ASSESSING THE IMPACT OF CONTROL INTERVENTIONS AND AWARENESS ON MALARIA: A MATHEMATICAL MODELING APPROACH

Malaria is a vector-borne contagious disease which remains a public health burden for decades. It is highly endemic in Sub-Saharan African, and an estimated number of two hundred and twenty-eight million cases was reported in 2018 around the world. We develop and examine a deterministic model which describes the transmission dynamics of malaria between mosquito and human populations and examine the impacts of control interventions with their level of awareness on its control. The malaria-free equilibrium of the model is shown to be locally asymptotically stable if the threshold quantity R0 < 1. We study the stability of the endemic equilibrium and the conditions for the existence of backward bifurcation are presented. A sensitivity analysis was done to measure the outcome of the control intervention parameters on the reproduction number. The result shows that residual spray and bed-net usage are the most important parameter on the reproduction number. A numerical simulation was carried out and the result shows that combining bed-net usage and residual spray will reduce the burden of malaria faster. Particularly, results suggest that awareness and proportion of bed-net usage and residual spray should be priorities and increased to at least 75% for the possibilities of eliminating malaria.


INTRODUCTION
Among the deadliest diseases with a highly challenging health burden in the tropical regions is malaria. It is a vector-borne transmissible disease that is highly endemic in Sub-Saharan Africa, especially in improvised and low hygienic environments [1,2,3]. Despite continuous research about malaria for the past decades, it remains a major public health burden for which it was declared endemic in one hundred and nine countries in 2008 [4]. An estimated number of two hundred and twenty-eight million cases were reported in 2018 around the world. As stated in the world malaria report released by the World Health Organization (WHO) in 2019, about four hundred and five thousand deaths were recorded [2,5]. Approximately three hundred to five hundred million cases occur globally annually, with over one million deaths yearly. The burden of malaria is tremendous in the Sub-Saharan African region such that eighty percent of these cases and ninety percent of these deaths occur in this region [6,7,8]. About 78% of deaths occurs in children below age five [1]. Malaria is caused by an infection with the protozoan parasite of genus Plasmodium. In humans, five different species of Plasmodium can cause infection, namely Plasmodium falciparum, Plasmodium malariae, Plasmodium ovale, Plasmodium vivax, and Plasmodium knowlesi [1,9,10]. It is transferred to humans through the bite of an infected female Anopheles mosquito [1]. After an effective bite, the parasite multiplies in the human liver and bloodstream to develop into an infectious form. After the incubation period of the disease (which is within 9-14 days), human begins to show symptoms.
The symptoms characterized by malaria include; rise in body temperature, headache, cold, shivering, pain, anemia, fatigue, and vomiting among other symptoms [2,11]. To present, there is no effective vaccine against malaria. However, it is preventable and curable. Treatments such as the use of anti-malaria drugs have been in use for decades. Though, some existing anti-malaria medications are losing their effectiveness as a result of the drug resistance evolved in the parasite [2,12]. Some control techniques have been employed to prevent the infection of malaria in the human population, such as insecticide-treated mosquito nets (ITNs), indoor residual spraying (IRS) and bed-nets use [4,11]. The occurrence of malaria has been increasing lately as a result of parasite drug resistance and mosquito insecticide resistance [9,13], thus causing the disease to remain endemic in many regions.
