On the Dynamics of a Viral Marketing Model with Optimal Control using Indirect and Direct Methods

  • João N.C. Gonçalves University of Minho, Portugal
  • M. Teresa T. Monteiro University of Minho
  • Helena Sofia Rodrigues Polytechnic Institute of Viana do Castelo
Keywords: Indirect Methods, Direct Methods, Optimal Control Theory, Viral Marketing, SIR Epidemiological Model

Abstract

The complexity of optimal control problems requires the use of numerical methods to compute control and optimal state trajectories for a dynamical system, aiming to optimize a particular performance index. Considering a real viral advertisement, this article compares the dynamics of a viral marketing epidemic model with optimal control under different cost scenarios and from two perspectives: using numerical methods based on the Pontryagin's Maximum Principle (indirect methods) and methods that treat the optimal control problem as a nonlinear constrained optimization problem (direct methods). Based on the trade-off between the maximization of information spreading and the minimization of the costs associated to it, an optimal control problem is formulated and studied. The existence and uniqueness of the solution are proved. Our results show not only that the cost of implement control policies is a crucial parameter for the spreading of marketing messages, but also that low investment costs in control strategies fulfill the proposed trade-off without compromising the financial capacity of a company.

References

J. Leskovec, L.A. Adamic and B.A. Huberman, The dynamics of viral marketing, ACM Transactions on the Web (TWEB), vol. 1,no. 1, pp. 1–39, 2007.

K. Kandhway and J. Kuri, Optimal control of information epidemics modeled as Maki Thompson rumors, Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 12, pp. 4135 4147, 2014.

K. Kandhway and J. Kuri, How to run a campaign: Optimal control of SIS and SIR information epidemics, Applied Mathematics and Computation, vol. 231, pp. 79–92, 2014.

A. Karnik and P. Dayama, Optimal control of information epidemics, in Proc. Fourth International Conference on Communication Systems and Networks (COMSNETS), 2012.

H.S. Rodrigues and M.J. Fonseca, Can information be spread as a virus? viral marketing as epidemiological model, Mathematical Methods in the Applied Sciences, vol. 39, no. 16, pp. 4780–4786, 2016.

L. Wang and B.C. Wood, An epidemiological approach to model the viral propagation of memes, Applied Mathematical Modelling,vol. 35, no. 11, pp. 5442–5447, 2011.

J. Yang, C. Yao, W. Ma and G. Chen, A study of the spreading scheme for viral marketing based on a complex network model,Physica A: Statistical Mechanics and its Applications, vol. 389, no. 4, pp. 859–870, 2010.

C. Long and R.C. Wong, Viral marketing for dedicated customers, Information systems, vol. 46, pp. 1–23, 2014.

A. Mochalova and A. Nanopoulos, A targeted approach to viral marketing, Electronic Commerce Research and Applications, vol.13, no. 4, pp. 283–294, 2014.

S. Rosa D.F.M. Torres, Parameter Estimation, Sensitivity Analysis and Optimal Control of a Periodic Epidemic Model with Application to HRSV in Florida, Statistics, Optimization & Information Computing, vol. 6, no. 1, pp. 139–149, 2018.

R. Denysiuk, H.S. Rodrigues, M.T.T. Monteiro, L. Costa, I. Esp´ ırito Santo and D.F.M. Torres, Multiobjective approach to optimal control for a dengue transmission model, Statistics, Optimization & Information Computing, vol. 3, no. 3, pp. 206–220, 2015.

L.S. Pontryagin, Mathematical theory of optimal processes, CRC Press, 1987.

R. Kheirandish, A.S. Krishen and P. Kachroo, Application of optimal control theory in marketing: What is the optimal number of choices on a shopping website?, International Journal of Computer Applications in Technology, vol. 34, no. 3, pp. 207–215, 2009.

J.N.C. Gonc ¸alves, H.S. Rodrigues and M.T.T. Monteiro, Optimal control strategies for an advertisement viral diffusion, in Proc.Congress of APDIO, the Portuguese Operational Research Society, vol. 223, pp. 135–149, 2017.

L. Stampler, How Dove’s’ Real Beauty Sketches’ became the most viral video ad of all time, Business Insider, 2017 (Accessed on September 2016).

W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, In Proc. of the Royal Society of London A: mathematical, physical and engineering sciences, vol. 115,no. 772, pp. 700–721, 1927.

S. Schroeder YouTube Now Has One Billion Monthly Users, 2016 (Accessed on September 2016).

J.C. Lagarias, J.A. Reeds, M.H. Wright and P.E. Wright, Convergence properties of the Nelder–Mead simplex method in low dimensions, SIAM Journal on Optimization, vol. 9, no. 1, pp. 112–147, 1998.

H.W. Hethcote, The mathematics of infectious diseases, SIAM Review, vol. 42, no. 4, pp. 599–653, 2000.

W.H. Fleming and R.W. Rishel, Deterministic and stochastic optimal control, Springer Science & Business Media, 2012.

K.R. Fister, S. Lenhart and J.S. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations,vol. 1998, no. 32, pp. 1–12, 1998.

D. Xu, X. Xu, Y. Xie and C. Yang, Optimal control of an SIVRS epidemic spreading model with virus variation based on complex networks, Communications in Nonlinear Science and Numerical Simulation, vol. 48, pp. 200–210, 2017.

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems Series B, vol. 2, no.4, pp. 473–482, 2002.

B. Passenberg, Theory and algorithms for indirect methods in optimal control of hybrid systems, Technischen Universität München,2012.

M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear programming: theory and algorithms, John Wiley & Sons, 2013.

E. Trélat, Optimal control and applications to aerospace: some results and challenges, Journal of Optimization Theory and Applications, vol. 154, no. 3, pp. 713–758, 2012.

S. Lenhart and J.T. Workman, Optimal control applied to biological models, CRC Press, 2007.

H.S. Rodrigues, M.T.T. Monteiro and D.F.M. Torres, Optimization of dengue epidemics: a test case with different discretization schemes, AIP Conference Proceedings, vol. 1168, no. 1, pp. 1385–1388, 2009.

R.H. Byrd, J. Nocedal and R.A. Waltz, KNITRO: An integrated package for nonlinear optimization, Large-scale nonlinear optimization, vol. 1168, no. 1, pp. 35–59, 2006.

A. Wätcher and L.T. Biegler, On the implementation of an interior point filter line-search algorithm for large-scale nonlinear programming, Mathematical programming, vol. 106, no. 1, pp. 25–57, 2006.

R.J. Vanderbei and D.F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Computational Optimization and Applications, vol. 13, no. 1, pp. 231–252, 1999.

R. Fourer, D.M. Gay and B.W. Kernighan, A modeling language for mathematical programming, Management Science, vol. 36, no.5, pp. 519–554, 1990.

J. Czyzyk, M.P. Mesnier and J.J. Moré, The NEOS server, IEEE Computational Science and Engineering, vol. 5, no. 3, pp. 68–75, 1998.

Published
2018-11-02
How to Cite
Gonçalves, J. N., T. Monteiro, M. T., & Sofia Rodrigues, H. (2018). On the Dynamics of a Viral Marketing Model with Optimal Control using Indirect and Direct Methods. Statistics, Optimization & Information Computing, 6(4), 633-644. https://doi.org/10.19139/soic.v6i4.441
Section
Research Articles