led to fewer infected mosquitoes.
5.3.2. Monocontrol Strategies for Resistance Mosquitoes: Infected Human Profiles
We compare control Strategies A and B when insecticide-resistant mosquitoes are present. As with the insecticide-sensitive case, the application of the controls, regardless of the strategy, can greatly reduce the total number of infected humans which is high in the absence of controls (Figures 14 and 18). The average level of all the controls , and was used as in the previous sections. Strategy B which involves using only control while setting controls performs better in reducing the number of infected human than Strategy A (which involves using controls , while setting ).
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The profile for infected mosquitoes is given in Figure 19, and it shows that Strategy A performs better at reducing infected mosquitoes in the presence of insecticide-resistant mosquitoes.
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Thus, personal protection against mosquito bite only as control strategy reduced malaria burden in humans better than mosquito-reduction strategies. However, mosquito-reduction strategies only led to fewer infected mosquitoes when mosquitoes have evolved resistance against insecticide.
6. Discussion
Since the 1950s, global temperature has been on the increase with great consequence for vector-borne diseases. It is expected that this warming leads to an expansion in the distribution of vectors into new regions. Several naive populations such as those in cities in the Americas and the Caribbeans have witnessed in recent times increase in emergence and re-emergence of vector-borne diseases such as Chikungunya, Dengue, Yellow Fever, and Zika. Results from Agusto et al. [2] indicate that malaria burden will decrease as temperature goes beyond the range suitable for mosquitoes. According to the World Health Organization (WHO), the global burden of malaria has been on the decline in the last ten years largely due to the use of insecticide-treated bednets [61].
The global cases of malaria increased from 237 million cases in 2010 to 216 million cases in 2016, an 18% global reduction in incidence rate [61]. This progress has, however, stalled; between 2014 and 2016, there was a substantial increase in malaria incidence cases in the Americas. For instance, in 2015, 211 million cases were reported. This increase in reported cases may be perhaps due to mosquitoes developing resistance to the insecticide used in the bednets [61], particularly pyrethroid [67].
The results from Agusto et al. [2] show that when the mean monthly temperatures are in the range , individuals in the community should use both mosquito-reduction strategy and personal protection against mosquito bites; however, when temperatures lie within the range , mosquito-reduction strategy should be used in the community (or household) over personal protection against mosquito bites. In this study, we applied optimal control theory to investigate the optimal control strategy necessary to control the disease in these temperature ranges. We found that in both temperature ranges, the use of mosquito-reduction strategy such as larvicides and adulticides on average should be maintained at relatively high levels which could lead invariably to the development of insecticide-resistant mosquitoes [59, 62].
This result is in line with Barnes et al.’s [59] findings which identified a gene region within Anopheles funestus that allows the mosquitoes to break down insecticides, thereby resulting in the development of insecticide-resistant mosquitoes. Barnes et al.’s study was a continent-wide population structure of Anopheles funestus mosquitoes. This resistance gene has swept through the entire mosquito populations in southern Africa [60]. According to the WHO, between 2010 and 2016, the frequency of pyrethroid resistance in malaria vectors increased globally [62]. However, the actual impact of insecticide resistance on the effectiveness of insecticidal vector control is not fully known; the development of insecticide resistance highlights a potential challenge to control and elimination of malaria, particularly in Africa where the burden of malaria remains highest [62]. It can undercut the over two decade gain made from malaria control leading to increase in malaria incidence and mortality [68].
The result of this study further shows that control measure aimed at only controlling mosquito population (mosquito-reduction strategies) in the temperature regions suitable for mosquitoes will lead to fewer infected mosquitoes. To effectively control the disease burden, the control measure should prevent or reduce the number of bites humans receive from mosquitoes since personal protection against mosquito bite is better at reducing malaria burden in humans than mosquito-reduction strategies.
When mosquitoes have evolved resistance against insecticide, personal protection against mosquito is equally better than mosquito-reduction strategies (larvacides and adulticides) in reducing malaria burden in humans. However, mosquito-reduction strategies only led to fewer infected mosquitoes. To effectively control the mosquito population, given the added pressure of insecticide resistance, a higher level of larvicides and adulticides is needed. This level is higher than the level used when mosquitoes are sensitive to insecticides. We also observed (not shown here) that as the ratio of sensitive to resistance mosquitoes increases in the resistance direction, the level of optimal control of larvacides and adulticides remained constant and close to 100 %.
As observed in this study of insecticide-resistant mosquitoes, control and elimination of malaria in the presence of resistance mosquitoes are possible only with the use of high level of insecticide use. This then leads to a vital question: “will the evolutionary adaptation to insecticides then lead to supermosquitoes developing resistance to all insecticide as observed with bacteria with antimicrobial resistance?”
Therefore, to prevent such catastrophic development, efforts should be geared towards exploring other efficient ways and strategies to control malaria and prevent the development of insecticide-resistant mosquitoes by targeting either the mosquito larvae or adult stages.
Efforts aimed at malaria control can help reduce the development of insecticide mosquitoes; for instance, active participation of the community and government through community-based interventions is essential for community wide disease elimination and control using strategies that interrupt disease transmission such as backyard draining of standing water [69], improve drainage, and community education regarding introducing a novel vector control strategy like the genetically engineered insects [70–73].
