MODELING, ANALYSIS AND OPTIMAL CONTROL OF VECTOR-BORNE DISEASES WITH AWARENESS FACTOR

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper we have developed an SEIRS model for vector borne diseases, taking awareness about the disease as a factor. The model has been formulated and analysed, along with assessment of creating awareness about preventive measures for the disease. The behaviour of the model, the effect of awareness and the effect of the control measures taken have been studied by carrying out numerical simulation using MATLAB.


INTRODUCTION
Vector borne diseases are transmitted through vectors which are organisms that transmit pathogens and parasites from one infected person (or animal) to another, causing serious diseases in human populations. These diseases are commonly found in tropical and sub-tropical regions and places where access to safe drinking-water and sanitation systems are problematic.
Vector-borne diseases account for 17% of the estimated global burden of all infectious diseases [23]. According to the World Malaria Report (WMR) 2020 released by World Health Organisation (WHO), there were 229 million malaria cases around the globe for the past four years [24]. The disease claimed around 409000 lives in 2019 alone. However, the world's fastest growing vector-borne disease is dengue, with a 30-fold increase in disease incidence over the last 50 years. There are many other vector borne diseases such as lymphatic filariasis, lyme disease, Chikungunya, Yellow fever etc.
One of the reasons for the transmission of these vector borne diseases is the lack of awareness among the human population. People need to know as to how these diseases spread and should take necessary precautions and use preventive measures to stop the spread of them. Hence different countries have been introducing various awareness programs to make the people aware of the cause of the disease and how to curb it. For example, April 25 is observed globally as Malaria Awareness Day and the theme for the year 2020 was "Zero malaria starts with me".
India has launched the National Dengue Day and it is observed on May 16 every year. On this day efforts are taken by the government to spread awareness about dengue and how to take necessary precaution to prevent it. The National Vector Borne Disease Control Programme (NVBDCP) in India which was launched in the year 2003 also takes various measures to make people aware of these diseases. The effect of awareness in controlling the spread of the diseases has been modeled mathematically and studied for various diseases [7,15,20,21]. Several SEIRS models for malaria and dengue have been modelled [2,10,13,16,18,22] and the optimal strategies for controlling them have been studied [3,6,8,9,11,12,17]. However, none of them have considered the population as aware population and unaware population.
In this paper, we develop an SEIRS model by dividing the susceptible population into two classes-the aware susceptible and the unaware susceptible population. In Sections 3, 4 and 5, we analyse the model and establish the stability of the model. In Section 6, we use optimal control theory and derive necessary conditions by applying Pontryagin's Maximum Principle to control the transmission of the disease efficiently . Finally we carry out numerical simulation using MATLAB and study the behaviour of the model, the effect of awareness and the effect of the various control measures used to prevent the transmission of the disease.

MODEL FORMULATION
Let N H denote the total human population which is divided into various components, Let us assume the susceptible human population to be of two classes where S H 1 denotes the susceptible human population who are unaware of the disease and S H 2 the susceptible human population who are aware of the disease. Let E H and I H denotes the exposed human population and the infected human population respectively. Then N H = S H 1 + S H 2 + E H + I H denotes the total human population.
Let S M and I M denote the susceptible and infected mosquito population respectively and N M = S M + I M , the total mosquito population.
Based on the above classification of the human population and the mosquito population, the dynamics of vector borne diseases are modelled as a system of non-linear differential equations.
The system of equations are as follows: Adding the equations corresponding to S H 1 , S H 2 , E H , I H and corresponding to S M and I M , we where Λ 1 = birth rate of human population, β 1 = contact rate of unaware susceptible humans with infective mosquitoes, δ =rate of transfer of unaware susceptible individual to aware susceptible class, µ H = natural death rate of the human population, γ= rate of progression of humans from the infected class to the susceptible class after recovery, k= a fraction of recovered persons going to the aware class, β 2 = contact rate of aware susceptible humans with infective mosquitoes, η= rate of progression of humans from the exposed to the infectious class, α 1 = disease induced death rate of humans, Λ 2 = recruitment rate of mosquitoes, β 3 = contact rate of infected human with susceptible mosquitoes, α 2 =death rate of mosquitoes due to control measures, µ M = natural death rate of mosquitoes.

