Study on the threshold of a stochastic SIR epidemic model and its extensions. (English) Zbl 1471.92370
Summary: This paper provides a simple but effective method for estimating the threshold of a class of the stochastic epidemic models by use of the nonnegative semimartingale convergence theorem. Firstly, the threshold \(R_0^{\mathrm{SIR}}\) is obtained for the stochastic SIR model with a saturated incidence rate, whose value is below 1 or above 1 will completely determine the disease to go extinct or prevail for any size of the white noise. Besides, when \(R_0^{\mathrm{SIR}}>1\), the system is proved to be convergent in time mean. Then, the threshold of the stochastic SIVS models with or without saturated incidence rate are also established by the same method. Comparing with the previously-known literature, the related results are improved, and the method is simpler than before.
MSC:
| 92D30 | Epidemiology |
References:
| [1] | Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J., A stochastic differential equation SIS epidemic model, SIAM J Appl Math, 71, 876-902 (2011) · Zbl 1263.34068 |
| [2] | Tornatore, E.; Buccellato, S. M.; Vetro, P., Stability of a stochastic SIR system, Phys A, 354, 111-126 (2005) |
| [3] | Tornatore, E.; Vetro, P.; Buccellato, S. M., SIVR epidemic model with stochastic perturbation, Neural Comput Appl, 24, 2, 309-315 (2014) |
| [4] | Ji, C.; Jiang, D., Threshold behaviour of a stochastic SIR model, Appl Math Model, 38, 5067-5079 (2014) · Zbl 1428.92109 |
| [5] | Lahrouz, A.; Omari, L., Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat Probab Lett, 83, 960-968 (2013) · Zbl 1402.92396 |
| [6] | Zhao, Y.; Jiang, D.; O’Regan, D., The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys A: Stat Mech Appl, 392, 4916-4927 (2013) · Zbl 1395.92180 |
| [7] | Lin, Y.; Jiang, D.; Wang, S., Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A, 394, 187-197 (2014) · Zbl 1395.34064 |
| [8] | Jiang, D.; Yu, J.; Ji, C.; Shi, N., Asymptotic behavior of global positive solution to a stochastic SIR model, Math Comput Model, 54, 221-232 (2011) · Zbl 1225.60114 |
| [9] | Yang, Q.; Jiang, D.; Shi, N.; Ji, C., The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J Math Anal Appl, 388, 248-271 (2012) · Zbl 1231.92058 |
| [10] | Zhao, Y.; Jiang, D., The threshold of a stochastic SIS epidemic model with vaccination, Appl Math Comput, 243, 718-727 (2014) · Zbl 1335.92108 |
| [11] | Schurz, H.; Tosun, K., Stochastic symptotic stability of SIR model with variable diffusion rates, J Dyn Differ Equ, 27, 69-82 (2015) · Zbl 1312.60077 |
| [12] | Liu, Q.; Chen, Q., Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 428, 140-153 (2015) · Zbl 1400.92515 |
| [13] | Mao, X., Stochastic Differential equations and applications (1997), Horwood Publishing: Horwood Publishing Chichester · Zbl 0892.60057 |
| [14] | Lipster, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228 (1980) · Zbl 0435.60037 |
| [15] | Mao, X.; Marion, G.; Renshaw, E., Environmental brownian noise suppresses explosions in populations dynamics, Stoch Process Appl, 97, 95-110 (2002) · Zbl 1058.60046 |
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