A stochastic epidemic model with nonmonotone incidence rate: Sufficient and necessary conditions for near-optimality

Abstract In this paper, we studied the near-optimal control of a stochastic susceptible–infected–recovered–susceptible (SIRS) model that includes a nonmonotone incidence rate. The near-optimal control problem was formulated by reducing the infected population while keeping the treatment cost to a minimum. We obtained the sufficient and necessary conditions for the near-optimality. We showed that any near-optimal control can be solved by using the Hamiltonian function to approximate the cost function. According to the adjoint equations, we derived the estimate for the error bound of the near-optimality. Numerical simulations were performed to illustrate the results and confirm the effect of treatment control on the disease dynamics.

[1]  Yun Kang,et al.  A stochastic SIRS epidemic model with infectious force under intervention strategies , 2015 .

[2]  Huaiping Zhu,et al.  An SIS Infection Model Incorporating Media Coverage , 2008 .

[3]  Wendi Wang.  Backward bifurcation of an epidemic model with treatment. , 2006, Mathematical biosciences.

[4]  Xun Yu Zhou.  Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality , 1998 .

[5]  S. Peng,et al.  Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control , 1999 .

[6]  Qing-Long Han,et al.  Optimal tracking control with feedforward compensation for offshore steel jacket platforms with active mass damper mechanisms , 2016 .

[7]  Oluwole Daniel Makinde,et al.  Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy , 2007, Appl. Math. Comput..

[8]  Deborah Lacitignola,et al.  On the backward bifurcation of a vaccination model with nonlinear incidence , 2011 .

[9]  Tadeusz Antczak.  A sufficient condition for optimality in nondifferentiable invex programming , 2001 .

[10]  Meijing Shan,et al.  A stochastic SIS epidemic model with vaccination , 2017 .

[11]  Baolin Zhang,et al.  Discrete feedforward and feedback optimal tracking control for offshore steel jacket platforms , 2014 .

[12]  Khalid Hattaf,et al.  Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment , 2015, Acta biotheoretica.

[13]  Liangjian Hu,et al.  A Stochastic Differential Equation SIS Epidemic Model , 2011, SIAM J. Appl. Math..

[14]  Xun Li,et al.  Near-optimal control problems for linear forward-backward stochastic systems , 2010, Autom..

[15]  S. C. Mpeshe,et al.  Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and Vaccination , 2011, Acta biotheoretica.

[16]  Juan Li,et al.  Observer-based optimal fault-tolerant control for offshore platforms , 2014, Comput. Electr. Eng..

[17]  T. K. Kar,et al.  Stability analysis and optimal control of an SIR epidemic model with vaccination , 2011, Biosyst..

[18]  W. Eckalbar,et al.  Dynamics of an epidemic model with quadratic treatment , 2011 .

[19]  Xianning Liu,et al.  Backward bifurcation of an epidemic model with saturated treatment function , 2008 .

[20]  Daqing Jiang,et al.  The threshold of a non‐autonomous SIRS epidemic model with stochastic perturbations , 2017 .

[21]  Swapan Kumar Nandi,et al.  Complex Dynamics of an SIR Epidemic Model with Saturated Incidence Rate and Treatment , 2016, Acta biotheoretica.

[22]  I. Ekeland.  Nonconvex minimization problems , 1979 .

[23]  Shigui Ruan,et al.  Global analysis of an epidemic model with nonmonotone incidence rate , 2006, Mathematical Biosciences.

[24]  Soovoojeet Jana,et al.  Complex dynamics of an epidemic model with vaccination and treatment controls , 2016 .

[25]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..