FEASIBLE SOLUTION
In this section we show that the model governed by the system of non-linear equations given by (1) is epidemiologically and mathematically well-posed in a region Ω.
Moreover, it is positively invariant and mathematically well posed in the domain Ω.
Proof. In order to prove the theorem, we need to prove the following: In order to prove (a), making use of equations (2) and (3) we have, . Taking limits as t → ∞, we have . This shows that the total population and each population class remains bounded for all finite time t ≥ 0 in Ω.
We now prove the second part of the theorem. Since all the parameters used in the system and the initial values of the compartments are greater than zero, we can place lower bounds on each of the equations given in the model.
Through basic differential equations methods we have, Proceeding similarly for S M (t) and I M (t), we have Combining (a) and (b), we conclude that the feasible solution is mathematically well posed in the region Ω. This proves the theorem.
We calculate the equilibrium points and discuss the local stability of the equilibrium points in the subsequent sections.

EXISTENCE OF EQUILIBRIUM POINTS
There are two equilibrium points for model (1) which are the disease free equilibrium point and the endemic equilibrium point. The existence of these points are given in the following theorem.
Theorem 2. There exists two equilibrium points for the system (1) which are as follows: (1) The disease free equilibrium point is given by Proof. The equilibrium points are calculated by equating the equations of the system (1) to zero.
The disease-free equilibrium point exists in the absence of exposed humans, infected humans and infected vectors in the system. This means that E H = I H = 0 and I M = 0. Solving the system (1) gives us the disease-free equilibrium point as This proves the first part of the theorem.
To prove the second part, let be the endemic equilibrium point of (1). Then all the components of P 1 should be positive. If we set the system of differential equations in (1) to zero, we get Now putting the value of S * M from equation (8) in equation (9), we get Putting the value of I * M from equation (10) in equation (4) (11) Substituting the value of I * M in equation (5), we have Substituting the value of E * H which is in terms of S * H 1 , S * H 2 and I * M in (7),we have where R * 0 can be called a threshold number.
We now calculate the basic reproduction number R 0 .

BASIC REPRODUCTION NUMBER R 0
The basic reproduction number R 0 is defined as the number of secondary infections that one infectious individual would generate on an average over the course of the infectious period.
There are many methods to calculate R 0 . We use the next generation operation approach as given in [1]. When R 0 < 1, the disease will decline and eventually die out. When R 0 > 1, the disease will spread in the population. Hence this means that the threshold quantity to be taken into account to eradicate the disease is to reduce the value of R 0 to be less than one.
F includes only infections that are newly arising, and V includes terms that describe the transfer of infectious from one infected compartment to another at the disease free equilibrium point.
Then according to [1], the matrix of FV −1 is called the next generation matrix for the model.
The basic reproduction number R 0 is given by R 0 = σ (FV −1 ) which is the dominant eigenvalue of FV −1 . Corresponding to the model (1), and the partial derivative of (1) with respect to (E H , I H , I M ) and the Jacobian matrix is The inverse of V : The dominant eigenvalue of the matrix .
Substituting for S H 1 , S H 2 and S M the reproduction number is given by,

LOCAL STABILITY ANALYSIS
In this section, we analyse the local stability of the disease free equilibrium point and the endemic equilibrium point.
Proof. Consider the Jacobian matrix L 0 of the system (1) at the disease free equilibrium point To calculate the eigenvalues, we consider the characteristic equation of L 0 . It is given by . The eigenvalues of the matrix L 0 are −µ H , −(µ M + α 2 ), −(µ H + δ ) and the roots of the cubic polynomial λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0.
For the equilibrium point to be locally stable, the roots of the polynomial λ 3 + a 1 λ 2 + a 2 λ + a 3 have to be negative. We use Routh-Hurwitz criterion which states that if a i ≥ 0, i = 1, 2, 3 and a 1 a 2 − a 3 > 0, the roots of the polynomial λ 3 + a 1 λ 2 + a 2 λ + a 3 are negative.
In the next section we use Optimal Control Theory to see the effect of awareness and the use of different control measures on the mosquito population.

