Global analysis of stochastic SIR model with variable diffusion rates

Pitchaimani M., Rajasekar S.P.


In this article, a stochastic SIR epidemic model with treatment rate in a population of varying size is proposed and investigated. For the stochastic version, we briefly discuss the existence of global unique solutions and using the Lyapunov function, the disease free equilibrium solution is globally asymptotic stabe if $\mathcal{R}_0\leq1$ and the endemic equilibrium solution is obtained when $\mathcal{R}_0>1$. The main attention is paid to the $p$th-moment exponentially stable on the system, proved under suitable assumptions on the white noise perturbations and the optimal control for the deterministic model. Finally numerical simulations are done to show our theoretical results and to demonstrate the complicated dynamics of the model.


SIR model, Stochastic Asymptotic stability, pth-moment exponentially sta- bility, Lyapunov function, Optimal control.

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L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974.

E. Allen, Modeling with Ito's Stochastic Differential Equations, Springer, The Netherrlands, 2007.

T. C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York and Basel, 1998.

R. Khas'minskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alpen, 1980.

Fima C Klebaner, Introduction to Stochastic Calculus with Applications, Second edition, Imperial College Press, London, 2005.

Bernt Oksendal , Stochastic Differential Equations: An Introduction with Applications, Sixth Edition Springer, Heidelberg , 2003


W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, part i. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics., 115(1927), 700-721.

H. Schurz and K. Tosun, Stochastic asymptotic stability of SIR model with variable diffusion rates, J. Dyn. Diff. Equat., 27(2014), 69-82.

Daqing Jiang and Jiajia Yu, Chunyan Ji, Ningzhong Shi, Asymptotic behavior of global positive solution to a stochastic SIR model,Mathematical and Computer Modelling, 54(2011), 221-232.

M. Pitchaimani and R. Rajaji, Stochastic Asymptotic stability of Nowak – May model with diffusion rates,Methodol Comput Appl Probab, 18(2016), 901-910.

A. Lahrouz, L. Omari and D. Kiouach, Global Analysis of a deterministic and stochastic nonlinear SIRS epidemic model,Nonlinear Analysis Modelling and Control, 16 (2011), 59-76

Yanli Zhou, Weiguo Zhang, Sanling Yuan and Hongxiao Hu,Persistence and extinction in stochastic SIRS models with general nonlinear incidence rate, Electronic Journal of Differential Equations,42(2014), 1-17.

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical theory of Optimal Processes, Johnwiley & Sons,Inc. Newyork. 1962.

Wendell H. Fleming and Raymond W. Rishel, Deterministic and Stochastic Optimal Control, Springer - Verlag, Newyork, 1975.

Donald E. Kirk,Optimal Control: An Introduction, Dover Publications, Inc., Newyork, 1998.

Gul Zaman, Yong Han Kang, Giphil Cho and I1 Hyo Jung, Optimal Strategy of Vaccination & treatment in an SIR epidemic model,Mathematics and Computer Simulation, 136(2016), 63-77, 2016.

Abid Ali Lashari, Optimal control of an SIR epidemic model with a saturated treatment, Appl. Math. Inf. Sci., 10(2016), 185-191.

Tunde Tajudeen Yusuf and Francis Benyah, Optimal control of Vaccination and treatment in an SIR epidemic model, World Journal of Modeling and Simulation, 8(2012), 194-204.

Hassan Laarabi, Mostafa Rachik, Ouata El Kahloui and El Houssine Labriji, Optimal Vaccination Strategy of an SIR epidemic model with a saturated treatment, Universal Journal of Applied Mathematics, 1(2013), 185-191.

H. S. Rodrigues, PhD Software codes., Mar 2012.


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