# Optimal Control and Temperature Variations of Malaria Transmission Dynamics

Authors: Folashade B. Agusto
Journal: Complexity
 (2020)

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## Abstract

Malaria is aPlasmodium 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.


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## Body

## 1. Introduction

Malaria is aPlasmodium 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 late19th century have increased by about 0.5°C-0.6°C [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 Nin˜o 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 ranges14°C–18°C and 35°C–40°C. 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 (14°C–18°C) 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 30°C,32°C [15]. Furthermore, temperature equally affects the malaria parasites inside the vector; at 22°C, it takes 19 days for these parasites to mature within the mosquito, but the maturation period decreases to eight days at 30°C [15, 16].Temperatures greater than34°C (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 22.61°C,28.58°C, in Central African cities as 16.68°C,27.92°C, in East African cities as 19.04°C,26.75°C, and in KwaZulu-Natal, South Africa, as 16°C,25°C. 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 KV, the mosquito biting rate bV, the transmission probability per contact for susceptible mosquitoes βV and susceptible humans βH, the maturity rate from larvae stage to adult mosquito θV, and the human recruitment rate ΠH.The study also carried out uncertainty and sensitivity analysis by incrementing temperature by2°C; they found that the larvae maturation rate θV has a minimum PRCC value at T=24°C and a maximum value at T=30°C. Furthermore, the study showed that the mosquito biting rate bV has a maximum PRCC value at T=26°C and a minimum PRCC value occurs at T=24°C. The implication of this result on control policy is that when the mean monthly temperatures are in the range 16.7,25°C, 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 26,34°C, 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 Section2, 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 NHt, is split into mutually exclusive subpopulations of individuals who are susceptible SHt, exposed EHt, infectious IHt, and recovered from the infection RHt so that(1)NHt=SHt+EHt+IHt+RHt.Similarly, the total mosquito population size at timet, denoted by NVt, is subdivided into immature mosquitoes (eggs, larvae, and pupae) LVt, susceptible mosquitoes SVt, exposed mosquitoes EVt, and infectious mosquitoes IVt.The simple model is given by the following deterministic, nonautonomous, system of nonlinear differential equations, where dot notation denotes the derivative with respect to timet and T and T^=+δT denote air and water temperature:(2)SH⋅t=ΠH+ωHRHt−λHT,IV,NHSHt−μHSHt,EH⋅t=λHT,IV,NHSHt−σH+μHEHt,IH⋅t=σHEHt−γH+μH+δHIHt,RH⋅t=γHIHt−ωH+μHRHt,LV⋅t=ϕVT1−LVtKVSVt+EVt+IVt−θVT^+μLT^LVt,SV⋅t=θVT^LVt−λVT,NHSVt−μVTSVt,EV⋅t=λVT,NHSVt−σVT+μVTEVt,IV⋅t=σVTEVt−μVTIVt.The flow diagram of malaria model (2) is shown in Figure 1; Table 1 describes the state variables and parameters.Figure 1
The malaria model (2) flow diagram.Table 1
Description of the variables and parameters of the malaria model (2).
VariableDescriptionSHtNumber of susceptible humansEHtNumber of exposed humansIHtNumber of infectious humansRHtNumber of recovered humansLVtNumber of immature (eggs, larva, and pupa stages) vectorsSVtNumber of susceptible vectorsEVtNumber of exposed vectorsIVtNumber of infectious vectorsParameterDescriptionΠHHumans recruitment rateβHTransmission probability for susceptible humans per contactβVTransmission probability for susceptible vectors per contactσHProgression rate of infectious humansμHHumans natural mortality rateδHHumans disease-induced mortality rateγHRecovery rateωHLoss of immunity rateKVCarrying capacity of immature mosquitoesThe parameterΠH in the system (2) is the human recruitment rate, λHT,IVt,NHt is the temperature-dependent force of infection of healthy susceptible humans (after contact with infectious mosquitoes), and μH is the natural death rate of humans. The infection rate, λHT,IVt,NHt, is defined as(3)λHT,IVt,NHt=βHbVTIVtNHt,where βH is the susceptible human transmission probability of malaria infection per bite from an infectious mosquito and bVT 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 λHT,IVt,NHt [23–25]. The parameter, σH, represents exposed humans progression rate into the infectious class IH. Also, the parameter γH represents the rate at which individual in the IH class recovers (and moves to the recovered RH class). The parameter ωH represents the rate at which the recovered individuals in RH loose their infection-acquired immunity and move to the SH class. The parameter δH is the rate at which individuals in the infectious human class IH 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 ϕVT. We assume that the immature mosquito population (comprising of the larvae and pupae) is limited by the carrying capacity KV; this parameter is related to the amount of available nutrients and space [28–30]. Thus, ϕVT1−LVt/KVSVt+EVt+IVt represents the logistic growth rate for the immature mosquitoes. The parameter θVT^ 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 μLT^ (it is worth noting that, in this study, the dynamics of the three aquatic stages of the mosquito is modeled using the single compartment, LVt, 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 by(4)λVT,IHt,NHt=βVbVTIHtNHt,where βV 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 σVT is the temperature-dependent progression rate of exposed mosquitoes [35, 36]. The parameter μVT 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, bVT, and egg deposition rate, ϕVT, are given by(5)bVT=−0.00014T2+0.027T−0.322,ϕVT=−0.153T2+8.61T−97.7.Similarly, the temperature-dependent parametersσVT,θVT^, and μVT are defined in [37] as follows:(6)σVT=−0.00083T2+0.044T−0.487,(7)θVT^=BTpEAT^τEAT^,(8)μVT=−ln−0.000828T2+0.0367T+0.522.The temperature-dependent functionBT 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 pEAT^, and the development time from egg to adult mosquito is represented by τEAT^ [37]. The temperature-dependent function, BT, is defined as(9)BT=EFDTμVT,where EFDT is the number of eggs laid per female per day. Hence, BT, 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 1/μVT [37]. These temperature-dependent parameters are expressed as [37](10)EFDT=−0.153T2+8.61T−97.7,(11)pEAT^=−0.00924T^2+0.453T^−4.77,(12)τEAT^=1−0.00094T^2+0.049T^−0.552.Hence, substituting the temperature-dependent function (9)–(12) into (7), we have(13)θVT^=−0.153T2+8.61T−97.7−0.00924T^2+0.453T^−4.77×−0.00094T^2+0.049T^−0.5521−ln−0.000828T2+0.0367T+0.522.Lastly, the parameterμLT^ [38] is defined as(14)μLT^=18.560+20.6541+T^/19.7596.827−1.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].


