An innovation diffusion model with two innovations under two simultaneous effects in a competitive market: co-existence through optimal control

Volume 35, Issue 1, pp 1--15 https://dx.doi.org/10.22436/jmcs.035.01.01

Publication Date: April 11, 2024 Submission Date: August 16, 2023 Revision Date: October 04, 2023 Accteptance Date: February 19, 2024

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Authors

S. Chugh - Department of Mathematics, D. M. College, Moga, 142001, Punjab, India. - Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, 148106, Punjab, India. J. Dhar - Department of Applied Science, ABV-Indian Institute of Information Technology and Management, Gwalior, 474015, Madhya Pradesh, India. R. K. Guha - Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, 148106, Punjab, India.

Abstract

Since mass media plays an important role in influencing the non-adopter population, the adoption rate changes with the media awareness rate. Motivated by this concept, in this paper, a model of three non-intersecting classes of non-adopters, adopters for product-I and adopters for product-II is proposed. Under the influence of media coverage and word-of-mouth, the dynamic behaviour of the system is investigated. The basic influence numbers \({\cal R}_{0_1}\) and \({\cal R}_{0_2}\) associated with the first and second innovations help in performing stability analysis. It is observed from stability analysis that adopter-free equilibrium is conditionally stable. Also, the system has no stable interior equilibrium point. The basic influence numbers determine the sustainability of a particular product in the market. The optimal control theory is used to reduce the frustration rate in both adopter classes. The Hamiltonian function is constructed using the extended optimum control model and is then solved according to Pontryagin's maximum principle to get the cost. Also, coexistence is possible with the implementation of optimal control. Sensitivity analysis has been performed for both the basic influence numbers \({\cal R}_{0_1}\) and \({\cal R}_{0_2}\). Lastly, numerical experimentations have been executed to assist analytical findings with distinct sets of parameters.

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Keywords

• Basic influence number
• word-of-mouth effect
• impact of media
• sensitivity analysis
• optimal control

MSC

•  37C75
•  93D20
•  37C20
•  35F21

References

• [1] F. M. Bass, A new product growth model for consumer durable, Manage. Sci., 15 (1969), 215–227
  • View Article


• [2] S. Chugh, J. Dhar, R. K. Guha, Stability and optimal control of two products innovation diffusion system, Results Control Optim., 14 (2024), 1–19
  • View Article
  • Google Scholar


• [3] S. Chugh, R. K. Guha, J. Dhar, An innovation diffusion model in partial competitive and cooperative market: Analysis with two innovations, J. Appl. Pure Math., 4 (2022), 27–36
  • View Article
  • Google Scholar


• [4] J. Dhar, M. Tyagi, P. Sinha, An innovation diffusion model for the survival of a product in a competitive market: Basic influence numbers, Int. J. Pure Appl. Math., 89 (2013), 439–448
  • View Article
  • Google Scholar
  • MATH


• [5] J. Dhar, M. Tyagi, P. Sinha, The impact of media on a new product innovation diffusion: A mathematical model, Bol. Soc. Parana. Mat. (3), 33 (2015), 171–182
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] W. H. Fleming, R. W. Rishel, Deterministic and stochastic optimal control, Springer-Verlag, New York (2012)
  • View Article
  • Google Scholar


• [7] C. Furlan, C. Mortarino, M. S. Zahangir, nteraction among three substitute products: An extended innovation diffusion model, Stat. Methods Appl., 30 (2021), 269–293
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] N. K. Goswami, B. Shanmukha, Dynamics of covid-19 outbreak and optimal control strategies: A model-based analysis, Adv. Syst. Sci. Appl., 21 (2021), 65–86
  • View Article
  • Google Scholar


• [9] H. Guo, Global dynamics of a mathematical model of tuberculosis, Can. Appl. Math. Q., 13 (2005), 313–323
  • View Article
  • Google Scholar
  • MATH


• [10] D. Horsky, L. S. Simon, Advertising and the diffusion of new products, Mark. Sci., 2 (1983), 1–17
  • View Article
  • Google Scholar


• [11] L. Huo, J. Xu, J. He, T. Lin, Maximizing the influence of the innovative products diffusion considering advertisement and promotion strategies, Discrete Dyn. Nat. Soc., 2021 (2021), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] D. C. Jain, R. C. Rao, Effect of price on the demand for durables: Modeling, estimation and findings, J. Bus. Econ. Stat., 8 (1990), 163–170
  • View Article
  • Google Scholar


• [13] J. M. Jones, C. J. Ritz, Incorporating distribution into new product diffusion models, Int. J. Res. Mark., 8 (1991), 91–112
  • View Article
  • Google Scholar


• [14] M. C. Kekana, O. D. Makinde, A deterministic model for (n = 2) competitive products in a market system, Am. J. Undergrad. Res., 10 (2011), 11–20
  • View Article
  • Google Scholar


• [15] J. Kim, D.-J. Lee, J. Ahn, A dynamic competition analysis on the korean mobile phone market using competitive diffusion model, Comput. Ind. Eng., 51 (2006), 174–182
  • View Article
  • Google Scholar


• [16] M. Kot, Elements of mathematical ecology, Cambridge University Press, Cambridge (2001)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] O. Koutou, B. Sangar´e, B. Traor´e, Optimal control of malaria transmission dynamics combining some usual strategies and an imperfect vaccine, Discuss. Math. Differ. Incl. Control Optim., 40 (2020), 33–59
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] R. Kumar, A. K. Sharma, K. Agnihotri, Dynamics of an innovation diffusion model with time delay, East Asian J. Appl. Math., 7 (2017), 455–481
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] R. Kumar, A. K. Sharma, K. Agnihotri, Stability and bifurcation analysis of a delayed innovation diffusion model, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 709–732
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] R. Kumar, A. K. Sharma, G. P. Sahu, Dynamical behavior of an innovation diffusion model with intra-specific competition between competing adopters, Acta Math. Sci. Ser. B (Engl. Ed.), 42 (2022), 364–386
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] D. L. Lukes, Differential equations: classical to controlled, Academic Press, New York (1982)
  • View Article
  • Google Scholar


• [22] T. Modis, Technological forecasting at the stock market, Technol. Forecast. Soc. Change, 62 (1999), 173–202
  • View Article
  • Google Scholar


• [23] L. S. Pontryagin, Mathematical theory of optimal processes, CRC Press, (1987)
  • View Article
  • Google Scholar


• [24] G. P. Sahu, J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate, Appl. Math. Model., 36 (2012), 908–923
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] C. J. Silva, H. Maurer, D. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321–337
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] R. Tuli, J. Dhar, H. S. Bhatti, Innovation diffusion model with interactions and delays in adoption for two competitive products in two different patches, Ric. Mat., 68 (2019), 705–726
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] R. Tuli, J. Dhar, H. S. Bhatti, Sustaining of two competing products under the impact of the media including the experience of adopters, J. Appl. Math. Comput., 60 (2019), 343–367
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] R. K. Upadhyay, S. Acharya, Modeling the recent outbreak of COVID-19 in India and its control strategies, Nonlinear Anal. Model. Control, 27 (2022), 254–274
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] P. Van den Driwssche, J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48
  • View Article
  • Google Scholar


• [30] Y. Yu, W. Wang, Y. Zhang, An innovation diffusion model for three competitive products, Comput. Math. Appl., 46 (2003), 1473–1481
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] D. Zhao, S.-F. Ji, H.-P. Wang, L.-W. Jiang, How do government subsidies promote new energy vehicle diffusion in the complex network context? a three-stage evolutionary game model, Energy, 230 (2021), 1–13
  • View Article
  • Google Scholar