STABILITY ANALYSIS OF SEIR MODEL USING NON-NEWBORN VACCINATION AND COST EFFECTIVE TREATMENT

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This study focused on the modification and probation of a Susceptible-Exposed-Infected-Recovered (SEIR) model for non-newborn vaccination and cost effective treatment. The system of differential equations has been derived from SEIR model to creates a bond between susceptible S, infected I, exposed E and recovered E participants for understanding the spread out of contagious diseases. Further, the local stabilities of both disease free equilibrium points and endemic equilibrium points were found stable at epidemic conditions i.e. epidemic (R0 > 1) and no epidemic (R0 ≤ 1). In addition, numerical simulation has been performed to investigate the proposed model at regular set of values of parameters. Moreover, our vaccination target is only non-newborn individuals to protect the population without effecting the economy of country.


INTRODUCTION
The construction of new models and modification of models in the field of mathematical epidemiology gives attainable approach for better future of science and technology. Up to now various research have been made in the field of mathematics among them mathematical biology has gained much of attraction because of its vast applications in the area of medicine [1]. The early developments in the field of mathematical biology have been carried out in 18th century [2]. Later, these studies flourished by many researchers to develop the most appropriate mathematical model to scrutinize biological diseases [2,11]. In recent years, among all the infectious diseases like influenza, dengue, measles, Spanish flue, etc., dengue virus which also known as vector-borne contagious disease considered as the major threat for public health in Pakistan [3,7]. Therefore, epidemic models such as: SI, SIS, SIR, SEIR and SEIRS are required to overcomes infectious problems of the whole world [6,10]. However, these epidemic models are based on three aims. Firstly, to understand the spreading and transmission of contagious disease, structure of epidemic model and behaviour of concern parameters. Secondly, to calculate threshold quantity which is also known as basic reproductive number which predict either epidemic occur or not. And third aim is to construct a strategy for the control and eradication of contagious disease [20]. In addition, these mathematical models have been derived on the base of first order differential equations, which are helpful in analyzing the spread and control of contagious diseases [12,15]. Usually, these mathematical models are the categorical models, which represents four compartments such as susceptible, exposed, infected and recovered, while each compartment represents a particular step of the epidemic. However, in these models the change rate from one class to another class is numerically presented with the help of derivatives [16]. Further, the system of ODE's (i.e. SIR, SEIR, SEIRS mathematical model etc.) is described by using classes of population and rate change derivatives as a function of time [4].
Herein, the SIR epidemiological model described the dynamics of infectious diseases with continue immunity and a qualitative discussion to analyze stability. More importantly, the diseasefree equilibrium points of SIR model are found locally and globally asymptotically stable if the reproduction number R 0 < 1, while the endemic equilibrium points of SIR models are locally asymptotically stable when reproduction number is R 0 > 1. However, in order to eradicate disease successfully by using SIR model, the vaccination level should be larger because disease preventation rely on vaccination proportion as well as efficiency of the vaccine [19]. Moreover, the SEIRS model also depicts the infectious diseases among with different parameters such as unequal birth and death rates, vaccinations for newborns and non-newborns and temporary immunity with the help of vital features of SI, SIS and SIR models. However, in case of SEIRS the mathematical approach in determined the disease-free and endemic equilibrium points with local stability were analyzed according to its epidemic conditions i.e. non-epidemic R 0 ≤ 1 and epidemic R 0 > 1) using the time-series and phase portraits of the susceptible S, exposed E, infected I, and recovered R individuals [16].
In our study we discussed four classes of proposed model i.e. S = susceptible, E = exposed, I = infected and R = recovered of with different parameters including birth rate, natural death rate, disease death rate, vaccine for non-newborn and treatment rate. In addition, the regular set of values will be used for these parameters in numerical simulation. Further, graphical study has been investigated based On the numerical values. Moreover, the treatment rate function is supposed which is directly proportional to number of infectious patients up to certain limit. Further more, the stability analysis has been carried out by use of multiple endemic equilibrium points.
Also, experimental work has been made on the bases of these equilibrium points and local stability. Thus, this research work will be helpful for the future study in the field of mathematical biology.