## 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 NHt, is split into mutually exclusive subpopulations of individuals who are susceptible SHt, exposed EHt, infectious IHt, and recovered from the infection RHt so that(1)NHt=SHt+EHt+IHt+RHt.Similarly, the total mosquito population size at timet, denoted by NVt, is subdivided into immature mosquitoes (eggs, larvae, and pupae) LVt, susceptible mosquitoes SVt, exposed mosquitoes EVt, and infectious mosquitoes IVt.The simple model is given by the following deterministic, nonautonomous, system of nonlinear differential equations, where dot notation denotes the derivative with respect to timet and T and T^=+δT denote air and water temperature:(2)SH⋅t=ΠH+ωHRHt−λHT,IV,NHSHt−μHSHt,EH⋅t=λHT,IV,NHSHt−σH+μHEHt,IH⋅t=σHEHt−γH+μH+δHIHt,RH⋅t=γHIHt−ωH+μHRHt,LV⋅t=ϕVT1−LVtKVSVt+EVt+IVt−θVT^+μLT^LVt,SV⋅t=θVT^LVt−λVT,NHSVt−μVTSVt,EV⋅t=λVT,NHSVt−σVT+μVTEVt,IV⋅t=σVTEVt−μVTIVt.The flow diagram of malaria model (2) is shown in Figure 1; Table 1 describes the state variables and parameters.Figure 1
The malaria model (2) flow diagram.Table 1
Description of the variables and parameters of the malaria model (2).
VariableDescriptionSHtNumber of susceptible humansEHtNumber of exposed humansIHtNumber of infectious humansRHtNumber of recovered humansLVtNumber of immature (eggs, larva, and pupa stages) vectorsSVtNumber of susceptible vectorsEVtNumber of exposed vectorsIVtNumber of infectious vectorsParameterDescriptionΠHHumans recruitment rateβHTransmission probability for susceptible humans per contactβVTransmission probability for susceptible vectors per contactσHProgression rate of infectious humansμHHumans natural mortality rateδHHumans disease-induced mortality rateγHRecovery rateωHLoss of immunity rateKVCarrying capacity of immature mosquitoesThe parameterΠH in the system (2) is the human recruitment rate, λHT,IVt,NHt is the temperature-dependent force of infection of healthy susceptible humans (after contact with infectious mosquitoes), and μH is the natural death rate of humans. The infection rate, λHT,IVt,NHt, is defined as(3)λHT,IVt,NHt=βHbVTIVtNHt,where βH is the susceptible human transmission probability of malaria infection per bite from an infectious mosquito and bVT 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 λHT,IVt,NHt [23–25]. The parameter, σH, represents exposed humans progression rate into the infectious class IH. Also, the parameter γH represents the rate at which individual in the IH class recovers (and moves to the recovered RH class). The parameter ωH represents the rate at which the recovered individuals in RH loose their infection-acquired immunity and move to the SH class. The parameter δH is the rate at which individuals in the infectious human class IH 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 ϕVT. We assume that the immature mosquito population (comprising of the larvae and pupae) is limited by the carrying capacity KV; this parameter is related to the amount of available nutrients and space [28–30]. Thus, ϕVT1−LVt/KVSVt+EVt+IVt represents the logistic growth rate for the immature mosquitoes. The parameter θVT^ 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 μLT^ (it is worth noting that, in this study, the dynamics of the three aquatic stages of the mosquito is modeled using the single compartment, LVt, 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 by(4)λVT,IHt,NHt=βVbVTIHtNHt,where βV 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 σVT is the temperature-dependent progression rate of exposed mosquitoes [35, 36]. The parameter μVT 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, bVT, and egg deposition rate, ϕVT, are given by(5)bVT=−0.00014T2+0.027T−0.322,ϕVT=−0.153T2+8.61T−97.7.Similarly, the temperature-dependent parametersσVT,θVT^, and μVT are defined in [37] as follows:(6)σVT=−0.00083T2+0.044T−0.487,(7)θVT^=BTpEAT^τEAT^,(8)μVT=−ln−0.000828T2+0.0367T+0.522.The temperature-dependent functionBT 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 pEAT^, and the development time from egg to adult mosquito is represented by τEAT^ [37]. The temperature-dependent function, BT, is defined as(9)BT=EFDTμVT,where EFDT is the number of eggs laid per female per day. Hence, BT, 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 1/μVT [37]. These temperature-dependent parameters are expressed as [37](10)EFDT=−0.153T2+8.61T−97.7,(11)pEAT^=−0.00924T^2+0.453T^−4.77,(12)τEAT^=1−0.00094T^2+0.049T^−0.552.Hence, substituting the temperature-dependent function (9)–(12) into (7), we have(13)θVT^=−0.153T2+8.61T−97.7−0.00924T^2+0.453T^−4.77×−0.00094T^2+0.049T^−0.5521−ln−0.000828T2+0.0367T+0.522.Lastly, the parameterμLT^ [38] is defined as(14)μLT^=18.560+20.6541+T^/19.7596.827−1.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 uHt,uLt, and uVt, representing personal protection, larvaciding, and adulticiding. Thus, model (2) becomes:(15)dSHtdt=ΠH+ωHRHt−λHT,1−uHtIV,NHSHt−μHSHt,dEHtdt=λHT,1−uHtIV,NHSHt−σH+μHEHt,dIHtdt=σHEHt−γH+μH+δHIHt,dRHtdt=γHIHt−ωH+μHRHt,dLVtdt=ϕVT1−LVtKVSVt+EVt+IVt−θVT^+μLT^LVt−uLtLVt,dSVtdt=θVT^LVt−λVT,1−uHtNHSVt−μVTSVt−uVtSVt,dEVtdt=λVT,1−uHtNHSVt−σVT+μVTEVt−uVtEVt,dIVtdt=σVTEVt−μVTIVt−uVtIVt,where(16)λHT,1−uHtIVt,NHt=βH1−uHtbVTIVtNHt,λVT,1−uHtNH=βV1−uHtbVTIHtNHt.The factor1−uHt reduces the force of infection associated with humans and mosquitoes, respectively [50–52].This study aims to minimize the objective function defined as(17)Ju=∫0tfA1IH+B1LV+B2SV+B3EV+B4IV+C1uH+εC2uH2+C3uL+εC4uL2+C5uV+εC6uV2dt,subject to the differential equations (15), where tf 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 uHt,uLt,uVt. 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 A1Bj,j=1,…,4,Ck,k=1,…,6.We seek to find the optimal controlsuH∗,uL∗,anduV∗ such that(18)JuH∗,uL∗,uV∗=minUJuH,uL,uV.The optimal control solution is given as(19)uH∗=12εC2−C1NH∗+SV∗bVTβVλEV∗−λSV∗IH∗+IV∗bVTβHλEH∗−λSH∗SH∗−λRH∗RH∗,uL∗=λLV∗LV∗−C32εC4,uV∗=λEV∗EV+λIV∗IV∗+λSV∗SV∗−C52εC6.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 usedSH0=9384291,EH0=8200,IH0=1200,RH0=1000,LV0=11277924,SV0=44300,EV0=1452,andIV0=2800. The weight factors are chosen as A1=Bi=Cj=1.00,i=1,…,4,j=1,…,6. 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 of25.32°C 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 3.3330×104 infected humans in the absence of control compared to approximately 8.7797×103 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 2.4505×104 infected mosquitoes in the absence of control on the thirtieth day compared to approximately 120 with the application of control.Figure 2
