Impact of Awareness to Control Malaria Disease: A Mathematical Modeling Approach

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Introduction
Malaria is an ancient disease with challenging health issues.
e tropical regions such as Africa, Asia, and America are favorable for the rapid spread of this disease [1]. In 2018, there are estimated two-hundred and twenty-eight million cases of malaria around the world. is deadly disease is the root cause of the death of four-hundred-five thousand people according to the World Health Organization (WHO) 2019 world malaria report [2]. is disease is originated by the plasmodium parasite. e transmission of this infection to human body is by the bite of a female mosquito. Medical symptoms such as a rise in the body temperature, fatigue, pain, shivering, and sweats may occur within a few days after an infected mosquito bite. Till the time, there is no effective vaccine developed and some existing antimalarial drugs are losing their effectiveness due to the drug resistance evolved in the parasite [3]. e literature on the mathematical model for vectorborne disease likewise malaria is vast. e first published model demonstrating the life cycle of the malaria parasite was developed by Sir Ross [4]. e model proposed by Sir Ronald Ross is one of the simplest models, known as the classical Ross model in the literature. It demonstrates the crosstalk between the number of mosquitoes and the proportion of bite that produced infection in the human body.
is model was frequently used in the past due to its simplicity. However, the simplicity of the mathematical model is at the cost of limitations in the analysis. erefore, considering the great challenge of malaria disease, several researchers have developed/modified existing models by introducing different factors/parameters in the model. e mathematical modeling of infectious disease has proved to play an important role in understanding the insights of the transmission dynamics and appropriate control strategies [5]. In past, several mathematical models have been proposed to analyze this deadly disease. e extension of Sir Ross's model includes the modification/addition of different important factors likewise, latent period of infection [6], immunity factor [7], the heterogeneity of human and mosquito [8,9], susceptibility to malaria in host population [10,11], exposed human and mosquito [3,12], and recovered human [8,13], among others.
Macdonald has introduced the effect of exposed mosquitoes by employing a separate differential equation catering the rate of change of exposed mosquitoes [7]. Similarly, Anderson and May proposed to add the rate of change of exposed human in the model [6]. e comparison of two epidemiological models of immunity to malaria shows that different characterizations of immunity, boosted by exposure to infection, generate qualitatively different results [8]. Nagwa and Shu analyzed the deterministic differential equation model for endemic malaria in the presence of the variable host and parasite population [13]. eir results suggested that disease is persistent if the threshold parameter exceeds the barrier of magnitude unity; otherwise, a disease-free equilibrium always exists. In a similar study, Chitnis and co-authors have presented the bifurcation analysis of the reproductive number (RN) [14]. RN is the number of secondary infections that one infectious individual would create over the time course of the disease period, provided that total population except infectious is susceptible [15]. Lashari and co-authors formulated the vector-borne disease environment in the form of an optimal control problem. ey have introduced three different control parameters including personal protection, disease medical treatment, and mosquito reduction strategies [3]. Ozair and co-authors have analyzed the transmission dynamics of malaria disease with a nonlinear incident rate [16]. Addawe and Lope proposed to divide the human population into two compartments: preschool (0-5 years) and over five age [17]. eir findings verify the existing results that asymptotical stability is guaranteed with RN magnitude less than unity. Mishra and co-authors, Samanta and co-authors, and Greenhalag and co-authors have proposed and analyzed a nonlinear mathematical model to assess the effect of awareness by media on the prevalence of infectious disease [18][19][20]. Cai et al. introduced a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations [21]. Sung Chan and co-authors have studied the vector-bias mathematical model and considered two different incidence areas: a high transmission area and a low transmission area [22]. Recently, Sung Chan and co-authors have developed a new transmission model to evaluate the rate of malaria relapse infections in the northern part of Korea and to examine its effect at the population-level on radical cure [23].
In this article, a deterministic vector-borne disease model is proposed. Previous studies suggested that prevention is a control parameter for such infectious diseases.
us, it shall be helpful to add awareness terms in the mathematical model of the disease. e whole infected host population is divided into two groups, aware and unaware infected individuals. We analyzed the model to study the impact of awareness programs conducted by awareness campaign through medical staff on the spread of malaria disease. In the modeling process, it is assumed that the growth rate of the cumulative density of awareness program will increase with an increase in the unaware infected individuals in the host population. We also assumed that both infected classes can spread disease when the mosquitoes contact them, but the aware class has very low chance to spread disease due to awareness campaigns. It is further assumed that due to awareness, the contact rate of infected mosquito interaction with aware humans will be reduced. e performance of the proposed model is evaluated by comparing the result with previous models. e result of the proposed model suggests that disease-free equilibrium is achieved earlier than the existing model. e rest of the article is as follows. In section 2, the formulation of a mathematical model is presented. Section 3 describes the positivity and boundedness of solution. In section 4, the existence of disease-free equilibrium including derivation of basic reproduction number and stability analysis of the model is presented. e results of numerical simulation are illustrated in Section 5, Section 6 provides discussion, and Section 7 provides conlclusion.