SEIR MODEL AND ITS BASIC REPRODUCTIVE NUMBER
The proposed SEIR model with limited (non-newborn only) vaccination and cost effective treatment will provide the whole portrait of contagious diseases and its corresponding with ecology. Figure 1 shows the block diagram of modified SEIR model, which is constructed by dividing the whole population Π in four epidemic categories (classes) those are Susceptible (S), Exposed (E), Infected (I) and Recovered (R) [4,9].
The four categories S, E, I and R of the SEIR model are depict for detail in Table 1. For the proposed SEIR model, the model permits with different birth and death rates, vaccinations only for non-newborns (i.e children and adults) and cost effective treatment for individuals from susceptible category.
The Table 2 summarizes the details of different +ive parameters lodge in the SEIR model for each of the four categories.   Arithmetically, the SEIR model is explicit as a system of ordinary differential equations given by [16,17]: In this research, the treatment function is defined as where c = kI 0 this tells that F(t) ∝ I as well as the number of infectious individuals are less or equal to a static value I 0 but later on treatment rate turn into constant. This research has main concern with cost effective treatment, in which medication and bedding in hospitals may or may not be sufficient.
The reduce system of (1) is enough to analyze because in first three equations of system (1) R is not use From system (2) Thus we have lim n→∞ sup(S + I + R) ≤ Π µ so the possible and reasonable region for the set of Therefore the system (2) is well posed by arithmetically and endemically in Λ because the region Λ is positively invariant w.r.t system (2).
Moreover for these disease free equilibrium points I < I 0 , therefore the system(2) turns to Then system(6) may be written as The Jacobian matrices of transmission matrix (Y ) and transition matrix ϒ(Y ) at disease free equilibrium points X * 0 are, respectively [5,18] To remove confusion we suppose Ω = FV −1 and the N.G.M of system (2) is Now we will find the spectral radius of N.G.M [14] that is defined as Hence the basic reproductive number R 0 of system (2) is given by

EQUILIBRIUM POINTS OF SEIR MODEL
Here we will find and discuss equilibrium points of our proposed model. First we known that, the disease free equilibrium points of system (2) X 0 d f e = (S 0 , E 0 , I 0 ) = ( Π µ , 0, 0) always exits when I ≤ I 0 [16,13]. Now we will find the endemic equilibrium points of system (2) which satisfies For above system, if 0 < I ≤ I 0 , then F(t) = kI and if I > I 0 , then F(t) = c, Moreover If R 0 > 1, system (8) confess a unique positive result i.e X * ee = (S * , E * , I * ) given by From system (10) From third equation of system (10) Put the value of I * in equation (12) we get (14) Put the value of E * from system (14) in equation (13) we get Putting the value of I * in equation (11) we get the value of S * i.e.
We know that from equation (11) S * = Π β I * +µ putting this value in equation (16) we get (17) Putting the value of I * in equation (13) we get Hence from equation (17) R 0 ≤ 1 + β I 0 µ ∼ = Q 0 iff I * ≤ I 0 . Therefore X * ee = (S * , E * , I * ) are endemic equilibrium points of system (2) iff 1 < R 0 ≤ Q 0 . In system (9) when I > I 0 , to get the positive solution of system (2), we solve S and E from first and third equation of system (9) respectively and substitute the value of S and E in second equation of system (9). We have S = Π µ+β I and E = (µ+γ+∆)I+c ε after substitution in second equation of system (9) we get After putting the values in equation (19) we get (21) dI 2 + eI + f = 0.
The system (21) gives us discriminant i.e. D = e 2 − 4d f with two positive real roots e < 0 and D ≥ 0.
As we know form equation (8) ( Putting the values of d, e and f in the equation of discriminant D. We get For positive real root D ≥ 0 we have (25) After simplification of above equation and for e < 0 the R 0 is equivalent to (26) Therefore for equation (21) then X * i = (S * i , E * i , I * i ), i = 1, 2 are endemic equilibrium points of system (2) if I * i > I 0 . As we know (30) In above equation right hand side is negative and greater than the value at left hand side so if negative value is greater therefore the left hand value is always less than zero i.e.
It follows the definition of e that is (32) Adding and subtracting kµ in above equation we get: Similarly if (34) Then By a comparable statement we get that I 2 < I 0 iff R 0 > Q 2 now we will sum up the above discussion as following: (36) and 1. Disease free equilibrium points i.e.X 0 d f e = (S 0 , E 0 , I 0 ) = ( Π µ , 0, 0) always exist in system (2).
3. There is existence of two more endemic equilibrium points i.e.

