DYNAMICAL RESPONSE BY THE INSTANT BUYER AND THINKER BUYER IN AN INNOVATION DIFFUSION MARKETING MODEL WITH MEDIA COVERAGE

In this exposition, a three compartment model has established to analyze the purchasing behavior of buyer. Population for a specific product marketing is classified into non-adopter group N(t), thinker group I(t) and adopter group A(t). The relation within adopter and the non-adopter group is word of mouth. For enhance the impact of word of mouth, media is the another dominating factor for impressing the non-adopter population, who has to become an adopter. In this model, there are two types of buyers instant buyers and thinker buyers. Instant purchasing is an unplanned decision to purchase a brand-new product, done just before purchase. These types of non-adopters become instant adopters. Thinker buyer is not immediate-adopter, and it will take the time for the buying of a brand-new product. Boundedness, Positivity and Basic influence numbers(BINs) of the model is studied. Stability analysis is carried out for all the possible equilibrium points. Hopf bifurcation analysis has also been determined at the adopter-free state and interior equilibrium. Sensitivity analysis for the basic reproduction number and variables about the parameters of the model has been examined for the internal steady state. Finally, numerically experimentations have been carried out to support our analytical findings with a different set of parameters.


Introduction
As the issues concerning the Bass model in [1] study about the modeling regarding the diffusion of innovations has received into a form of research consisting of several handful studies, publications, also different other papers.Analysis has been established to examine the basic and logical opinions and evaluation issues underlying the diffusion models of current product recognition.The critics estimate that development during the earlier two decades.Others find with an investigation plan on offer of diffusion models assumption-based more quality and realistically higher efficient including real to be.Diffusion is the plan of action by that innovation is transmitted within specific channels across time between the part of a social arrangement.Since a hypothesis of developments, diffusion system has the major objective on transmission channels, that are the means through which knowledge regarding reform is imparted into the social arrangement.Researchers in consumer behavior has been associated with assessing the applicability of hypothesis extended in the global diffusion field to buyer analysis [2].Throughout its introduction into marketing during the 1960s, [1,3,4,5], innovation diffusion system possesses flashed considerable investigation among customer response, marketing administration including supervision and marketing science critics.Specific marketing management articles become targeted toward the involvement of those theories which target new product possibilities also as discovering marketing strategies directed to possible adopters [6,7].Researchers within management plus retailing science should be devoted to these extensions from diffusion system by suggesting models analytically for explaining and projecting the diffusion about innovation within any social order.Currently, research, as mentioned earlier more, has been concerned growing standardizing rules as for how a modification should be diffused into the social arrangement.As a special edition regarding particular Bass model, an investigation concerning the modeling of the diffusion about reform within marketing has the outcome of extensive research.The contribution regarding the analysis during the 1970s was observed by [8].However, during the ensuing decade, a plethora regarding investigations has provided to ours knowledge regarding particular fundamental, evaluation and conceptual opinions holding diffusion models.Modeling concerning innovation diffusion has been significantly performed to the obstacles from a wide field of disciplines [9].Models recommended dealing with the aspect through the aggregate and non-aggregate levels must evolve over a significant period.Status of uncertainty into the models like innovation diffusion can be incorporated within these model parameters, its arrangement, or both.The resulting models should describe the business to the aggregate or disaggregate level also be employed as predictive or standardizing purposes [10].The author introduced an epidemic model as the infection spread from an interaction of infected individual to premature individual.Similarly, in the diffusion process non-adopter population can become the user of innovation by the direct interaction with the adopter population [11].Further, The author proposed an epidemic model; the infection can be control by vaccination and media information as a key factor to aware the people.Several researchers analyzed the epidemics models without consolidating the media awareness in controlling the diseases [12,13,14,15].Many researchers studied the influence of media coverage in controlling the infectious diseases [16,17,18].At present scenario, media play a significant role in product expansion [19].In the present study, we make some modifications in the model presented by [20].This paper is planned as follows: In section 1, describe the introduction of the research study.
In section 2, representation of the proposed model.In section 3, positivity along with boundedness of the proposed model has been studied.In section 4, basic influence number of the model, asymptotic stability for every possible steady state has been analyzed the stipulation for the occurrence of Hopf bifurcation about the adopter-free equilibrium and the interior steady state.
In section 5, the sensitivity analysis of the basic reproduction number and essential variables at interior steady state concerning model parameters has been achieved.At last, in section 6, numerically experimentations have been carried out to support our analytical findings with a different set of parameters.