Model Formulation
Mosquito-borne diseases, e.g., yellow fever, dengue fever, and malaria, are frequently observed in tropical and subtropical countries. ese illnesses spread widely in a short period with the life-threatening impact of many human lives. Among these vector-borne diseases, malaria is one of the serious and major illnesses caused by several species of mosquito-borne parasite (Plasmodium falciparum, Plasmodium malaria, Plasmodium ovale, Plasmodium vivax, etc.) [24]. Anopheles female mosquito is responsible for the transmission of disease to the human body through a bite [3]. is blood meal of mosquito converts a healthy human being (susceptible) into a category called infected hosts. e human population that are not infected but under the threat that they can catch malaria infection are known as the exposed population. Individuals recovered from the infected population through medical treatment without threat to their life fall in the group of recovered ones. Figure 1 shows the schematic diagram of the proposed malaria disease model for particular population of humans and mosquitoes. e human population is divided into five different compartments (susceptible, exposed, unaware infected, aware infected, and recovered) representing the total population at any time t. Let N h be the total population in a region under consideration for malaria disease analysis.

Complexity
Suppose S h (t) denotes the numbers of humans susceptible to disease, E h (t) is the count for the exposed hosts to disease, I uh (t) and I Ah (t) are showing unaware and aware infected human population, and R h (t) is the numbers of individual who have temporarily recovered from disease. Similarly, mosquito population has been grouped into three compartments. Let S m (t) is the count for susceptible mosquito population, E m (t) is the numbers for exposed mosquito population, I m (t) represents the numbers of an infected member of the mosquito population, and N m (t) is the total count of the mosquito population in a particular area of interest. Also, let M(t) be the cumulative density of awareness programs driven in the region at time t. e growth rate of the density of awareness programs is assumed to be proportional to unaware infected individuals. It is assumed that the contact of infected mosquitoes with aware individuals will be reduced. e constant μ 1 represents the rate at which awareness campaigns are being implemented and μ 0 represents the depletion rate of these campaigns due to ineffectiveness, a social problem in the population.
For host population, φ h is the recruitment rate of human population in a particular region, b is the contact rate of mosquito to human population, β mh is the probability that bite of infectious mosquito result in the transmission of disease to susceptible human, c 1 is the rate of loss of immunity in recovered human, α h is the rate of progression from exposed to unaware infected class, c is the treatment rate in the region, c 0 is the recovery rate of unaware infected population with temporary immunity, u 1 represents the successful efforts of treatment resulting recovered humans, ρ i is the rate of awareness to unaware infected human, a LI is the rate of loss of awareness of aware infected human, c m is the recovery rate of aware infected human (that rate is greater than the normal recovery rate due to awareness), and δ h and μ h are the disease and natural death rate of the human, respectively.
For mosquito population, φ m and μ m are the mosquito recruitment and natural death rate, δ m represents the strategies to kill mosquito after awareness, β hm is the probability that a blood meal of an infectious mosquito result in the transmission of disease to susceptible individual, β Ahm is the probability of disease transmission from aware infected human to susceptible mosquitoes, b 1 is the contact rate between unaware infected human to susceptible mosquitoes, and α m is the rate of progression from exposed mosquitoes to infected mosquitoes. e definitions of the mathematical parameters with their values are summarized in Table 1. ese definitions lead to a set of coupled nonlinear differential equations describing the proposed model, for the infectious vector-borne disease. e mathematical form of the model is described as follows:  Figure 1: Schematic diagram of the proposed model algorithm.