Other useful strategies that can reduce the development of insecticide-resistant mosquitoes target the larvae stage through the use of larvicides such as larviciding with larvivorous fish and bacterial larvicide [40–43, 45–48].
Larvivorous fish have been employed as biological control of mosquito larvae since 1937 [74–79]. They target mosquito larvae and are safe for humans since they are devoid of insecticide [44]. Larvivorous fish are well suited for urban areas where the density of humans needing protection is higher than mosquito breeding sites [76] and places with seasonal transmission [44].
Bacterial larvicides is another mosquito larvae targeting the control method using Bacillus thuringiensis var. israelensis (Bti) and Bacillus sphaericus (Bs) [45–48]. These microbial larvacides selectively kills the mosquito larvae when they ingest the bacterial spores leaving unharmed other cohabiting organisms [80].
Thus, implementing these larvae control strategies in addition to the use of bednets and adult control will not only reduce the chance of developing insecticide-resistant mosquitoes, it will ultimately lead to control and elimination of malaria in a community.
7. Conclusions
In this study, we applied optimal control theory to the model developed in Agusto et al. [2] in order to determine if individuals in the community should use both mosquito-reduction strategy and personal protection against mosquito bites for mean monthly temperatures are in the range or they should emphasize mosquito-reduction strategy in the community (or household) ahead of personal protection for higher mean monthly temperatures in the range .
Using the simple model and results in [2], we summarized below some of the findings of this study. Detailed theoretical and epidemiological findings of the model are given in [2]:(i)Numerical simulations of the model (2) show that the total number of new cases of infection (malaria-associated burden) increases with increasing mean monthly temperature in the following ranges:(a) in the three West African cities.(b) in the three Central African cities.(c) in the three East African cities.(d) in KwaZulu-Natal, South Africa.(ii)Malaria burden in the presence of temperature variations can be decreased via the application of time-dependent controls.(iii)The solution profile of the personal protection control mimics the response of infected mosquitoes to temperature in the absence of control.(iv)Malaria burden can be decreased in the presences of insecticide-resistant and temperature variations.(v)With monocontrol strategies, personal protection against mosquito bite is better at reducing malaria burden in humans than mosquito-reduction strategies, and mosquito-reduction strategies led to fewer infected mosquitoes.
Thus, we conclude that personal protection, particularly the use of bednets, should be encouraged not only at low temperatures but particularly at high temperatures when individuals are driven not to use the nets. Furthermore, control and elimination of malaria are possible even with mosquitoes developing resistance to the insecticide. In future work, we will incorporate the kinetics of insecticide-treated bednets, larvacides (with either larvivorous fish or bacterial larvicide), and adulticides (which will decay in some functional form). We will also incorporate rainfall and seasonality to address the possibility of control and resistance prevention using optimal control in areas with seasonal malaria.
Appendix
A. Characterization of Optimal Controls
We seek to find the optimal controls such thatwhere the control set
The controls are bounded Lebesgue integrable functions [50–54].
The optimal control triple (, , and ) satisfy necessary conditions from Pontryagin’s maximum principle [81]. This principle converts (15) and (16) into a problem of minimizing pointwise a Hamiltonian , with respect to the controls (, , and ). The optimality conditions are obtained using the Hamiltonian formulated from the cost functional (16) and the governing dynamics (15). Thus, Hamiltonian is given aswhere are the associated adjoints for the state variables . The adjoint equation system is determined from the partial derivatives of the Hamiltonian (A.3) with respect to the state and control variables.
Theorem 1. Given the optimal controls and the solutions of the corresponding state system (15) which minimizes the objective functional over . Then, there exist adjoint variables , satisfying the systemand with transversality conditionsThe optimality conditions is given asFurthermore, the control is given as
Proof. Using the result by Fleming and Rishel [82], the existance of the optimal control can be shown. Thus, taking the partial derivative of the Hamiltonian function with respect to the state variables leads to the system of adjoint equations. Thus, the adjoint system is given asevaluating the adjoint system at the optimal controls and corresponding state variables, the stated adjoint systems (A.4) and (A.5) are given. In addition, the controls are obtained by taking the partial derivative of the Hamiltonian function, , with respect to the control variables in the interior of the control set and then solving for controls ; hence, we have the result for the optimality conditions given asSolving for , and , we haveThe characterization (A.7) can be derived by using the bounds on the controls.
Remark 1. The uniqueness of the optimal control for a small time can be shown using the a priori boundedness of the state and adjoint functions along with the resulting Lipschitz structure of the ODE’s. Therefore, the uniqueness of the optimal control follows from the uniqueness of the optimality system, which consists of (15) and (A.4), (A.5) with characterization (A.7). The restriction on the length of the time to a small interval is to guarantee the uniqueness of the optimality system. The small length of time is due to the opposite time orientations of (15), (A.4), and (A.5); the state problem has initial values, and the adjoint problem has final values. This restriction is very common in optimal control problems [51, 53, 54].
Data Availability
The temperature data are within the plots. And they can be made available upon request.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This research was supported by the Strategic Environmental Research and Development Program under grant RC-2639 and University of Kansas General Research Grant 2301-2105075.