Optimal control of an SIVRS epidemic spreading model with virus variation based on complex networks

Abstract A novel SIVRS mathematical model for infectious diseases spreading is proposed and investigated in this paper. In this model virus variation factors are considered in the process of epidemic spreading based on complex networks, which can describe different contact status for different agents including the susceptible, the infectious, the variant and the recovered in a network. An optimal control problem is formulated to maximize the recovered agents with the limited resource allocation and optimal control strategies over the susceptible, the infected and the variant are investigated. Then the existence of a solution to the optimal control problem is given based on Pontryagin's Minimum Principle and modified forward backward sweep technique. Numerical simulations are provided to illustrate obtained theoretical results.

[1]  S. Lenhart,et al.  OPTIMIZING CHEMOTHERAPY IN AN HIV MODEL , 1998 .

[2]  Jingjing Cheng,et al.  SIHR rumor spreading model in social networks , 2012 .

[3]  Erik I. Verriest,et al.  Control of epidemics by vaccination , 2005, Proceedings of the 2005, American Control Conference, 2005..

[4]  M. Brandeau,et al.  Resource allocation for control of infectious diseases in multiple independent populations: beyond cost-effectiveness analysis. , 2003, Journal of health economics.

[5]  Renat M. Yulmetyev,et al.  How chaosity and randomness control human health , 2005 .

[6]  Jiajia Wang,et al.  SIRaRu rumor spreading model in complex networks , 2014 .

[7]  Hem Raj Joshi,et al.  Optimal control of an HIV immunology model , 2002 .

[8]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[9]  Hans J. Herrmann,et al.  Traffic gridlock on complex networks , 2012 .

[10]  Suzanne Lenhart,et al.  Optimal control of treatments in a two-strain tuberculosis model , 2002 .

[11]  Degang Xu,et al.  Novel SIVR epidemic spreading model with virus variation in complex networks , 2015, The 27th Chinese Control and Decision Conference (2015 CCDC).

[12]  Wanlin Xie,et al.  A rumor spreading model with variable forgetting rate , 2013 .

[13]  Weihua Gui,et al.  Global Stability of a Variation Epidemic Spreading Model on Complex Networks , 2015 .

[14]  Joy Kuri,et al.  How to run a campaign: Optimal control of SIS and SIR information epidemics , 2014, Appl. Math. Comput..

[15]  Rui Xu,et al.  Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity , 2010, Comput. Math. Appl..

[16]  O. Hölder.  Ueber einen Mittelwerthabsatz , 1889 .

[17]  Holly Gaff,et al.  Optimal control applied to vaccination and treatment strategies for various epidemiological models. , 2009, Mathematical biosciences and engineering : MBE.

[18]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[19]  Bo Wang,et al.  Epidemic dynamics behavior in some bus transport networks , 2012 .

[20]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[21]  Ali Muhammad,et al.  Practical Scientific Computing , 2011 .

[22]  Jiajia Wang,et al.  Rumor spreading model considering forgetting and remembering mechanisms in inhomogeneous networks , 2013 .

[23]  Saswati Sarkar,et al.  Optimal Patching in Clustered Malware Epidemics , 2014, IEEE/ACM Transactions on Networking.

[24]  Daniela Iacoviello,et al.  Optimal control for SIRC epidemic outbreak , 2013, Computer Methods and Programs in Biomedicine.

[25]  Chenquan Gan,et al.  An epidemic model of computer viruses with vaccination and generalized nonlinear incidence rate , 2013, Appl. Math. Comput..

[26]  Franciszek Rakowski,et al.  Influenza epidemic spread simulation for Poland — a large scale, individual based model study , 2010 .

[27]  Xiaofan Yang,et al.  The spread of computer viruses over a reduced scale-free network , 2014 .

[28]  Yoshiharu Maeno,et al.  Profiling of a network behind an infectious disease outbreak , 2009, ArXiv.

[29]  Yong Han Kang,et al.  Stability analysis and optimal vaccination of an SIR epidemic model , 2008, Biosyst..

[30]  Alessandro Vespignani,et al.  Epidemic dynamics and endemic states in complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  A. Lahrouz,et al.  Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model , 2011 .

[32]  Piet Van Mieghem,et al.  In-homogeneous Virus Spread in Networks , 2013, ArXiv.

[33]  M. Newman,et al.  Epidemics and percolation in small-world networks. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[34]  Y. Moreno,et al.  Epidemic outbreaks in complex heterogeneous networks , 2001, cond-mat/0107267.

[35]  Lu-Xing Yang,et al.  The optimal dynamic immunization under a controlled heterogeneous node-based SIRS model , 2016 .

[36]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[37]  Toshikazu Kuniya,et al.  Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model , 2011 .

[38]  M. Petit.  Dynamic optimization. The calculus of variations and optimal control in economics and management : by Morton I. Kamien and Nancy L. Schwartz. Second Edition. North-Holland (Advanced Textbooks in Economics), Amsterdam and New York, 1991. Pp. xvii+377. ISBN0-444- 01609-0 , 1994 .

[39]  Qi Xuan,et al.  Reaction-diffusion processes and metapopulation models on duplex networks , 2013 .

[40]  Steve R. White,et al.  Computers and epidemiology , 1993, IEEE Spectrum.

[41]  Jinde Cao,et al.  Impact of media coverage on epidemic spreading in complex networks , 2013, Physica A: Statistical Mechanics and its Applications.

[42]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .

[43]  Alessandro Vespignani,et al.  Epidemic dynamics in finite size scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Zhong Zhao,et al.  Transmission dynamics and optimal control of measles epidemics , 2015, Appl. Math. Comput..

[45]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[46]  Jiajia Wang,et al.  SIR rumor spreading model in the new media age , 2013 .

[47]  Qi Xuan,et al.  Emergence of heterogeneous structures in chemical reaction-diffusion networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  O. Diekmann,et al.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation , 2000 .

[49]  B. Ran,et al.  Stability analysis of the mixed traffic flow of cars and trucks using heterogeneous optimal velocity car-following model , 2014 .

[50]  Qi Xuan,et al.  Structural control of reaction-diffusion networks. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Suzanne Lenhart,et al.  Optimal control of vaccine distribution in a rabies metapopulation model. , 2008, Mathematical biosciences and engineering : MBE.

[52]  Margaret L Brandeau,et al.  Dynamic resource allocation for epidemic control in multiple populations. , 2002, IMA journal of mathematics applied in medicine and biology.

[53]  Horst R. Thieme,et al.  Mathematics in Population Biology , 2003 .