Positivity and Boundedness of Solutions
e mathematical model presented in the system of equations (1)-(9) describes the rate of change of different compartments of human and mosquito population. erefore, it is important to verify that all solutions with nonnegative initial conditions shall remain nonnegative for all time. All the solutions of proposed system, which initiates inside region D, remain in region D. Mathematically, is result can be summarized in the following theorem.
All the solutions of the system of equations (1)- (9) are bounded in domain D: , I m (t)) be any solution with positive initial conditions Notice that the sum of the first five compartments (9) is equal to the total human population of size N h and the sum of compartments (S m (t), E m (t), I m (t)) is equal to the total mosquito population of size N m . Hence adding these equations yields the time derivative along with the solution of equations (1)-(9) given as Equations (11) and (12) can be written as Solving these different inequalities yields Consequently, taking the limit as t ⟶ ∞

Existence of Disease-Free Equilibrium Point
Disease-free equilibrium points are the steady-state solution, when there is no malaria infection. us, the disease-free equilibrium point for the system of equations (1)- (9) implies that E h � 0, I uh � 0, I Ah � 0, R h � 0, E m � 0, and I m � 0, and after solving the equations (1) and (6) . us, we obtain the disease-free equilibrium point, E 1 :

Basic Reproductive Number.
e basic reproductive number R 0 measures the average number of new malaria infections generated by a single infected individual in a completely susceptible population [26].
To obtain R 0 for system of equations (1)-(9), we used the next-generation matrix technique described in [15,25,27,28]. Let x � (E h , I Uh , I Ah , E m , I m , S h , R h , S m ) T , then the model system of equations (1)-(9) can be written as where Finding the partial derivative of F and V at disease-free equilibrium point E 1 gives F and V, respectively, as follows: So that where v − is the spectral radius of the product FV − 1 . us, In equation (25), (α h /(α h + μ h ))is the probability that a human will survive the exposed state to become infectious; (1/(δ h + μ h + c + c 0 u 1 )) is the average duration of the infectious period of the human; (α m /(μ m + δ m + α m )) is the probability that a mosquito will survive the exposed state to become infectious; and (1/(μ m + δ m )) is the state to become infectious. Let the basic reproductive number, R 0 , be written Here, R h describes the number of humans that one infectious mosquito infects over its expected infection period in a completely susceptible human population and R m describes the number of mosquitoes infected by one infectious human during the period of infectiousness in a completely susceptible mosquito population.

Stability Analysis of Disease-Free Equilibrium.
We analyzed the stability of disease-free equilibrium of the system of equations (1)-(9) by using the basic reproductive number R 0 in the following theorem.