Many researchers have employed mathematical models from different fields to investigate the spread of infectious diseases in a given population using diverse approaches (see [14,15,16,17,18,19,20]). Modeling of malaria has helped in understanding the transmission dynamics of this disease, including appropriate control strategies to mitigate it. Sir Ross published the first model in 1911 to demonstrates the development of malaria [2,21]. Over the years, this model has been used in the past due to its simple nature. However, as the burden of malaria increases over the decades, numerous researchers have modified the existing model of Sir Ross, and have developed models by introducing different factors, parameters, and variables to further understand its dynamics spread in the population. For example, the model by Sir Ross's was improved by incorporating; the latent period of infection [2,22]; the heterogeneity of human and mosquito [23,24]; immunity factor [2,25,26]; susceptibility to malaria in the host population [27,28]; a model with exposed human and exposed mosquito [12,29] and recovered human [23,30], among many other study. Furthermore, some modeler has developed models that integrate the effect of climate change (such as temperature and rainfall), and seasonality [31,32,33,34,35]. In addition, some researchers have investigate the transmission dynamics of malaria within-host level [36,37,38,39,40]. A mathematical model has been utilized in the decision-making process of intervention programs for the prevention and control of malaria in the populace. Thus, a huge number of researchers have employed mathematical models to predict effective control of malaria using optimal control theory. These studies assess the optimal interventions strategies required for effective control of malaria. Examples of these studies includes [11,41,42,43,44,45]. Although numerous researches have been performed on the spread and control of malaria, however, it remains a health burden in some regions, especially Sub-Saharan Africa. Thus, continuous efforts must be encouraged in modeling the dynamics of this disease and its control in the endemic regions. Among many models that have been developed, we discuss the methods, results, and limitation of few studies that stands as the foundation to the model proposed in this work.
In [30], the authors proposed a five compartmental deterministic model to describe the transmission dynamics of malaria between the mosquito and human populations. The model allows the transmission from the recovered humans due to incomplete immunity to reinfection. In this study, the authors employ both the standard incidence and the mass action incidence malaria model to evaluate the effect of incomplete immunity to reinfection in the spread of the disease in the human population. The result from this study shows that the standard incidence model shows the phenomenon of backward bifurcation as a result of the reinfection of individuals who recovered from malaria. Furthermore, the result shows that this phenomenon can be eliminated by using the mass action incidence instead of the standard incidence function. Thus, the global dynamics of malaria disease with reinfection is determined by the threshold quantity reproduction number. In addition, numerical simulations result suggests that increasing the rate of incomplete protection of recovered humans and decreasing the life expectancy of mosquitoes, will increase the region of backward bifurcation. It must be noted that the model developed and analyzed in [30] did not consider the exposed humans and the aquatic stage of the mosquito population. Also, it is not always true that a recovered individual must be re-infected by an infected mosquito before progressing to the infected human population. Immune human individuals progress to the susceptible population following the loss of their immunity [11,29].
A study on the effect of bed-net usage on malaria commonness is presented in [46]. The authors formulated and analyzed the basic susceptible -infectious (SI) model, consisting of human and mosquito populations, to examine the effect of bed-net use on the spread of malaria infection in the population. The model incorporates the effect of human behavior such as the lack of effective usage of bed-net, on the spread of the disease. Results from the model analysis show the existence of backward bifurcation. This implies that reducing the reproduction number only is not sufficient in eliminating the disease, except when the initial cases of malaria infection in both populations are insignificant. Furthermore, results illustrate that bed-net usage decreases the reproduction number. Specifically, results reveal that if seventy-five percent of the human population effectively uses the bed-net, then malaria may be eradicated. The limitation of this model includes the absence of the exposed individuals who can transfer the malaria infection to mosquitoes when they come for their blood meal. Also, since the presence of backward bifurcation nullifies the guarantee that reduction of the threshold quantity below unity will eliminate the disease, it is important to encourage additional strategies like indoor residual spraying, and early treatment of infected individuals to lessen the burden of malaria in the population. In addition, a model that accounts for the control of immature mosquitoes in the aquatic stage will help in studying the effective control of malaria in the population.