Simulation results of model (15) with and without controls using the maximum mean temperature of 25.32°C in KwaZulu-Natal, South Africa. (a) Infectious individuals. (b) Infectious mosquitoes.
(a)(b)The corresponding time-dependent controls (uLt,uVt, and uHt) are depicted in Figure 3. The time-dependent controls uLt start at the upper bound of unity for the most part of the simulation period before reducing to the lower bound. Control uVt starts to be low at 0.165 but quickly rises to the upper bound for most of the simulation period. The control uHt starts at the upper bound for about eight days before gradually reducing to the lower bound.Figure 3
Simulation results of the control profile of model (15) using the maximum mean temperature of 25.32°C in KwaZulu-Natal, South Africa.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 controlsuLt,uVt, and uHt. We take only a single value for the total number of infected humans and mosquitoes and the mean optimal control level for each control uLt,uVt, and uHt 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 16.7°C,25.3°C. Figure 4(a) shows that there are fewer infected individuals with the application of the control measures.Figure 4
Simulations of the model (2) as a function of time plotting the monthly total number of infected humans and mosquitoes in KwaZulu-Natal, South Africa. (a) Total number of infected humans. (b) Total number of infected mosquitoes. (c) Optimal controls uHt,uLt, and uVt. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)Table 2
Values and ranges of the temperature-independent parameters of the model (2).
ParameterBaseline valueRangeReferenceΠH400/day10−800/dayVariableβH0.24/day0.072−0.64/day[25, 57]μH0.00004/day1/80×365−1/70×365/day[25, 57]δH0.0003454/day1.0×10−15−4.1×10−4/day[25, 57]σH12/365×3.04/day0.067−0.20/day[25, 57, 58]γH0.0023±0.0005/day0.0014−0.017/day[25, 57]ωH1.7×10−5/day5.5×10−5−1.1×10−2/day[25, 57]KV400005.0×101−3.3×106[29, 30]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 uLt and uVt on average are close to the upper bound throughout the year for all temperature values, while the time-dependent control uHt starts relatively low for low temperature and is relatively high for high temperature. It should be noted that the solution profile for uHt 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.Table 3
Cities in West, Central, and East African regions.
West African citiesCentral African citiesEast African citiesNZerekore, GuineaLubumbashi, DRCKigali, RwandaBamako, MaliTchibanga, GabonGulu, UgandaNiamey, NigerAti, ChadLodwar, KenyaFigure 5
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in NZerekore, Guinea; Bamako, Mali; and Niamey, Niger; in West Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 6
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with controls and (b, d, f) optimal controls uHt,uLt, and uVt, in Lubumbashi, DRC; Tchibanga, Gabon; and Ati, Chad; all Central Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 7
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with controls and (b, d, f) optimal controls uHt,uLt, and uVt, in Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya; in East Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figures5(a)–5(e) show that in West Africa, malaria burden increases for temperatures in the range 22.61°C,28.58°C, while in Central Africa (see the infected humans in Figure 6), malaria burden increases for temperatures in the range 16.68°C,27.92°C. Lastly, the range in East Africa is 19.04°C,26.75°C (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 uHt mimics the trajectory of the infected mosquitoes as the mosquitoes respond to variations in temperature (Figures 8–10).Figure 8
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in NZerekore, Guinea; Bamako, Mali; and Niamey, Niger; in West Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 9
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in Lubumbashi, DRC; Tchibanga, Gabon; and Ati, Chad; in Central Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 10
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya; in East Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Specifically, in West Africa, Figures5(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 uLt and uVt 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 uHt starts relatively low for low temperatures and is relatively high for high temperatures. It should be noted that the solution profile for uHt varies from about 0.25 to 0.7 as the temperature gradient varies across West Africa. The control uHt 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 Figure6). The average time-dependent controls uLt and uVt 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 uHt 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 uHt 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 Figure7). The average time-dependent control uHt varies from about 0.3 to 0.7, while the controls uLt and uVt 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 uHt 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 22.61°C,28.58°C, and in Central Africa, the temperature is 16.68°C,27.92°C. The temperature range in East Africa is 19.04°C,26.75°C, and in South Africa, the range is 16.7°C,25.3°C.We observed, in Figure11(a) in the absence of control, an increase in the total number of infected humans in West Africa until the midtemperature of 25.59°C 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.Figure 11
Simulations of the malaria model (2) as a function of time in temperature ranges suitable for mosquitoes: (a, c, e, g) the monthly total number of infected humans with control and (b, d, f, h) optimal controls uHt,uLt, and uVt, in West, Central, East, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 12
Simulations of the malaria model (2) as a function of time in the suitable temperature ranges. (a, c, e) The monthly total number of infected mosquitoes with control, and (b, d, f) optimal controls uHt,uLt, and uVt, in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)We observed in Figures11(b), 11(d), 11(f), and 11(h) that the average time-dependent controls uLt and uVt are close to the upper bound for all temperature values in all the four regions. In West Africa, the average control uHt increases until the midtemperature of 25.59°C before decreasing at a higher temperature, while in East, Central, and South Africa, the time-dependent control uHt 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.


