Optimal control analysis of deterministic and stochastic epidemic model with media awareness programs

Shrishail Ramappa Gani, Shreedevi Veerabhadrappa Halawar

Abstract


The present study considered the optimal control analysis of  both deterministic differential equation modeling and stochastic differential equation modeling of infectious disease by taking effects of media awareness programs  and treatment of infectives on the epidemic into account. Optimal media awareness strategy under the quadratic cost functional using Pontrygin's Maximum Principle  and Hamiltonian-Jacobi-Bellman equation are derived for both deterministic and stochastic optimal problem respectively. The Hamiltonian-Jacobi-Bellman equation is used to solve stochastic system, which is fully non-linear equation, however it ought to be pointed out that for stochastic optimality system it may be difficult to obtain the numerical results. For the analysis of the stochastic optimality system, the results of deterministic control problem are used to find an approximate numerical solution for the stochastic control problem.  Outputs of the simulations shows that media awareness programs place important role in the minimization of infectious population with minimum cost.

Keywords


Epidemic model, Awareness campaigns, Optimal control, Stochastic perturbation.

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References


Misra, A.K., Sharma, A. & Shukla, J.B. (2015). Stability analysis and optimal control of an epidemic

model with awareness program by media. J Bio Sys., 138, 53-62.

Liu, W. & Zheng, O. (2015). A stochastic SIS epidemic model incorporating media coverage in a two patch setting. Applied Mathematics and Computation, 262, 160-168.

Durrett, R. & Levin, S.A. (1994). Stochastic spatial models:The users guide to ecological application. Philosophical Transactions: Biological Sciences, 343, 329 - 350.

Tchunche,J.M., Khamis, S.A., Agusto, F.B. & Mpeshe, S.C. (2010). Optimal control and sensitivity analysis of an influenza model with treatment and vaccination. Acta Biotheoretica, 59, 1-28.

Okosun, K.O., Makide, O. D. & Takaidza, I. (2013). The impact of optimal control on the treatment of HIV/AIDS and screening of unaware infective. Applied Mathematical Modeling, 37, 3802 - 3820.

Ishikawa, M. (2012). Optimal strategies for vaccination using the stochastic SIRV model. Transactions of the Institute of the Systems, Control and Information Engineers, 25, 343 - 348.

Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V. & Mishchenko, E.F. (1962). The mathematical theory of optimal processes, Wiley, New York.

Fleming, W.H. & Rishel, R.W. (1975). Deterministic and stochastic optimal control, Springer Verlag, New York.

Zhao, Y., Jiang, D. & O’Regan, D. (2013). The extinction and persistance of the stochastic SIS epidemic model with vaccination, Physica A, 392, 4916-4927.

Carletti, M. (2002). On stability properties of stochastic model for phase-bacteria interaction in open marine environment, Math. Biosci., 175, 117-131.

Sulem, A., & Tapiero, C.S. (1994). Computational aspects in applied stochastic control, Computational Economics, 7, 109146.

Tornatore, E., Buccellato, S.M., & Vetro, P. (2006). On a stochastic disease model with vaccination, Rendiconti del Circolo Matematicodi Palermo. Serie II, 55, 223240.

Tornatore, E., Vetro, P. & Buccellato, S. M. (2014). SIVR epidemic model with stochastic perturbation, Neural Computing and Applications, 24, 309315.

Witbooi, P.J., Muller, G.E. & Van Schallkwyk, G.J. (2015). Vaccination Control in a Stochastic SVIR Epidemic Model, Computational and Mathematical Methods in Medicine, Article ID 271654, 9 pages.

Lukes, D.L. (1982). Differential equations: classical to control, Academic press.

Dalal, N., Greenhalgh, D. & Mao, X. (2007). A stochastic Model of AIDS and Condom use, J. Math. Anal. Appl. 325, 36-53.

Gray,A., Greenhalgh, D., Hu, L., Mao, X. & Pan, J. (2011). A stochastic differential equation SIS epidemic model, SIAM J.Appl. Math., 71, 876-902.

Mao, X. (1997). Stochastic differential equations and applications. Horwood.

Oksendal,B. (1998). Stochastic differential equations: an introduction with applications, Universitext, Springer, Berlin, Germany, 5th edition.

Higham, D. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525546.




DOI: http://dx.doi.org/10.11121/ijocta.01.2019.00423

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Copyright (c) 2018 Shreedevi V Halawar

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