MODEL FORMULATION
Motivated by the model presented in [30,41,46], we present a deterministic model to investigate the impact of awareness about preventive measures and treatment care on the spread of malaria in a population. Malaria is a disease that is preventable, treatable, and curable in the human population. Among many other preventative measures against malaria is the use of insecticides treated nets (henceforth refer to as bed-nets); use of residual spray; intermittent preventive treatment; and use of repellent that contains diethyltoluamide [41,47]. However, if an individual is infected with malaria, the use of anti-malaria medications has been shown to regulate malaria in humans [48]. Sadly, the incidence of malaria is increasing due to drug and mosquito insecticide resistance in regions where malaria is endemic [9,13]. Preventive and treatment healthcare has shown to lessen the burden of malaria in the endemic regions [48,49], thus increasing the chance of eradicating malaria in these regions, it is important to increase the awareness or educational campaign about prevention and treatment strategies against this disease. Individuals' awareness about malaria and its mode of transmission will allow the human population to take precautionary measures such as personal prevention against mosquito bites, and control of mosquito population. As a result of this awareness, it is expected that the human population embraces prevention and treatment of malaria, thus leading to a decrease in the cases of malaria. In this study, we incorporate a saturated function for the level of awareness on bednets usage, residual spray, and treatment of infected humans in the model to study the outcome of preventive and treatment measures enhanced by awareness on the dynamic transmission of malaria among humans. We denote the awareness about the use of bed-nets as A 1 , awareness about residual spray as A 2 and awareness about the mode of treatment for malaria disease as A 3 , Transmission of malaria can only occur between two interacting hosts namely human and mosquito, thus we group the interacting hosts into human and mosquito populations. The human population is further sub-group into susceptible S h , exposed E h , infectious I h , and recovered R h , based on their disease status. There are mainly four stages in the development of mosquitoes namely; egg, larva, pupa, and adult stage. However, for simplicity purposes, we classify the mosquito population into the immature and mature stage, such that (egg, larva, and pupa) are classified as immature mosquito population denoted by M m . Furthermore, we sub-divide the mature mosquito population into susceptible S m , exposed E m , and infectious I m . Hence, the total human and mosquito populations at time t ≥ 0 are given as N h (t) = S h + E h + I h + R h , and N m (t) = M m + S m + E m + I m respectively. The susceptible human population is produced through birth or immigration at the recruitment rate π h , followed by the loss of immunity of recovered humans at a rate ω. All human populations are reduced by natural mortality with a constant rate µ h . The susceptible human population is more reduced by the force of infection rate λ h (A 1 + A 2 ) (defined in 4), following an effective bite by an infected mosquito, thus susceptible humans moved to the exposed human population after infection. The exposed human population decreased by the progression rate of exposed individuals to the infectious population at a rate σ h . The infectious population is generated by the progression of exposed humans to their infectious state at a rate σ h and is reduced by disease-induced death (death caused by malaria) at a rate δ h , and recovery of infectious individuals at a rate τ h (A 3 ). The recovery rate of infectious humans is model as a function of awareness such that where the recovery rate of individuals is denoted by τ h , and τ max q is the recovery rate due to awareness. Lastly, on the human population, the recovered human population is produced by the recovery rate of infectious humans. The recovered human population is depopulated as a result of the loss of immunity at a rate ω. The mosquito population is grouped into four subpopulations namely immature, susceptible, exposed, and infectious mosquito populations. The immature mosquito population is generated by mosquito egg deposition at a rate π m (A 2 ). This population is reduced due to the development of immature mosquitoes at a rate φ and mosquito death at rate µ m (A 2 ). We model the egg deposition rate as a function of awareness such that where π max and π min are the maximum and minimum egg deposition rates of mosquitoes. Residual spraying is expected to reduce the recruitment of mosquitoes to its minimum rate such that if p = 0, the egg deposition rate of mosquito will remain in its maximum value π max . The susceptible mosquito population is created by the maturation rate of immature mosquitoes. All the mature mosquitoes (susceptible, exposed, and infectious) are reduced by mosquito mortality with a rate µ m (A 1 + A 2 ). After effective contact with an infected human, the susceptible mosquito population is further reduced by the force of infection rate λ m (A 1 + A 2 ) (defined in 4), and thus move to the exposed mosquito population. This population is decreased by the movement rate of exposed mosquitoes to the infectious population at a rate σ m . The infectious mosquito population is generated by the progression of exposed mosquitoes to their infectious state at a rate σ m . The awareness compartment A is populated by a saturated function F(I h ), where a 0 , a 1 , and a 2 are information growth rate. This class is reduced by fading of memory about awareness, or human sentiment about awareness information at a rate a 3 . The saturated function F(I h ) depends on the infectious human population density since the awareness about the disease and the need to protect individuals is proportional to the number of infected humans. This kind of function has been used in [50] to model the role of information in disease prevalence. Following the above model formulation descriptions and assumptions, the deterministic model used in studying the dynamics of malaria in this study is given as The description of the model variables and parameters are presented in Table 1 and Table 2 respectively, while the schematic illustration is provided in Figure 1.