Proof.
e Jacobian of the system of equations (1)-(9) evaluated at the disease-free equilibrium point E 1 is obtained as follows: where 6 Complexity e DFE, E 1 , is locally asymptotically stable (LAS) if we show that all the eigenvalues of equation (26) have negative real part. Since the first, sixth, and ninth columns of equation (26) contain only the diagonal terms, three eigenvalues − μ h , − (μ m + δ m ), and − μ 0 can be obtained from these columns. Remaining eigenvalues of equation (26) can be obtained from the submatrix J 1 formed by excluding the first, sixth, and ninth rows and columns of equation (26). Hence, we have Similarly, the fourth column of equation (27) contains only the diagonal term which forms negative eigenvalues − (μ h + c 1 ). All the remaining eigenvalues of equation (27) can be calculated by the submatrix J 2 : e eigenvalues of equation (28) are the roots of the following characteristic equation: From equation (29), we can get five eigenvalues. One of the eigenvalues j 44 � − (μ m + c m + δ m + a LI ) has a negative real part. e other four eigenvalues can be obtained from equation (30): If we let b 1 � − j 22 , b 2 � − j 33 , b 3 � − j 77 , and b 1 � − j 88 , then equation (30) becomes where where a 0 is in the terms of reproduction number and R 0 , can be written as We employ the Routh-Hurwitz criterion, which states that all the roots of equation (31), has negative real parts, if and only if the coefficients a i are positive and matrices H i > 0 for i � 0, 1, 2, 3, and 4. From equation (32), it is easy to see that a 1 > 0, a 2 > 0, a 3 > 0, and a 4 > 0 since all b i ′ s > 0. Moreover, if R 0 < 1, it follows from equation (33) that a 0 > 0. Also, the Hurwitz matrices for equation (31) are found to be positive that are given as follows: Complexity erefore, all the eigenvalues of the Jacobian matrix in equation (26) have negative real parts, when R 0 < 1 and DFE is LAS. However, when R 0 > 1 we see that a 0 < 0 and by Descartes's rule of signs, there is exactly one sign-change in the sequence a 4 , a 3 , a 2 , a 1 , and a 0 of the coefficient of equation (31). So, there is one eigenvalue with the positive real part and DFE is unstable.
e case for R 0 � 1 is possible when a 0 � 0 in the characteristic equation. By considering the special case of Routh-Hurwitz stability criterion [29], and replacing this term with very small value, i.e., ε, the result could be interpreted as first case, i.e., R 0 < 1.
Proof. To prove this theorem, we adopt the method described in [30][31][32]. Castillo-Chavez and co-authors described this method to prove the GAS of DFE in their research article [32]. We begin the proof by defining new variables and breaking our system of equations (1)-(9) into two subsystems. With X � (S h , R h , S m ) and I � (E h , I uh , I Ah , E m , I m ), this system can be written as where X ∈ R 3 denotes the number of uninfected compartments and I ∈ R 5 denotes the number of infected compartments. e two vector-valued functions F(X, I) and G(X, I) are given as E 0 � (X * , 0) denotes the disease-free equilibrium of the subsystems, where e conditions (H 1 ) and (H 2 ) below must be met for the global stability: where A is an M-matrix (the off-diagonal elements of A are nonnegative). Now consider the reduced system (dX/dt) � F(X, 0), is the GAS equilibrium point for the reduced system (dX/dt) � F(X, 0). To see this, solve equation (43) to obtain where is completes the proof.

Existence of Endemic Equilibrium Point.
Besides the disease-free equilibrium point, we shall show that the system of equations (1)-(9) has a unique endemic equilibrium point E 2 . e endemic equilibrium point is a steady-state solution, where the disease persists in the population.
is is a positive solution of equation given by where e systems (1)-(9) have no positive solution; when R 0 < 1, A 1 > 0, A 2 > 0, A 3 > 0, and A 4 > 0 However, when R 0 > 1, A 4 < 0, by using Descartes's rule of sign, there is exactly one sign-change in the equation, and there exists exactly one positive root. is implies that a unique endemic equilibrium exists. is completes the proof.