## 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 16.7°C,25.3°C. Figure 4(a) shows that there are fewer infected individuals with the application of the control measures.Figure 4
Simulations of the model (2) as a function of time plotting the monthly total number of infected humans and mosquitoes in KwaZulu-Natal, South Africa. (a) Total number of infected humans. (b) Total number of infected mosquitoes. (c) Optimal controls uHt,uLt, and uVt. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)Table 2
Values and ranges of the temperature-independent parameters of the model (2).
ParameterBaseline valueRangeReferenceΠH400/day10−800/dayVariableβH0.24/day0.072−0.64/day[25, 57]μH0.00004/day1/80×365−1/70×365/day[25, 57]δH0.0003454/day1.0×10−15−4.1×10−4/day[25, 57]σH12/365×3.04/day0.067−0.20/day[25, 57, 58]γH0.0023±0.0005/day0.0014−0.017/day[25, 57]ωH1.7×10−5/day5.5×10−5−1.1×10−2/day[25, 57]KV400005.0×101−3.3×106[29, 30]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 uLt and uVt on average are close to the upper bound throughout the year for all temperature values, while the time-dependent control uHt starts relatively low for low temperature and is relatively high for high temperature. It should be noted that the solution profile for uHt 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.Table 3
Cities in West, Central, and East African regions.
West African citiesCentral African citiesEast African citiesNZerekore, GuineaLubumbashi, DRCKigali, RwandaBamako, MaliTchibanga, GabonGulu, UgandaNiamey, NigerAti, ChadLodwar, KenyaFigure 5
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in NZerekore, Guinea; Bamako, Mali; and Niamey, Niger; in West Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 6
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with controls and (b, d, f) optimal controls uHt,uLt, and uVt, in Lubumbashi, DRC; Tchibanga, Gabon; and Ati, Chad; all Central Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 7
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with controls and (b, d, f) optimal controls uHt,uLt, and uVt, in Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya; in East Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figures5(a)–5(e) show that in West Africa, malaria burden increases for temperatures in the range 22.61°C,28.58°C, while in Central Africa (see the infected humans in Figure 6), malaria burden increases for temperatures in the range 16.68°C,27.92°C. Lastly, the range in East Africa is 19.04°C,26.75°C (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 uHt mimics the trajectory of the infected mosquitoes as the mosquitoes respond to variations in temperature (Figures 8–10).Figure 8
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in NZerekore, Guinea; Bamako, Mali; and Niamey, Niger; in West Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 9
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in Lubumbashi, DRC; Tchibanga, Gabon; and Ati, Chad; in Central Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 10
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya; in East Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Specifically, in West Africa, Figures5(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 uLt and uVt 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 uHt starts relatively low for low temperatures and is relatively high for high temperatures. It should be noted that the solution profile for uHt varies from about 0.25 to 0.7 as the temperature gradient varies across West Africa. The control uHt 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 Figure6). The average time-dependent controls uLt and uVt 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 uHt 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 uHt 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 Figure7). The average time-dependent control uHt varies from about 0.3 to 0.7, while the controls uLt and uVt 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 uHt starts relatively low at 0.3 for low temperature and is relatively high at 0.7 for high temperature.


