×

How to run a campaign: optimal control of SIS and SIR information epidemics. (English) Zbl 1410.92124

Summary: Information spreading in a population can be modeled as an epidemic. Campaigners (e.g., election campaign managers, companies marketing products or movies) are interested in spreading a message by a given deadline, using limited resources. In this paper, we formulate the above situation as an optimal control problem and the solution (using Pontryagin’s Maximum Principle) prescribes an optimal resource allocation over the time of the campaign. We consider two different scenarios – in the first, the campaigner can adjust a direct control (over time) which allows her to recruit individuals from the population (at some cost) to act as spreaders for the Susceptible-Infected-Susceptible (SIS) epidemic model. In the second case, we allow the campaigner to adjust the effective spreading rate by incentivizing the infected in the Susceptible-Infected-Recovered (SIR) model, in addition to the direct recruitment. We consider time varying information spreading rate in our formulation to model the changing interest level of individuals in the campaign, as the deadline is reached. In both the cases, we show the existence of a solution and its uniqueness for sufficiently small campaign deadlines. For the fixed spreading rate, we show the effectiveness of the optimal control strategy against the constant control strategy, a heuristic control strategy and no control. We show the sensitivity of the optimal control to the spreading rate profile when it is time varying.

MSC:

92D30 Epidemiology
91D10 Models of societies, social and urban evolution
49N90 Applications of optimal control and differential games
91F10 History, political science

References:

[1] Asano, E.; Gross, L. J.; Lenhart, S.; Real, L. A., Optimal control of vaccine distribution in a rabies metapopulation model, Math. Biosci. Eng., 5, 219-238, (2008) · Zbl 1158.92324
[2] Behncke, H., Optimal control of deterministic epidemics, Opt. Contr. Appl. Meth., 21, 269-285, (2000) · Zbl 1069.92518
[3] Gaff, H.; Schaefer, E., Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6, 469, (2009) · Zbl 1169.49018
[4] Lashari, A. A.; Zaman, G., Optimal control of a vector borne disease with horizontal transmission, Nonlinear Anal.: Real World Appl., 13, 203-212, (2012) · Zbl 1238.93066
[5] Ledzewicz, U.; Schättler, H., On optimal singular controls for a general SIR model with vaccination and treatment, Discr. Contin. Dyn. Syst., 981-990, (2011) · Zbl 1306.49056
[6] Morton, R.; Wickwire, K. H., On the optimal control of a deterministic epidemic, Adv. Appl. Probab., 622-635, (1974) · Zbl 0324.92029
[7] Sethi, S. P.; Staats, P. W., Optimal control of some simple deterministic epidemic models, J. Oper. Res. Soc., 129-136, (1978) · Zbl 0383.92017
[8] Yan, X.; Zou, Y., Optimal Internet worm treatment strategy based on the two-factor model, Electron. Telecommun. Res. Inst. J., 30, (2008)
[9] Zhu, Q.; Yang, X.; Yang, L. X.; Zhang, C., Optimal control of computer virus under a delayed model, Appl. Math. Comput., 218, 11613-11619, (2012) · Zbl 1278.49045
[10] Castilho, C., Optimal control of an epidemic through educational campaigns, Electron. J. Differ. Equat., 1-11, (2006) · Zbl 1108.92035
[11] Karnik, A.; Dayama, P., Optimal control of information epidemics, Proc. IEEE Commun. Syst. Networks Conf., 1-7, (2012)
[12] Chierichetti, F.; Lattanzi, S.; Panconesi, A., Rumor spreading in social networks, Automata Lang. Program., 375-386, (2009) · Zbl 1247.05227
[13] Pittel, B., On spreading a rumor, SIAM J. Appl. Math., 213-223, (1987) · Zbl 0619.60068
[14] S. Boyd, A. Ghosh, B. Prabhakar, D. Shah, Gossip algorithms: design, analysis and applications, in: Proceedings of IEEE International Conference on Computer Communications, 2005, pp. 1653-1664.
[15] S. Belen, The Behaviour of Stochastic Rumours, Ph.D. dissertation, University of Adelaide, Australia, 2008.
[16] Sethi, S. P.; Prasad, A.; He, X., Optimal advertising and pricing in a new-product adoption model, J. Optim. Theory Appl., 351-360, (2008) · Zbl 1159.49035
[17] Krishnamoorthy, A.; Prasad, A.; Sethi, S. P., Optimal pricing and advertising in a durable-good duopoly, Eur. J. Oper. Res., 486-497, (2010) · Zbl 1177.90232
[18] Khouzani, M. H.R.; Sarkar, S.; Altman, E., Optimal control of epidemic evolution, Proceedings of IEEE International Conference on Computer Communications, 1683-1691, (2011)
[19] Barrat, A.; Barthlemy, M.; Vespignani, A., Dynamical processes on complex networks, (2008), Cambridge University Press · Zbl 1198.90005
[20] Fleming, W. H.; Rishel, R. W., Deterministic and stochastic optimal control, (1975), Springer-Verlag New York · Zbl 0323.49001
[21] Birkhoff, G.; Rota, G. C., Ordinary differential equations, (1989), John Wiley and Sons · Zbl 0183.35601
[22] Kamien, M. I.; Schwartz, N. L., Dynamic optimization: the calculus of variations and optimal control in economics and management, (1991), North-Holland Amsterdam · Zbl 0727.90002
[23] Kutz, N. J., Practical scientific computing, AMATH 581 course notes, (2005), Department of Applied Mathematics, University of Washington
[24] Fister, K. R.; Lenhart, S.; McNally, J. S., Optimizing chemotherapy in an HIV model, Electron. J. Differ. Equat., 1-12, (1998) · Zbl 1068.92503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.