Optimal Control of Heterogeneous Mutating Viruses

Different strains of influenza viruses spread in human populations during every epidemic season. As the size of an infected population increases, the virus can mutate itself and grow in strength. The traditional epidemic SIR model does not capture virus mutations and, hence, the model is not sufficient to study epidemics where the virus mutates at the same time as it spreads. In this work, we establish a novel framework to study the epidemic process with mutations of influenza viruses, which couples the SIR model with replicator dynamics used for describing virus mutations. We formulated an optimal control problem to study the optimal strategies for medical treatment and quarantine decisions. We obtained structural results for the optimal strategies and used numerical examples to corroborate our results.

[1]  Tamer Basar,et al.  Deterministic and stochastic models for the detection of random constant scanning worms , 2008, TOMC.

[2]  D. E. Matthews.  Evolution and the Theory of Games , 1977 .

[3]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[4]  Eitan Altman,et al.  Dispatch then stop: Optimal dissemination of security patches in mobile wireless networks , 2010, 49th IEEE Conference on Decision and Control (CDC).

[5]  Eitan Altman,et al.  Maximum Damage Malware Attack in Mobile Wireless Networks , 2010, 2010 Proceedings IEEE INFOCOM.

[6]  V. Capasso.  Mathematical Structures of Epidemic Systems , 1993, Lecture Notes in Biomathematics.

[7]  Piet Van Mieghem,et al.  Protecting Against Network Infections: A Game Theoretic Perspective , 2009, IEEE INFOCOM 2009.

[8]  C. Cannings,et al.  Evolutionary Game Theory , 2010 .

[9]  Declan Butler,et al.  Flu surveillance lacking , 2012, Nature.

[10]  P. Francis,et al.  Optimal tax/subsidy combinations for the flu season , 2004 .

[11]  Quanyan Zhu,et al.  Optimal control of influenza epidemic model with virus mutations , 2013, 2013 European Control Conference (ECC).

[12]  Gerardo Chowell,et al.  Perspectives on model forecasts of the 2014–2015 Ebola epidemic in West Africa: lessons and the way forward , 2017, BMC Medicine.

[13]  Daniel I. S. Rosenbloom,et al.  Imitation dynamics of vaccination behaviour on social networks , 2011, Proceedings of the Royal Society B: Biological Sciences.

[14]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[15]  H. Behncke.  Optimal control of deterministic epidemics , 2000 .

[16]  M A Nowak,et al.  Virus dynamics and drug therapy. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Eitan Altman,et al.  Optimal control of epidemic evolution , 2011, 2011 Proceedings IEEE INFOCOM.

[18]  Quanyan Zhu,et al.  Optimal Security Policy for Protection Against Heterogeneous Malware , 2016 .

[19]  Elena Gubar,et al.  Structure of optimal control in the model of propagation of two malicious softwares , 2015, 2015 International Conference "Stability and Control Processes" in Memory of V.I. Zubov (SCP).

[20]  Piet Van Mieghem,et al.  Epidemic processes in complex networks , 2014, ArXiv.

[21]  M. Newman.  Threshold effects for two pathogens spreading on a network. , 2005, Physical review letters.

[22]  Gerardo Chowell,et al.  The role of vaccination in the control of sars. , 2005, Mathematical biosciences and engineering : MBE.

[23]  K. Wickwire,et al.  A note on the optimal control of carrier-borne epidemics , 1975, Journal of Applied Probability.

[24]  Quanyan Zhu,et al.  Game-theoretic analysis of node capture and cloning attack with multiple attackers in wireless sensor networks , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[25]  Samuel Rota Bulò,et al.  Infection and immunization: A new class of evolutionary game dynamics , 2011, Games Econ. Behav..

[26]  E. Blum,et al.  The Mathematical Theory of Optimal Processes. , 1963 .

[27]  James M. Hyman,et al.  The Role of Network Topology on the Initial Growth Rate of Influenza Epidemics , 2003 .