Remark 2.
ere are more than one endemic equilibrium existing in case, when

Results
e general overview of the proposed model has been shown in Figure 1. e definitions and corresponding numeric values of the parameters/variables in the system of equations (1)- (9) have been summarized in Table 1. e definitions of the compartment variables are summarized in Table 2. Figure 2 represents the 3D-plot for the proposed model. In this figure, the proposed models converge to the same point on different initial conditions. A comparison result of proposed model (incorporating awareness) and existing model (without awareness) has been shown in Figures 3-7 for human and mosquito populations. In each figure, reduction/growth of human and mosquito population in each class has been presented.
In Figures 3-5, the population of susceptible, unaware infected, and aware infected human population with and without awareness has been shown. We can easily see that, when there is no awareness, the population of susceptible humans is decaying rapidly (the red line) and there is a rapid increase in unaware infected human growth (the red line). But, when we give awareness to unaware infected humans, we experienced susceptible humans are increasing and unaware infected humans are decreasing significantly. It is due to the fact that, when we provide awareness to unaware infected humans, the contact rate of infected human and infected mosquitoes has also been reduced. e result revealed that a significant reduction in the population of unaware infected humans, for different timespans, is achieved through the proposed model. It has also been shown that the corresponding growth of unaware infected human population and infected mosquitoes increases without awareness as compared with that of the proposed model. e results also revealed that a significant reduction in the population of susceptible mosquitoes and infected mosquitoes for different timespans are achieved through the proposed model. It has also been shown that the corresponding growth of infected mosquito population is higher without awareness as compared with the proposed model.
In Figures 8 and 9, we show that, when we give awareness, it also affects the contact rate between human and mosquito population. Figures 8 and 9 describe that, when there is no awareness, the biting rate is also high in the region. And, infected human and mosquito population increases rapidly, but when we add awareness into the system, the contact rate between aware infected humans and mosquitoes has been reduced. We show the results on the different biting rates in Figures 8 and 9 with low and high awareness rates.

Discussion
Malaria is a mosquito vector-borne disease spread in around hundred countries worldwide. e highest mortality rates are reported in tropical countries likewise sub-Saharan Africa [33].
During the past decade, several prevention measures have been used to reduce the transmission of this deadly disease. e most frequently used measures include indoor spraying and bed nets. e vaccine of this deadly disease has not been prepared till date. e measures that can reduce the spread of this dangerous disease are preventive measures and awareness. In this paper, a mathematical model for the vector-borne disease has been proposed incorporating the awareness against this disease.
Vector-borne disease models have been proposed/modified in the past. Tumwiinw and co-authors proposed a mathematical model that tracks the dynamics of malaria in the human host and mosquito vector [34]. ey incorporated infected humans recovered from infections and immune humans after the loss of immunity against this deadly disease to rejoin the susceptible class again. Addawe and Lope have modified and analyzed the mathematical model of Tumwiinw and co-authors [17]. ey divided the human population into two compartments: preschool (up to five years of age) and the rest of human population (older than five years of age). Lashari and co-authors introduced three types of control  parameters, namely, personal protection, treatment, and mosquito-reduction strategies to reduce the spread of malaria disease [3]. us, the previous literature suggests that the reduction of the spread of this dangerous disease could be achieved through preventive measures. erefore, a mathematical model has been developed in this study to cater the awareness strategy as a variable in the model. e awareness could be addressed to infected humans, and the aware infected humans could reduce contact rate with mosquitoes due to awareness. e proposed model has been compared with existing model. e comparative results show that a significant improvement in the reduction of vector-borne infection could be achieved with awareness.

Conclusion
Malaria is a tropical infectious disease. Scientists have not succeeded till date to develop an effective vaccine to retaliate this deadly disease. us, the mathematical modeling of this disease has a crucial role to understand the insights of the transmission dynamics and corresponding appropriate prevention strategies. A novel mathematical model has been proposed in this study to prevent malaria disease. e simulation results have significantly shown that awareness is an important factor to fight against this deadly disease. us, the spread of this illness could be prevented through effective awareness strategies in regions, where it has rapid spread.

Data Availability
No data has been used in this study. t (days) Figure 9: Population of infected mosquitoes with different biting rates (without awareness (red line); with low to high awareness rate (blue, green, magenta, and yellow line)).