## 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 16.7°C,25.3°C. Figure 4(a) shows that there are fewer infected individuals with the application of the control measures.Figure 4
Simulations of the model (2) as a function of time plotting the monthly total number of infected humans and mosquitoes in KwaZulu-Natal, South Africa. (a) Total number of infected humans. (b) Total number of infected mosquitoes. (c) Optimal controls uHt,uLt, and uVt. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)Table 2
Values and ranges of the temperature-independent parameters of the model (2).
ParameterBaseline valueRangeReferenceΠH400/day10−800/dayVariableβH0.24/day0.072−0.64/day[25, 57]μH0.00004/day1/80×365−1/70×365/day[25, 57]δH0.0003454/day1.0×10−15−4.1×10−4/day[25, 57]σH12/365×3.04/day0.067−0.20/day[25, 57, 58]γH0.0023±0.0005/day0.0014−0.017/day[25, 57]ωH1.7×10−5/day5.5×10−5−1.1×10−2/day[25, 57]KV400005.0×101−3.3×106[29, 30]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 uLt and uVt on average are close to the upper bound throughout the year for all temperature values, while the time-dependent control uHt starts relatively low for low temperature and is relatively high for high temperature. It should be noted that the solution profile for uHt 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.Table 3
Cities in West, Central, and East African regions.
West African citiesCentral African citiesEast African citiesNZerekore, GuineaLubumbashi, DRCKigali, RwandaBamako, MaliTchibanga, GabonGulu, UgandaNiamey, NigerAti, ChadLodwar, KenyaFigure 5
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in NZerekore, Guinea; Bamako, Mali; and Niamey, Niger; in West Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 6
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with controls and (b, d, f) optimal controls uHt,uLt, and uVt, in Lubumbashi, DRC; Tchibanga, Gabon; and Ati, Chad; all Central Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 7
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected humans without and with controls and (b, d, f) optimal controls uHt,uLt, and uVt, in Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya; in East Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figures5(a)–5(e) show that in West Africa, malaria burden increases for temperatures in the range 22.61°C,28.58°C, while in Central Africa (see the infected humans in Figure 6), malaria burden increases for temperatures in the range 16.68°C,27.92°C. Lastly, the range in East Africa is 19.04°C,26.75°C (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 uHt mimics the trajectory of the infected mosquitoes as the mosquitoes respond to variations in temperature (Figures 8–10).Figure 8
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in NZerekore, Guinea; Bamako, Mali; and Niamey, Niger; in West Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 9
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in Lubumbashi, DRC; Tchibanga, Gabon; and Ati, Chad; in Central Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Figure 10
Simulations of the model (2) as a function of time: (a, c, e) the monthly total number of infected mosquitoes without and with control and (b, d, f) optimal controls uHt,uLt, and uVt, in Kigali, Rwanda; Gulu, Uganda; and Lodwar, Kenya; in East Africa. The mean monthly temperature is plotted in blue lines and is superimposed onto each figure.
(a)(b)(c)(d)(e)(f)Specifically, in West Africa, Figures5(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 uLt and uVt 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 uHt starts relatively low for low temperatures and is relatively high for high temperatures. It should be noted that the solution profile for uHt varies from about 0.25 to 0.7 as the temperature gradient varies across West Africa. The control uHt 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 Figure6). The average time-dependent controls uLt and uVt 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 uHt 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 uHt 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 Figure7). The average time-dependent control uHt varies from about 0.3 to 0.7, while the controls uLt and uVt 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 uHt 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 22.61°C,28.58°C, and in Central Africa, the temperature is 16.68°C,27.92°C. The temperature range in East Africa is 19.04°C,26.75°C, and in South Africa, the range is 16.7°C,25.3°C.We observed, in Figure11(a) in the absence of control, an increase in the total number of infected humans in West Africa until the midtemperature of 25.59°C 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.Figure 11
Simulations of the malaria model (2) as a function of time in temperature ranges suitable for mosquitoes: (a, c, e, g) the monthly total number of infected humans with control and (b, d, f, h) optimal controls uHt,uLt, and uVt, in West, Central, East, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 12
Simulations of the malaria model (2) as a function of time in the suitable temperature ranges. (a, c, e) The monthly total number of infected mosquitoes with control, and (b, d, f) optimal controls uHt,uLt, and uVt, in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)We observed in Figures11(b), 11(d), 11(f), and 11(h) that the average time-dependent controls uLt and uVt are close to the upper bound for all temperature values in all the four regions. In West Africa, the average control uHt increases until the midtemperature of 25.59°C before decreasing at a higher temperature, while in East, Central, and South Africa, the time-dependent control uHt 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.