Following the approach in [46] and [51], we define the forces of infection λ h (A 1 + A 2 ) and λ m (A 1 + A 2 ) as a function of the level of awareness such that where β hm is the likelihood that a susceptible individual will be infected due to a bite by infectious mosquitoes, β mh is the likelihood that a susceptible mosquito will be infected as a result of contact with infectious human, and ε(A 1 + A 2 ) is the contact rate of humans and mosquitoes, which is dependent on awareness about bed-net usage and residual spray respectively. The contact rate associated with awareness on bed-net usage ε(A 1 ) is given by a decreasing function where b represents the proportion of bed-net usage, and ε min and ε max are the minimum and maximum mosquito biting rate respectively. Note that bed-net usage is expected to decrease the contact rate to its minimum such that if the proportion of bed-net usage b = 0, then transmission would be at its maximum level ε max . In addition, the saturated function for the awareness of bed-net usage reduces the contact rate as the awareness increases. Similarly, the contact rate associated with awareness on the use of outdoor or indoor residual spray ε(A 2 ) is given by a decreasing function where ε min and ε max are the minimum and maximum mosquito biting rate respectively, and p is the proportion of residual spray. Note that residual spraying is expected to reduce the contact rate to its minimum such that if the proportion of residual spray p = 0, then transmission would be at its maximum level (ε max ). The saturated function for the awareness of residual spraying Similarly, the mosquito's death rate as a result of residual spray is given as where µ m is the natural death rate of mosquito, while µ max b and µ max p are the death rate as a result of insecticide on bed-nets and residual spray respectively.   Proof (3) is written as By using the integrating factor method, the above expression is further given as Hence,  so that, In the same way, it can be shown that Furthermore, for the malaria model (3) to be mathematically and epidemiologically meaningful, it is necessary to analyze system (3) in a biologically feasible region Ω = Ω h × Ω m ∈ and Using the standard technique (see [50,52]), the feasible region Ω can be shown to be positively invariant. Hence, all the solutions are in the feasible region Ω where the malaria model (3) is said to be mathematically and epidemiologically wellposed [53,52]. We claim the following result in the theorem below

MODEL ANALYSIS
Here, we investigate the existence and stability of the steady states, and the nature of bifurcation exhibited by system (3) is examined. The model presented in (3) , 0, 0, 0 We compute the reproduction number R 0 to study the stability of the model. Using the approach and notations in [52,54], the matrix F (new infections) and matrix V (transition terms) are respectively given as The next generation matrix (NGM) with large domain K L = FV −1 is given below as It is obvious from the model equation (3) that, there are only two states-at-infection among the four infected states. This can also be seen by looking at matrix F and observing that the entire second and fourth row contains zeros. Hence, the NGM K for the small domain is therefore two-dimensional. Thus, using the approach of [55] with an auxiliary matrix E, the NGM K is obtained as Thus, it follows that the reproduction number for the system (3), which is the spectral radius of K given by R 0 = ρ(K), is obtained as where .
The reproduction number R 0 is a threshold quantity that characterizes the average number of new secondary infections generated by a single infected individual during an infectious period, in a completely susceptible population [41,54]. Consequently, the threshold quantity given in equation (11) represents the average number of malaria infections that one malaria-infected individual can reproduce in an entirely susceptible population. Using Theorem 2 in [56], the local stability of the malaria-free equilibrium M 0 is summarized in the theorem below. Proof. To establish Theorem 3, we obtain the Jacobian matrix of system (3) at malaria free- Such that we can show that all the eigenvalues of J (M 0 ) are negative. The first five eigenvalues are obtained as, −µ h , −k 3 , −k 4 , −k 6 and −a 3 . Thus, the remaining four eigenvalues are obtained from the sub-matrix S , given as As stated by the Routh-Hurwitz criterion, the matrix S will be real and negative if

Existence of endemic equilibria and backward bifurcation.