Representation of the Proposed Model
The supposition of the Proposed model is discussed below: (i) Assuming that N(t), I(t), A(t) the population densities of the non-adopter group, thinker group and adopter group.
(ii) The relationship among non-adopter and adopter group is through and word of mouth.
For enhance the impact of word of mouth, media plays a significant role in influencing the non-adopter population, who has to become an adopter of a new product.
(iii)Due to media coverage and active command by word of mouth, some non-adopters will purchase the new product and meet the adopter group instantaneous.That type of the nonadopter is known as the instant buyer.On the other side, the non-adopter who are not immediately affected through media and word of mouth, but they can buy the product in eventuality.This type of non-adopter will join into the thinker group and known as thinker buyer.
(iv) Here, β is the instantaneous transformation rate non-adopter N(t) to adopter A(t).
(v) Here α fraction of non-adopter N(t) people will join to thinker group I(t), we have recognized τ is the lag period in thinker group as the effect of media and word of mouth, in this group is not instantaneous.There is some time gap between non-adopter to the adopter.The part of thinker group γ I will meet the adopter group after the delay τ.
(vi) Assume K is the recruitment rate of non-adopter N(t).ξ is the frustration rate of adopter A(t) and join the non-adopter group N(t) and m is coefficient of media coverage.Finally, µ is the death rate of non-adopter N(t) and adopter A(t).

Positivity and boundedness
For the positivity along with the boundedness of the proposed system (1) − (3), we state and evaluate the below mentioned lemmas: Lemma 3.1.The solution for the proposed system (1) − (3), with initial values are nonnegative, for all t ≥ 0.
Proof.Let N, I, A be the solution for the proposed system (1) − (3), with positive initial populations.We suppose N for t ∈ [0, τ].On solving the expression (1), may be written as which exhibit that For the expression (2), t ∈ [0, τ] may be recast as which exhibit that For the expression (3), t ∈ [0, τ] may be recast as With the above discussion, we prove that N(t) > 0, I(t) > 0 and A(t) > 0 for all t ≥ 0.
Lemma 3.2.The solution for the proposed system (1) − (3), with initial conditions is bounded uniformly in ζ , where with respect to t, we have On solving, we notice 0 ≤ W (t) ≤ K µ as t → ∞.Hence, W (t) is bounded.Therefore, the proposed system is bounded.