## 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 22.61°C,28.58°C, and in Central Africa, the temperature is 16.68°C,27.92°C. The temperature range in East Africa is 19.04°C,26.75°C, and in South Africa, the range is 16.7°C,25.3°C.We observed, in Figure11(a) in the absence of control, an increase in the total number of infected humans in West Africa until the midtemperature of 25.59°C 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.Figure 11
Simulations of the malaria model (2) as a function of time in temperature ranges suitable for mosquitoes: (a, c, e, g) the monthly total number of infected humans with control and (b, d, f, h) optimal controls uHt,uLt, and uVt, in West, Central, East, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 12
Simulations of the malaria model (2) as a function of time in the suitable temperature ranges. (a, c, e) The monthly total number of infected mosquitoes with control, and (b, d, f) optimal controls uHt,uLt, and uVt, in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)We observed in Figures11(b), 11(d), 11(f), and 11(h) that the average time-dependent controls uLt and uVt are close to the upper bound for all temperature values in all the four regions. In West Africa, the average control uHt increases until the midtemperature of 25.59°C before decreasing at a higher temperature, while in East, Central, and South Africa, the time-dependent control uHt 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 NHt, is split into mutually exclusive subpopulations of individuals who are susceptible SHt, exposed EHt, infectious IHt, and recovered from the infection RHt so that(20)NHt=SHt+EHt+IHt+RHt.Similarly, the total mosquito population size at timet, denoted by NVt, is subdivided into immature mosquitoes (eggs, larvae, and pupae) LVt, susceptible insecticide-sensitive and insecticide-resistant mosquitoes SVSt,SVRt, exposed insecticide-sensitive and insecticide-resistant mosquitoes EVSt,EVRt, and infectious insecticide-sensitive and insecticide-resistant mosquitoes IVSt,IVRt. 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 timet and T and T^=T+δT denote air and water temperature, respectively):(21)SH⋅t=ΠH+ω1RHt−λHST,IVS,NHSHt−λHRT,IVR,NHSHt−μHSHt,EH⋅t=λHST,IVS,NHSHt+λHRT,IVR,NHSHt−σH+μHEHt,IH⋅t=σHEHt−γH+μH+δHIHt,RH⋅t=γHIHt−ωH+μHRHt,LV⋅t=ϕVT1−LVtKVSVSt+EVSt+IVSt+SVRt+EVRt+IVRt−θVT^+μLT^LVt,SVS⋅t=fθVT^LVt−λVST,NHSVSt−μVSTSVSt,EVS⋅t=λVST,NHSVSt−σVST+μVSTEVSt,IVS⋅t=σVSTEVSt−μVSTIVSt,SVR⋅t=1−fθVT^LVt−λVRT,NHSVRt−μVRTSVRt,EVR⋅t=λVRT,NHSVRt−σVRT+μVRTEVRt,IVR⋅t=σVRTEVRt−μVRTIVRt,where(22)λHST,IVSt,NHt=βHbVTIVStNHt,λHRT,IVRt,NHt=ηHRβHbVTIVRtNHt,λVST,NH=βVbVTIHtNHt,λVRT,NH=ηVRβVbVTIHtNHt.Furthermore, we assume thatσVRT=ηR1σVST and μVRT=ηR2μVST. The parameters ηHR,ηVR,ηR1, and ηR2 are modification parameters indicating the difference between the insecticide-sensitive and insecticide-resistant-related functions. We have assumed that ηHR=ηVR=ηR1=1; 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 ηR2=0.5. However, infected insecticide-resistant mosquitoes are more susceptible to insecticide [64, 66]. Note, σVST and μVST are given by the same functions as those in equations (6) and (8), respectively.The flow diagram of the model is depicted in Figure13, the associated state variables are described in Table 4, and parameters are described in Table 1.Figure 13
Flow diagram of the malaria-resistant model (21).Table 4
Description of the variables of model (21).
VariableDescriptionSHtNumber of susceptible humansEHtNumber of exposed humansIHtNumber of infectious humansRHtNumber of recovered humansLVtNumber of immature (eggs, larva, and pupa stages) vectorsSVStNumber of susceptible sensitive vectorsEVStNumber of exposed sensitive vectorsIVStNumber of infectious-sensitive vectorsSVRtNumber of susceptible resistance vectorsEVRtNumber of exposed resistance vectorsIVRtNumber of infectious-resistant vectors


