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Innovation Diffusion Model for the Marketing of a Product with Interactions and Delay in Adoption for Two Different Patches

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Abstract

In this paper, an innovation diffusion model with the delay in adoption is planned to study the dynamics of the adopter of the same product within two different patches. It is determined that solutions are positive and bounded for the proposed system. Asymptotic stability analysis is carried out for all possible equilibrium points. The critical value of the delays \(\tau _1, \tau _2\) are determined. It is observed that for the interior equilibrium remains stable if either (or both) the adoption delays is (are) less than the threshold values, i.e., \(\tau _1<\tau _{10}^+,\tau _2>\tau _{20}^+\) or \(\tau _1>\tau _{10}^+,\tau _2<\tau _{20}^+\). If both \(\tau _1\) and \(\tau _2\) cross its thresholds, system perceived oscillating behavior, and Hopf bifurcation occurs. Sensitivity analysis for the basic influence number of the model has been examined. Subsequently, numerical simulations have been carried out to support our analytical findings with the different set of parameters.

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References

  1. Rogers, E.M.: Diffusion of Innovation, 4th edn. Free Press, New York (1995)

    Google Scholar 

  2. Bass, F.M.: A new product growth model for consumer durable. Manag. Sci. 15(5), 215–227 (1969)

    Article  Google Scholar 

  3. Lekvall, P., Wahlbin, C.: A study of some assumptions underlying innovation diffusion functions. Swed. J. Econ. 75, 362–377 (1973)

    Article  Google Scholar 

  4. Muller, E.: Trial/Awareness advertising decisions, a control problem with phase diagrams with non-stationary boundaries. J. Econ. Dyn. Control 6, 333–350 (1983)

    Article  Google Scholar 

  5. Singh, H., Dhar, J., Bhatti, H.S., Chandok, S.: An epedemic model of childhood disease dynamics with maturation delay and latent period of infection. Model. Earth Syst. Environ. 2(2), 1–8 (2016)

    Google Scholar 

  6. Tuli, R., Dhar, J., Bhatti, H.S., Singh, H.: Dynamical response by the instant buyer and thinker buyer in an innovation diffusion marketing model with media coverage. J. Math. Comput. Sci. 7(6), 1022–1045 (2017)

    Google Scholar 

  7. Kalish, S.: Monopolist pricing with dynamic demand and production cost. Mark. Sci. 2, 135–159 (1983)

    Article  Google Scholar 

  8. Sethi, S.P.: Optimal advertising policy with the conagion model. J. Optim. Theory. Appl. 29, 615–626 (1979)

    Article  MathSciNet  Google Scholar 

  9. Horsky, D., Simon, L.S.: Advertising and the diffusion of new products. Mark. Sci. 2, 1–17 (1983)

    Article  Google Scholar 

  10. Maleknejad, k, Mirzaee, F.: Numerical solution of stochastic linear heat conduction problem by using new algorithims. Appl. Math. Comput. 163(1), 97–106 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Mirzaee, F., Bimesl, S., Tohidi, E.: A numerical framework for solving high-order pantography-delay Volterra integro-differential equations. Kuwait J. Sci. Eng. 43(1), 69–83 (2016)

    Google Scholar 

  12. Mirzaee, F., Bimesl, S.: A uniformly convergent Euler matrix method for telegraph equations having constant coefficients. Mediterr. J. Math. 13(1), 497–515 (2016)

    Article  MathSciNet  Google Scholar 

  13. Maurer, S.M., Huberman, B.A.: Competitive dynamics of websites. J. Econ. Dyn. Control 27, 2195–2206 (2003)

    Article  Google Scholar 

  14. Kim, J., Lee, D.J., Ahn, J.: A dynamic competition analysis on the Korean mobile phone market using competitive diffusion model. Comput. Ind. Eng. 51, 174–182 (2006)

    Article  Google Scholar 

  15. Lopez, L., Sanjuan, M.F.A.: Defining strategies to win in the internet market. Physica A 301, 512–534 (2001)

    Article  Google Scholar 

  16. Mahajan, V., Peterson, R.A.: Innovation diffusion in a dynamic potential adopter population. Manag. Sci. 24, 1589–1597 (1978)

    Article  Google Scholar 

  17. Wendi, W., Fergola, P., Tenneriello, C.: An innovation diffusion model in patch environment. Appl. Math. Comput. 134, 51–67 (2003)

    MathSciNet  MATH  Google Scholar 

  18. Sisodiya, O.S., Mishra, O.P., Dhar, J.: Pathogen induced infection and Its control by vaccination: a mathematical model for cholera disease. Int. J. Appl. Comput. Math. 4, 74 (2018)

    Article  MathSciNet  Google Scholar 

  19. Giovangis, A.N., Skiadas, C.H.: A stochastic logistic innovation diffusion model studying the electricity consumption in Greece and the United States. Technol. Forecast. Soc. Change 61, 235–246 (1999)

    Article  Google Scholar 

  20. Gruber, H.: Competition and innovation: the diffusion of mobile telecommunication in central and eastern Europe. Inf. Econ. Policy 3, 19–34 (2001)

    Article  Google Scholar 

  21. Jun, D.B., Kim, S.K.: Forecasting telecommunication service subscribers in substitutive and competitive environments. Int. J. Forecast. 18, 561–581 (2002)

    Article  Google Scholar 

  22. Dhar, J., Tyagi, M., Sinha, P.: An innovation diffusion model for the survival of a product in a competitive market: basic influence number. Int. J. Pure Appl. Math. 89(4), 439–448 (2013)

    Article  Google Scholar 

  23. Dhar, J., Tyagi, M., Sinha, P.: The impact of media on a new product innovation diffusion: a mathematical model. Bol. Soc. Parana. Mat. 33(1), 169–180 (2015)

    Article  MathSciNet  Google Scholar 

  24. Kalish, S., Mahajan, V., Muller, E.: Waterfall and sprinkler new-product strategies in competitve global markets. Int. J. Res. Mark. 2, 105–119 (1995)

    Article  Google Scholar 

  25. Sahu, G.P., Dhar, J.: Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate. Appl. Math. Model. 36(3), 908–923 (2012)

    Article  MathSciNet  Google Scholar 

  26. Driwssche, P.Vanden, Watmough, J.: Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  27. Ruan, S.: Absolute stabilty, conditional stability and bifurcation in Kolmogrov-type predator-prey systems with discrete delays. Q. Appl. Math. 59(1), 159–174 (2001)

    Article  Google Scholar 

  28. Singh, H., Dhar, J., Bhatti, H.S.: Dynamics of a prey generalized predator system with disease in prey and gestation delay for predator. Model. Earth Syst. Environ. 2, 52 (2016)

    Article  Google Scholar 

  29. Lin, X., Wang, H.: Stability analysis of delay differential equations with two discrete delays. Can. Appl. Math. Q. 20(4), 519–533 (2012)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

I express my warm thanks to I.K.G. Punjab Technical University, Punjab for providing me the facilities for the research being required.

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Correspondence to Rishi Tuli.

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Tuli, R., Dhar, J. & Bhatti, H.S. Innovation Diffusion Model for the Marketing of a Product with Interactions and Delay in Adoption for Two Different Patches. Int. J. Appl. Comput. Math 4, 149 (2018). https://doi.org/10.1007/s40819-018-0583-x

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