LOCAL STABILITY OF EQUILIBRIUM POINTS
In this section we analyze the eigenvalues of Jacobian matrices of system (2) and check the local stability of disease free equilibrium points and endemic equilibrium points [16]. The Jacobian matrix is calculated from equilibrium points as: Where Jacobian matrix w.r.t equilibrium points is by using the jacobian matrix we evaluate the eigenvalues from |J(X 0 d f e orX * ee ) − λ I| = 0. then we check our system either it is stable or not. If all the eigenvalues are negative then system is stable otherwise if at least one eigenvalue is positive the system is unstable.

Disease free equilibrium points X 0
d f e . By using the Disease free equilibrium points i.e. X 0 d f e = (S 0 , E 0 , I 0 ) = ( Π µ , 0, 0) , for system (6) the jacobian matrix J(X 0 d f e ) is given as: . Now eigenvalues are found by using the characteristic equation which we discuss below: Similarly when F(t) = 0 All the eigenvalues are negative in above result, so the disease free equilibrium points are locally stable for system (2).

4.2.
Endemic equilibrium points X * ee . As researcher knows For system (4) For our requirement to check the local stability of endemic equilibrium points Routh-Hurwitz criteria is used [15,16]. Suppose Then equation (41) becomes It is clear that, 0 > 0, 1 > 0, 2 > 0, 3 Above discussion satisfies the three conditions of Routh-Hurwitz criteria given below: Hence by Routh-Herwitz criteria, all the eigenvalues of J(X * ee ) are negative therefore endemic equilibrium points are locally stable for proposed model.

NUMERICAL SIMULATION
The proposed SEIR model with non-newborn vaccination and cost effective treatment was estimated in Matlab. Table 3   The value of R 0 depends upon above values of parameters and computed for SEIR model in Table 4. Moreover, we will get two values of R 0 because of two conditions i.e. epidemic and no epidemic.  Table 3. So as to distinguish between the occurrence for the epidemic conditions of R 0 i.e. no epidemic (R 0 ≤ 1) and epidemic (R 0 > 1) are analyze separately. The local stability of disease free equilibrium points and endemic equilibrium points for both cases i.e. no epidemic and epidemic are estimated with help of corresponding eigenvalues and jacobian matrix.
5.1.1. No epidemic. The epidemic required condition R 0 is calculated as R 0 = 0.8615 implies no epidemic for the contagious disease because R 0 ≤ 1. The value of R 0 depends upon the mathematical values of model parameters and constants with β = 1 5 for our SEIR model. Here β = 1 5 means that 0.2 susceptible participants becomes exposed because of infected participants and left the susceptible category and enter the exposed category per day. The disease free equilibrium points X 0 d f e and endemic equilibrium points X * ee and eigenvalues λ i of their jacobian matrices i.e. J(X 0 d f e ) and J(X * ee ) in company with local stabilities.  Table 5 shows that the disease free equilibrium points i.e. X 0 de f are locally stable because all the eigenvalues i.e. λ 1 , λ 2 and λ 3 are negative with β = 1 5 . Where as the endemic equilibrium points i.e. X * ee are also locally stable (because all eigenvalues are negative in it) with β = 1 5 . Figure  means that 5 susceptible participants becomes exposed because of infected participants and left the susceptible category and enter the exposed category per day. Table 6 shows the disease free equilibrium points X 0 d f e and endemic equilibrium points X * ee and eigenvalues λ i of their jacobian matrices i.e. J(X 0 d f e ) and J(X * ee ) in company with local stabilities.  Table 6 shows that the disease free equilibrium points i.e. X 0 de f are locally stable because all the eigenvalues i.e. λ 1 , λ 2 and λ 3 are negative with β = 5. Where as the endemic equilibrium points i.e. X * ee are also locally stable (because all three eigenvalues are negative in it. Moreover, if only one value is negative then it will locally unstable) with with β = 5. Figure 3 describes the two dimensional phase portraits of four categories with initial condition 0.25.

CONCLUSION
In this research work, modified SEIR model based on non-newborn vaccination and cost effective treatment for mutual benefits is proposed. Herein, we generalize models of vaccination and treatment for large population. The main focus was to handle a problem when hospitals has lack of bedding and medication some time in war like conditions and in our rural areas and villages. Further, by using basic reproductive number R 0 , the behaviour of our proposed model has been found. The Disease free equilibrium points X 0 d f e and endemic equilibrium points X * ee exists and model is locally stable. Moreover, the proposed model is epidemic when R 0 ≤ 1 and endemic when R 0 > 1. For future, we may modified the model w.r.t to age limit structure, vital dynamics and isolations in climate behaviour to produce suitable epidemic models in the field of mathematical biology.

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