### 5.1. Optimal Control Problem with Resistance Mosquitoes

The goal in this section is to minimize the objective function defined as(23)JRuHt,uLt,uVt=∫0tfA1IH+B1LV+B2SVS+SVR+B3EVS+EVR+B4IVS+IVR+C1uH+εC2uH2+C3uL+εC4uL2+C5uV+εC6uV2dt,that is, we seek to find the optimal controls uH∗,uL∗,anduV∗, such that(24)JRuH∗,uL∗,uV∗=minUJRuH,uL,uV.The optimal control solutionsuHt,uLt,uVt 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 controlsuHt,uLt,uVt in these regions of sub-Sahara Africa. We will also investigate the impact of monocontrol strategies where the time-dependent controls uHt,uLt,uVt 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 is22.16°C,28.58°C, and in Central Africa, the temperature is 16.68°C,27.92°C. The temperature range in East Africa is 19.04°C,26.75°C, and in South Africa, the range is 16.7°C,25.3°C.In Central, East, and South Africa, the mean total number of infected humans increases with increasing temperature (Figures14(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 uLt and uVt close to the upper bound and the average control uH increases with the mean monthly temperature. Similar dynamics are observed in the mosquito population, and the plots are given in Figure 15.Figure 14
Simulations of the malaria-resistant model (21) as a function of time: (a, c, e, g) the monthly total number of infected humans with control and (b, d, f, h) optimal controls uHt,uLt, and uVt, in West, Central, East, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 15
Simulations of the malaria-resistant model (21) as a function of time in the suitable temperature ranges. (a, c, e) The monthly total number of infected mosquitoes with control and (b, d, f) optimal controls uHt,uLt, and uVt, in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)


### 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: controlsuLtanduVt—only while setting uHt=0(ii)
Strategy B: controluHt—only while setting uLt=uVt=0We compare the solution profiles obtained from these strategies to the profile obtained from the control strategy when all the controlsuLt,uVt, and uHt are used (Figure 11).


#### 5.3.1. Monocontrol Strategies for Sensitive Mosquitoes: Infected Human Profiles

Comparing the mean total number of infected humans in Figures11 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 uLt,uVt, and uHt. 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.Figure 16
Simulations of the malaria model (2) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, and Central Africa.
(a)(b)(c)(d)We observed in Figure11(a) increase in the mean total number of infected humans in West Africa in the absence of control until the midtemperature of 25.59°C before decreasing at the highest temperature of 28.58°C. 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 uLt and uVt is close to the upper bound for all temperature values in West Africa. The profile for control uHt 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 Figure17). These strategies perform relatively poorer than when the controls uLt,uVt, and uHt are all used.Figure 17
Simulations of the malaria model (2) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, and Central Africa.
(a)(b)(c)(d)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 (Figures14 and 18). The average level of all the controls uLt,uVt, and uHt was used as in the previous sections. Strategy B which involves using only control uHt while setting controls uLt=uVt=0 performs better in reducing the number of infected human than Strategy A (which involves using controls uLtanduVt, while setting uHt=0).Figure 18
Simulations of the malaria-resistant model (21) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)The profile for infected mosquitoes is given in Figure19, and it shows that Strategy A performs better at reducing infected mosquitoes in the presence of insecticide-resistant mosquitoes.Figure 19
Simulations of the malaria-resistant model (21) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)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.


## 5.1. Optimal Control Problem with Resistance Mosquitoes

The goal in this section is to minimize the objective function defined as(23)JRuHt,uLt,uVt=∫0tfA1IH+B1LV+B2SVS+SVR+B3EVS+EVR+B4IVS+IVR+C1uH+εC2uH2+C3uL+εC4uL2+C5uV+εC6uV2dt,that is, we seek to find the optimal controls uH∗,uL∗,anduV∗, such that(24)JRuH∗,uL∗,uV∗=minUJRuH,uL,uV.The optimal control solutionsuHt,uLt,uVt 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 controlsuHt,uLt,uVt in these regions of sub-Sahara Africa. We will also investigate the impact of monocontrol strategies where the time-dependent controls uHt,uLt,uVt 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 is22.16°C,28.58°C, and in Central Africa, the temperature is 16.68°C,27.92°C. The temperature range in East Africa is 19.04°C,26.75°C, and in South Africa, the range is 16.7°C,25.3°C.In Central, East, and South Africa, the mean total number of infected humans increases with increasing temperature (Figures14(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 uLt and uVt close to the upper bound and the average control uH increases with the mean monthly temperature. Similar dynamics are observed in the mosquito population, and the plots are given in Figure 15.Figure 14
Simulations of the malaria-resistant model (21) as a function of time: (a, c, e, g) the monthly total number of infected humans with control and (b, d, f, h) optimal controls uHt,uLt, and uVt, in West, Central, East, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 15
Simulations of the malaria-resistant model (21) as a function of time in the suitable temperature ranges. (a, c, e) The monthly total number of infected mosquitoes with control and (b, d, f) optimal controls uHt,uLt, and uVt, in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)


