A survey on two viruses extensions of epidemic model with continuous and impulse control

Elena Gubar ; Vladislav Taynitskiy

Contributions to game theory and management, Tome 14 (2021), pp. 127-154.

Résumé

The current study represents a survey on several modifications of compartment epidemic models with continuous and impulse control policies. The main contribution of the survey is the modification of the classical Susceptible Infected Recovered (SIR) model with the assumption that two types of viruses are circulating in the population at the same time. Moreover, we also take into consideration the network structure of the initial population in two-virus SIIR models and estimate the effectiveness of protection measures over complex networks. In each model, the optimal control problem has been formalized to minimize the costs of the virus spreading and find optimal continuous and impulse antivirus controllers. All theoretical results are corroborated by a large number of numerical simulations.

Détail

Keywords: epidemic process, compartment epidemic models, SIR model, optimal control, evolutionary games, virus mutation, structured population.

Bibliographie

[1] Agur Z. et al., “Pulse mass measles vaccination across age cohorts”, Proceedings of the National Academy of Sciences of the United States of America, 90 (1993), 11698–11702

[2] Altman E. et al., “Dispatch then stop: Optimal dissemination of security patches in mobile wireless networks”, Proceedings of the 48th IEEE Conference on Decisions and Control, CDC, 2010, 2354–2359 | MR

[3] Balter M., “New HIV strain could pose health threat”, Science, 281:5382 (1998), 1425–1426

[4] Barabási A. L., R. Albert, “Emergence of scaling in random networks”, Science, 286:5439 (1999), 509–512 | MR | Zbl

[5] Barabási A.-L., R. Albert, “Statistical mechanics of complex networks”, Rev. Mod. Phys., 74 (2002), 47 | MR | Zbl

[6] Behncke H., “Optimal control of deterministic epidemics”, Optimimal Control Appl. Methods, 21 (2000), 269–285 | MR | Zbl

[7] Blaquière A., “Impulsive Optimal Control with Finite or Infinite Time Horizon”, Journal Of Optimization Theory And Applications, 46 (1985) | MR | Zbl

[8] Bomze I., S. Rota Bulò, “Infection and immunization: A new class of evolutionary game dynamics”, Games Econ. Behav., 71 (2011), 193–211 | MR | Zbl

[9] Bonhoeffer S. et al., “Virus dynamics and drug therapy”, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971–6976

[10] Butler D., “Flu surveillance lacking”, Nature, 483 (2012), 520–522

[11] Capasso V., Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97, Springer, Berlin–Heidelberg, 1993 | MR | Zbl

[12] Castillo-Chavez C. et al., “Competitive exclusion in gonorrhea models and other sexually transmitted diseases”, SIAM J. on App. Math., 56:2 (1996), 494–508 | MR | Zbl

[13] Chahim M. et al., “A tutorial on the deterministic Impulse Control Maximum Principle: Necessary and sufficient optimality conditions”, European Journal of Operational Research, 219 (2012), 18–26 | MR | Zbl

[14] Chowell G. et al., “Perspectives on model forecasts of the 2014–2015 Ebola epidemic in West Africa: Lessons and the way forward”, BMC Med., 15 (2017), 42 | DOI

[15] Conn M., Handbook of Models for Human Aging, Academic Press, London, UK, 2006

[16] Dykhta V. A., O. N. Samsonyuk, “A maximum principle for smooth optimal impulsive control problems with multipoint state constraints”, Computational Mathematics and Mathematical Physics, 49 (2009), 942–957 | MR | Zbl

[17] Francis P. J., “Optimal tax/subsidy combinations for the flu season”, J. Econ. Dyn. Control, 28 (2004), 2037–2054 | MR | Zbl

[18] Fu X. et al., “Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization”, Phys. Rev. E., 77:3 (2008), 036113 | MR

[19] Fu F. et al., “Imitation dynamics of vaccination behaviour on social networks”, Proc. R. Soc. Lond. Ser. B, 278 (2011), 42–49

[20] Gjorgjieva J. et al., “The role of vaccination in the control of SARS”, Math. Biosci. Eng., 2 (2005), 753–769 | MR | Zbl

[21] Gubar E., Q. Zhu, “Optimal Control of Influenza Epidemic Model with Virus Mutations”, Proceedings of the 12th Biannual European Control Conference (Zurich, Switzerland, 2013), IEEE Control Systems Society, New York, NY, USA, 2013, 3125–3130

[22] Gubar E. et al., “Impact of Propagation Information in the Model of Tax Audit”, Recent Advances in Game Theory and Applications. “Static and Dynamic Game Theory: Foundations and Applications” (Switzerland, 2015), 91–110 | Zbl

[23] Kermack W. O., A. G. McKendrick, “A contribution to the mathematical theory of epidemics”, Proceedings of the Royal Society of London. Series A, 115:772 (1927), 700–721 | Zbl

[24] Kharraz A. et al., “Cutting the gordian knot: A look under the hood of ransomware attacks”, International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment, Springer, 2015, 3–24

[25] Khatri S. et al., The Role of Network Topology on the Initial Growth Rate of Influenza Epidemic, Technical Report BU-1643-M, KeAi, China, 2003

[26] Khouzani M. H. R. et al., “Maximum damage malware attack mobile wireless networks”, Proceedings of the 29th International Conference on Computer Communications, INFOCOM (San Diego, CA, USA, 2010), 749–757