Here, we examine the possibilities of the existence of endemic equilibria and a backward bifurcation. A model is known to exhibit the phenomenon of backward bifurcation when a small positive unstable equilibrium appears while the disease-free equilibrium and a larger positive equilibrium are locally asymptotically stable when the threshold quantity is less than unity. In other words, this phenomenon is possible when the stable disease-free equilibrium coexists with a stable endemic equilibrium, under some given values for which the reproduction number is less than unity.
The endemic equilibria denoted by the steady-state solution in the presence of malaria, and is obtained as , with the force of infections given as (15) substituting equation 14 and the value of λ * * m (A 1 + A 2 ) into (15) and expanding in λ * * h (A 1 + A 2 ) results to the following polynomial equation where the polynomial coefficients z i for i = 1..., 3 are given as where: , and P 6 = P 1 π h µ m (A 1 + A 2 ) + P 5 . From the polynomial (16) above, the coefficient z 1 is always positive, and the constant term z 3 is negative or positive depending on the value of R 0 . This implies that, if R 2 0 > 1, z 3 is negative and if R 2 0 < 1,then z 3 is positive. Thus, the following result hold.
Hence, backward bifurcation will occur for the value R c 0 such that R 0 < 1. The result is summarized in the theorem below. Following the result above, the backward bifurcation phenomenon implies that the epidemiological condition of having the reproduction number less than unity to eradicate a disease although necessary is no longer enough for disease eradication. Thus, to effectively control malaria in the population, additional control measures will be needed to enable epidemic control. That is, the condition R 0 < R c 0 < 1 must be satisfied.

NUMERICAL RESULTS AND DISCUSSION
We examine the effect of control interventions (bed-net usage, residual spray, and treatment) and their level of awareness on the dynamics of malaria. To accomplish this, we performed a sensitivity analysis to investigate the impact of control interventions on the reproduction number. Furthermore, we simulate the proposed model (3) under different scenarios, using the baseline parameter values as given in Table 3, except otherwise stated.
4.1. Impact of interventions on R 0 . Since the threshold quantity R 0 given in (11) determines the control of malaria in the population (except for scenario where the bifurcation phenomenon occurs), we assess the impact of the interventions (bed-net usage, insecticide residual spray, and treatment) on the reproduction number R 0 . To accomplish this, we use the normalized forward sensitivity indices to investigate the relationship of each parameter on R 0 . Using the method in [59,60], the normalized forward sensitivity index X R 0 i for each of the intervention parameter {b, p, q ∈ i}, is defined as By using the formula presented in (19), the numerical values for the normalized forward sensitivity indices of the three intervention parameters are given in Table 3. It must be noted that, p < 0, and X R 0 q < 0. This implies that an increase in the respective intervention parameters will reduce the value of the reproduction number. For instance, increasing the number of individuals who use bed-net will reduce the reproduction number and vice versa.
In Figure 2, we simulate the effect of each intervention on the reproduction number. Figure 2 shows a decrease in the reproduction number with increasing interventions as expected. However, Figure 2(c) shows that the proportion of treatment is less significant on the reproduction number. Thus, control interventions such as residual spray, and bed-net usage should be prioritized in reducing the burden of malaria in the population.
To examine the effect of control intervention and awareness on malaria burden, we obtained some contour plots for the reproduction number R 0 , as a function of control interventions and their respective level of awareness. As shown in Figure 3, an increase in control interventions with the level of awareness reduces the reproduction number. Specifically, Figure 3(          shows that increase in residual spray and its level of awareness reduces the infected human and mosquito population respectively. In Figure 4(c) and Figure 4(d), it is obvious that residual spray reduces the burden of malaria faster than bed-net usage. This supports the result from the sensitivity analysis as presented in Table 4. From the result, it is noted that increasing the The parameter values used are as given in Table 3. The parameter values used are as given in Table 3 proportion of bed-net usage or residual spray to 100% will effectively reduce the burden of malaria to the barest minimum. Since it is not realistic for the total human population to use a single control strategy, we simulate the impact of double control interventions on the infected population in Figure 5. We simulate the effect of treatment and its level of awareness on the total infected human and mosquito population in Figure 4(e) and Figure 4(f) respectively. The result shows that there is an insignificant effect of treatment on the total infected population.