OPTIMAL CONTROL ANALYSIS OF THE MODEL
In this section we reformulate the model (1) to estimate the effect of the control strategies used to control the mosquitoes. The three main control strategies are: use of bed nets for personal protection denoted by u 1 (t), treatment of infected individuals with drugs denoted by u 2 (t) and the spraying of insecticides on the breeding ground of mosquitoes denoted by u 3 (t). Taking these controls into account, model (1) is reformulated as follows: where 1 − w is the fraction of reduced mosquito population and hence the mosquitoes are reduced at the rate u 3 (1 − w). Moreover, 0 ≤ u 1 ≤ 1, 0 ≤ u 2 ≤ a 2 where a 2 is the efficacy of the drug used for treatment, 0 ≤ u 3 ≤ a 3 where a 3 is the efficacy of the insecticide at reducing mosquito population.
Our objective is to minimize the number of infected individuals through the optimal control strategies u 1 (t), u 2 (t) and u 3 (t). Define where t f is final time and l, m, n are positive weights to balance the factor and p, q and r denote the weighting constants, mI H is the cost of infection, pu 2 1 is the cost of use of bed nets, qu 2 2 is the cost of treatment efforts and ru 2 3 is the cost of use of insecticides. We need to find an optimal control u * 1 , u * 2 and u * 3 such that, where the control set, Theorem 5. There exists an optimal control u * 1 , u * 2 , u * 3 and corresponding solutions S H 1 , S H 2 , E H , I H , S M , I M of the system (15) that minimises J(u 1 , u 2 , u 3 ) over Ω 1 . Furthermore, there exists adjoint variables λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 such that with transversality conditions, The controls u * 1 , u * 2 and u * 3 are given by u * 1 = max{0, min(1, Proof. The optimal control exists since the integrand of J is convex with respect to u over a convex and closed control set Ω 1 . Moreover the system satisfies Lipschitz property with respect to the state variables since the state solutions are bounded [4]. The Pontryagin's Maximum Principle [19] converts, (15), with (16) and (17) into a problem of minimising a Hamiltonian H, with respect to u 1 , u 2 and u 3 . Define where f i , i = 1, 2, ..., 6 are right hand side of the system (15).
We also have the adjoint equations Evaluating the six equations given by (21) at the optimal control and the corresponding states will give the adjoint system (18) and (19).

NUMERICAL SIMULATION
In this section, we use simulation and see how model (1) behaves. Further in order to see the effect of the various control measures on the model (15) for vector borne diseases, we carry out the simulations for malaria. The parameters for the model are taken from [14] and [5].  Figures 2(a) and 2(b) show the behaviour of model (15) when no control measures are taken. In this case, we have u 1 = u 2 = u 3 = 0. It can be noted that when (15) reduces to the model given by (1). Figure 2 shows that the various compartments are stable after 150 days. Hence the endemic equilibrium point is stable. The stability of the endemic equilibrium point can also be proved numerically by calculating the value of R 0 . The expression for R 0 is given in (13). Using the above parameters, the value of R 0 is calculated and it is found to be 4.89 which is greater than 1. The eigenvalues of this matrix are -59.999, -0.558, -0.001, -0.0311, -0.074, -0.076 and they are all negative. Since the eigenvalues are negative, the system is stable.   Figure 4(b)). From Figures 4(a) and 5(b), it can be seen that using the bed nets is a better control measure than spraying of insecticides to reduce the interaction between mosquitoes and humans. However, from Figure 5(b) it can be seen that when both the control measures are used against the mosquitoes there is a considerable decrease in the mosquito population rather than using just one of them. Figure 6 compares the infected population when no control measures

CONCLUSION
In this paper we formulated an SEIRS model for the human population in which the susceptible population is divided into two compartments as those who are aware of the disease and those who are unaware of the disease. Moreover the various control measures were introduced in the model. These control measures were introduced to reduce the vector population as well Hence it remains to be seen whether the infected human population decreases if awareness of the disease among the human population is taken as a control measure.

ACKNOWLEDGEMENT
The infrastructural support provided by FORE School of Management, New Delhi in completing this paper is gratefully acknowledged.

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