[27] Khouzani M. H. R. et al., “Optimal control of epidemic evolution”, Proceedings of the 30th International Conference on Computer Communications, INFOCOM (Shanghai, China, 2011), 1683–1691

[28] Leander R. et al., “Optimal control of continuous systems with impulse controls”, Optimal Control Applications and Methods Optim. Control Appl. Meth., 36 (2015), 535–549 | MR | Zbl

[29] Luo X., Q. Liao, “Ransomware: A new cyber hijacking threat to enterprises”, Handbook of research on information security and assurance, 2009, 1–6

[30] Masuda N., N. Konno, “Multi-state epidemic processes on complex networks”, J. Theor. Biol., 243:1 (2006), 64–75 | MR | Zbl

[31] Newman M. E. J., “Threshold effects for two pathogens spreading on a network”, Phys. Rev. Lett., 95 (2005), 108701

[32] Newman L. H., “What we know about Friday's massive east coast Internet outage”, Wired Magazine, 2016

[33] Nuno M. et al., “Dynamics of two-strain influenza with isolation and partial cross-immunity”, SIAM J. on App. Math., 65:3 (2005), 964–982 | MR | Zbl

[34] Omic J. et al., “Protecting against network infections: A game theoretic perspective”, Proceedings of the 28th IEEE Conference on Computer Communications, INFOCOM (Rio de Janeiro, Brazil, 2009)

[35] Pappas G. J. et al., “Analysis and Control of Epidemics: A Survey of Spreading Processes on Complex Networks”, IEEE Control Systems Magazine, 36:1 (2016), 26–46 | DOI | MR | Zbl

[36] Pastor-Satorras R., A. Vespignani, “Epidemic spreading in scale-free networks”, Phys. Rev. Lett., 86:14 (2001), 3200

[37] Perkins T. A., G. Espana, “Optimal Control of the COVID-19 Pandemic with Non-pharmaceutical Interventions”, Bulletin of mathematical biology, 82:9 (2020), 118 | DOI | MR | Zbl

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

[39] Pragyan D. et al., The Economic Effects of Covid-19 Containment Measures, IMF Working Paper WP/20/158, 2020

[40] Rohloff K. R., T. Başar, “Deterministic and stochastic models for the detection of random constant scanning worms”, ACM Trans. Model. Comput. Simul. (ACM TOMACS), 18 (2008), 1–24

[41] Rowson T. et al., “How and When to End the COVID-19 Lockdown: an Optimization approach”, Frontiers of Public Health, 10 (2020) | DOI

[42] Rowthorn R., F. Toxvaerd, The Optimal Control of Infectious Diseases via Prevention and Treatment, Technical Report 2013, Cambridge-INET Working Paper, 2020

[43] Sethi S. P., G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Springer, Berlin, 2006 | MR

[44] Smith G. et al., “Origins and evolutionary genomics of the 2009 swine-origin H1 N1 influenza A epidemic”, Nature, 459:7250 (2009), 1122–1125

[45] Strogatz S. H., “Exploring complex networks”, Nature, 410:6825 (2001), 268–276 | Zbl

[46] Taynitskiy V., Gubar E., E. Zhitkova, “Structure of optimal control in the model of propagation of two malicious softwares”, Proceedings of the International Conference "Stability and Control Processes» in Memory of V.I. Zubov, SCP (Saint-Petersburg, Russia, 2015), 261–264 | MR

[47] Taynitskiy V., Gubar E., Q. Zhu, “Optimal Security Policy for Protection Against Heterogeneous Malware”, Proceedings of the International Conference on «Network Games, Control and Optimization», NETGCOOP 2016 (Avignon, France, 2016), 199–210

[48] Taynitskiy V., Gubar E., E. Zhitkova, “Optimization of protection of computer networks against malicious software”, Proc. of International Conference “Stability and Oscillations of Nonlinear Control Systems”, Pyatnitskiy's Conference, 2016

[49] Taynitskiy V., Gubar E., Q. Zhu, “Optimal Impulse Control of Bi-Virus SIR Epidemics with Application to Heterogeneous Internet of Things”, Constructive Nonsmooth Analysis and Related Topics, Abstracts of the International Conference. Dedicated to the Memory of Professor V.F. Demyanov, 2017, 113–116 | MR

[50] Taynitskiy V., Gubar E., Q. Zhu, “Optimal Control of Multi-strain Epidemic Processes in Complex Networks”, Game Theory for Networks, GameNets 2017, Lecture Notes of the Institute for Computer Sciences, 212, Social Informatics and Telecommunications Engineering, Springer, 2017, 108–117 | MR | Zbl

[51] Thomasey D. H., M. Martcheva, “Serotype replacement of vertically transmitted diseases through perfect vaccination”, J. Biol. Sys., 16:02 (2008), 255–277 | Zbl

[52] Vespignani A. et al., “Epidemic processes in complex networks”, Rev. Mod. Phys., 87 (2015), 925–979 | MR

[53] Wickwire K. H., “A note on the optimal control of carrier-borne epidemics”, J. Appl. Probab., 12 (1975), 565–346 | MR

[54] Zaccour G., Reddy P., S. Wrzaczek, “Quality effects in different advertising models - An impulse control approach”, European Journal of Operational Research, 255 (2016), 984–995 | MR | Zbl

[55] Zhu Q., Bushnell L., T. Başar, “Game-theoretic analysis of node capture and cloning attack with multiple attackers in wireless sensor networks”, Proceedings of the 51st IEEE Conference on Decision and Control, CDC'12 (Maui, HI, USA, 2012)