This result is similar to the one in Figure 3(c). Since bed-net usage and residual spray are preventive healthcare, the results from Figure 4 show that preventive healthcare is better than treatment healthcare. Thus, to reduce the burden of malaria, it is important to facilitate the use of preventive healthcare such as bed-net usage or residual spray among the populace.
In Figure 5 we simulate the effect of double control interventions with their respective level of awareness on the infected human and mosquito population. Overall, the result shows that combined control interventions reduce the total human and mosquito population faster than the use of single control intervention. Particularly, it is obvious from Figure 5(a) and Figure   5(b) that a combination of bed-net usage and the residual spray reduces the burden of malaria faster than any other double combined intervention. In Figure 6, we simulate the effect of all the control interventions with their respective level of awareness on the infected human and mosquito population. It is noted that the result is alike to the one presented in Figure 5(a) and Figure 5(b). Thus, it is recommended that awareness and proportion of bed-net usage and residual spray should be priorities to mitigate the burden of malaria in the population. In addition, it is recommended that the proportion of bed-net usage and residual spray should be increased to at least 75% to eliminate malaria in the population.

CONCLUSIONS
Malaria is one of the deadliest diseases with highly challenging health issues in tropical regions. It is highly endemic in Sub-Saharan Africa, especially in an improvised and low-hygiene environment. Malaria is an infectious disease that is preventable, treatable, and curable in the human population, thus, understanding the influence of mitigation strategies such as bed-net usage, residual spray and treatment can help us inform public health policy. In this study, we developed a deterministic model to investigate the dynamical features of malaria in the population, and we assessed the impacts of control interventions with their level of awareness to effectively mitigate the burden of the disease. We obtained the malaria-free equilibrium and the endemic equilibrium of the model. The malaria-free equilibrium is shown to be locally asymptotically stable whenever the reproduction number R 0 is less than unity, and unstable otherwise. Epidemiologically, this result implies that malaria can be effectively controlled in the population whenever the reproduction number is less than unity if the initial sub-populations of the infected compartments of the model system (3) are small enough. In other words, malaria can be effectively controlled in the population if the control strategies implemented can reduce and maintain the reproduction number below unity. We obtained the endemic equilibrium of the model, and the criteria for the existence of the phenomenon of bifurcation are investigated.
The model is said to undergo a backward bifurcation phenomenon when the critical value of R c 0 < R 0 < 1. The existence of the backward bifurcation phenomenon suggests that reducing reproduction number R 0 below unity is not enough to eliminate malaria, thus, a combination of control strategies may be needed to control the spread of malaria in the population. A sensitivity analysis was performed to examine the effect of bed-net usage, residual spray, and treatment on the reproduction number. The result shows that increase in any of the control interventions will decrease the reproduction number. Furthermore, the result shows that residual spray is the most influential intervention in reducing the reproduction number as presented in Table 4. Following this result, we simulate the effect of control interventions and their level of awareness on the infected human and mosquito population, under three different scenarios. The overall result from the numerical simulations is that combining bed-net usage and residual spray as a preventive healthcare measure will reduce the burden of malaria faster. Particularly, results suggest that awareness and proportion of bed-net usage and residual spray should be priorities and increased to at least 75% for the possibilities of eliminating malaria. Thus, we recommend that malaria control programs should focus on increasing bed-net usage to reduce the bite of humans by an infected mosquito. In addition, the use of residual spray should be encouraged to reduce the mosquito population. All these can be achieved by increasing awareness about preventive care for malaria and increasing the distribution of bed-net and residual spray in regions where malaria is endemic.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.