Reproduction Number and Dynamical Nature of the system
Now, we will explain one valuable, significant parameter which explains mathematical problems regarding new product adoption is the basic influence rate, basic influence number or basic influence ratio.The basic influence number is stated as the required fraction of subsequent adoption created by an adopted person that is related to basic influence number in disease spreading models [15].The advantage of that matrix is to define either or no the products will diffuse into people.For the projected model equation (1) − (3) has an adopter-free equilibrium position, i.e.E 0 ( K µ , 0, 0).Using the notation in [15,21], where F and V are two vectors to express the new adopters with the personal touch of adopter community with the non-adopter community and outlasting transfer terms, sequentially concerning the model expression of adopter sections.
Now, the jacobian of F and V by using the adopter-free equilibrium E 0 .
Where F is positive and V is a non-singular Matrix.Consequently, FV −1 is positive.
Basic influence number is represented by R A , which is the largest eigenvalue.Where, The nature of stability for possible equilibrium points of the system (1) − (3), is as follows: (i)The adopter-free equilibrium E 0 ( K µ , 0, 0) always exists.
(ii) For m > 0 the interior equilibrium E * (N * , I * , A * ) exists, where N * , I * , A * is given by when m = 0, the interior equilibrium E * (N * , I * , A * ) exists if R A > 1, where N * , I * , A * is given by As the transcendental polynomials equation concerning the 2nd degree of the pattern was studied by [22,23] and examined the mentioned below results: (i) If (B1) − (B3) exists, entire roots of (7) possess negative real parts for all τ ≥ 0.
Proof.Now, we apply the specific method depends upon the central manifold system to investigate the nature of local stability of interior steady state using ξ as the bifurcation parameter [24].
We consider the adopter-free equilibrium E 0 ( K µ , 0, 0) and recognize the significant condition of the bifurcation parameter about The eigenvalue of the matrix (τ = 0) is .
On solving λ = −µ, 0, −(γ + αK µ ).Thus λ = 0 is the simple eigenvalue and remaining two eigenvalues have negative real parts.Therefor, we can use the central manifold theory.It can be simply determined that the Jacobian J 0 about ξ = ξ * posses right eigenvector provided with W = (w 1 , w 2 , w 3 ) T so that J 0 W = 0, where The left eigenvector is presented with V = (v 1 , v 2 , v 3 ) that meet V J 0 = 0 and V.W = 1, where Here we take the notation x 1 = N, x 2 = I, x 3 = A.
Evaluating the 2nd order partial derivatives about the adopter-free equilibrium, we get and rest 2nd order partial derivatives are equal to zero.Consequently, we compute the coefficient of a and b by applying the resulting formula mentioned below.
Presently, we state the following lemma as related as presented within Song et al. [23,25]; (ii) If m > 0 or m = 0, then interior equilibrium E * is locally asymptotically stable for all τ ∈ (0, τ + 0 ] for τ > 0. (iii) If τ ≥ τ + 0 , then the interior equilibrium E * is unstable and the Hopf bifurcation occurs in the system, for m > 0 or m = 0.
Proof.This corresponding characteristic expression for the Jacobian matrix about the interior equilibrium point E * (N * , I * , A * ), is given as: where Case I: When τ = 0 and m > 0, the transcendental equation ( 20), reduces so By using Routh-Hurwitz method, entire roots of equation (22), have negative real parts and the interior equilibrium E * with media effect is locally asymptotically stable because Case II: When τ = 0 and m = 0, the transcendental equation (20), reduces so By using Routh-Hurwitz method, entire roots of equation (23), have negative real parts and the interior equilibrium E * without media effect is locally asymptotically stable because Case III: When τ > 0 and m > 0.
Let λ = iw is root of (20), as we make compare real and imaginary parts from (22), we have Simplifying ( 25) − (26), we make On substituting w 2 = z in equation (27), we state With Lemma (4.3), there holds at least one non-negative root w = w 0 of equation ( 27) satisfying ( 25) and (26), that means the equation (20) possess a pair of purely imaginary roots ±iw 0 .
The equation (21), reduces to On solving, by Lemma (4.4) the condition [iii] holds good.We can get the value of critical τ + k by the similar method as mentioned in case(III).

Sensitivity Analysis
In this section, sensitivity analysis for the basic influence number R A and state variables for the proposed system (1) − (3), concerning the context of the model parameters at the interior equilibrium position, has achieved.The corresponding sensitive parameters for the basic influence number R A and for the state variables about the interior equilibrium are presented in Table 2 and Table 3 respectively.Using parametric conditions shown in Table 1.It is established that K, α, β are the positive influence parameters to R A and ξ , µ are the negative influence parameters to R A .Next, α, β are the negative influence parameters and ξ , µ, m are the positive influence parameters for N * , So we notice large changes in N * by a small variation in the parameter ξ .
Succeeding, K is the most delicate parameter into I * also the resting parameters are less delicate as compared to K in I * .On end, µ is the extremely delicate parameter within A * including all the additional parameters are less delicate to A * as contrasted to µ.

Numerical Simulations
To verify the analytic conclusions of the model (see Fig. 1).
The effect of media coverage m on the fraction of thinker population is shown in figure (10) and (11).We established that the level of the interior steady state is significantly affected by the media coefficient m.  coefficient of media coverage (m).For the adopter-free state, basic influence number is R A < 1.

Conclusion
A transcritical bifurcation occurs at R A = 1 and for the interior steady state basic influence number is R A > 1.On introducing delay parameters τ Hopf-bifurcation appears in E 0 and E * .It is   examined that E 0 and E * is stable in the range τ ∈ (0, τ + 0 ).After this point, the system is an unstable and oscillatory character of the system occurs.Sensitivity analysis for basic influence number R A and for the state variables at interior steady state is also studied.At last, to defend the  analytical findings, we perform the numerical simulations with the different set of parameters.
We observe that as the effect of media coverage(m) increases, the fraction of thinker population is going on increases.

TABLE 1 .
Parametric conditions applied for sensitivity analysis

TABLE 2 .
The sensitivity listsγ R A y g = ∂ R A ∂ y g ×

TABLE 3 .
The sensitivity lists γ x h y g = ∂ x h ∂ y g × y gx h of the state variables for the proposed model equations (1) − (3), to the parameters y g as the parameter condi-