## 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 is22.16°C,28.58°C, and in Central Africa, the temperature is 16.68°C,27.92°C. The temperature range in East Africa is 19.04°C,26.75°C, and in South Africa, the range is 16.7°C,25.3°C.In Central, East, and South Africa, the mean total number of infected humans increases with increasing temperature (Figures14(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 uLt and uVt close to the upper bound and the average control uH increases with the mean monthly temperature. Similar dynamics are observed in the mosquito population, and the plots are given in Figure 15.Figure 14
Simulations of the malaria-resistant model (21) as a function of time: (a, c, e, g) the monthly total number of infected humans with control and (b, d, f, h) optimal controls uHt,uLt, and uVt, in West, Central, East, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 15
Simulations of the malaria-resistant model (21) as a function of time in the suitable temperature ranges. (a, c, e) The monthly total number of infected mosquitoes with control and (b, d, f) optimal controls uHt,uLt, and uVt, in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)(e)(f)(g)(h)


## 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: controlsuLtanduVt—only while setting uHt=0(ii)
Strategy B: controluHt—only while setting uLt=uVt=0We compare the solution profiles obtained from these strategies to the profile obtained from the control strategy when all the controlsuLt,uVt, and uHt are used (Figure 11).


### 5.3.1. Monocontrol Strategies for Sensitive Mosquitoes: Infected Human Profiles

Comparing the mean total number of infected humans in Figures11 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 uLt,uVt, and uHt. 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.Figure 16
Simulations of the malaria model (2) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, and Central Africa.
(a)(b)(c)(d)We observed in Figure11(a) increase in the mean total number of infected humans in West Africa in the absence of control until the midtemperature of 25.59°C before decreasing at the highest temperature of 28.58°C. 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 uLt and uVt is close to the upper bound for all temperature values in West Africa. The profile for control uHt 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 Figure17). These strategies perform relatively poorer than when the controls uLt,uVt, and uHt are all used.Figure 17
Simulations of the malaria model (2) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, and Central Africa.
(a)(b)(c)(d)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 (Figures14 and 18). The average level of all the controls uLt,uVt, and uHt was used as in the previous sections. Strategy B which involves using only control uHt while setting controls uLt=uVt=0 performs better in reducing the number of infected human than Strategy A (which involves using controls uLtanduVt, while setting uHt=0).Figure 18
Simulations of the malaria-resistant model (21) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)The profile for infected mosquitoes is given in Figure19, and it shows that Strategy A performs better at reducing infected mosquitoes in the presence of insecticide-resistant mosquitoes.Figure 19
Simulations of the malaria-resistant model (21) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)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.


## 5.3.1. Monocontrol Strategies for Sensitive Mosquitoes: Infected Human Profiles

Comparing the mean total number of infected humans in Figures11 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 uLt,uVt, and uHt. 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.Figure 16
Simulations of the malaria model (2) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, and Central Africa.
(a)(b)(c)(d)We observed in Figure11(a) increase in the mean total number of infected humans in West Africa in the absence of control until the midtemperature of 25.59°C before decreasing at the highest temperature of 28.58°C. 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 uLt and uVt is close to the upper bound for all temperature values in West Africa. The profile for control uHt 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 Figure17). These strategies perform relatively poorer than when the controls uLt,uVt, and uHt are all used.Figure 17
Simulations of the malaria model (2) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, and Central Africa.
(a)(b)(c)(d)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 (Figures14 and 18). The average level of all the controls uLt,uVt, and uHt was used as in the previous sections. Strategy B which involves using only control uHt while setting controls uLt=uVt=0 performs better in reducing the number of infected human than Strategy A (which involves using controls uLtanduVt, while setting uHt=0).Figure 18
Simulations of the malaria-resistant model (21) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)The profile for infected mosquitoes is given in Figure19, and it shows that Strategy A performs better at reducing infected mosquitoes in the presence of insecticide-resistant mosquitoes.Figure 19
Simulations of the malaria-resistant model (21) as a function of time plotting the average control levels for controls uL,uV, and uH in West, Central, East, Central, and South Africa.
(a)(b)(c)(d)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 16.7,25°C, individuals in the community should use both mosquito-reduction strategy and personal protection against mosquito bites; however, when temperatures lie within the range 26,34°C, 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 ratiof 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 usingBacillus 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 16.7,25°C or they should emphasize mosquito-reduction strategy in the community (or household) ahead of personal protection for higher mean monthly temperatures in the range 26,34°C.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)
22.61°C,28.58°C in the three West African cities.(b)
16.68°C,27.92°C in the three Central African cities.(c)
19.04°C,26.75°C in the three East African cities.(d)
16°C,25°C in KwaZulu-Natal, South Africa.(ii)
Malaria burden in the presence of temperature variations can be decreasedvia 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.



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*Source: 5056432-2020-11-28.xml*