Stability analysis of a SIR epidemic model with random parametric perturbations. (English) Zbl 1498.92198
Summary: This paper is concerned with a SIR model for the spread of an epidemic amongst a population of individuals with random additive perturbations of the transmission rate. Recently, many papers are devoted to the case of the Gaussian white noise perturbation. However, this model violates the condition of positivity of the transmission rate. In the paper we consider three models of the random perturbation which do not change this condition. The two of them are the telegraphic noise, trichotomous noise and the third is the bounded noise. Explicit conditions of the amost sure asymptotic stability of disease-free equilibrium state are obtained in the case of the first two models. An efficient numerical procedure is proposed for the construction of stability charts in the case of bounded noise. The effect of random perturbations on the stability behavior of disease-free equilibrium is discussed. Some transient mean-square properties of the SIR stochastic epidemic model are also presented.
MSC:
| 92D30 | Epidemiology |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60J28 | Applications of continuous-time Markov processes on discrete state spaces |
Keywords:
SIR stochastic epidenic model; mean-squre stability; telegraphic noise; trichotomous noise; bounded noiseReferences:
| [1] | Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc Roy Soc Lond A, 115, 700-721 (1927) · JFM 53.0517.01 |
| [2] | Iannelli, M., Mathematical theory of age-structured population dynamics (1995), Pisa: Giardini editori e stampatori |
| [3] | Anderson, R. M.; May, R. M., Population biology of infectious diseases, Part I, Nature, 280, 361-367 (1979) |
| [4] | Spagnolo, B.; Valenti, D.; Fiasconaro, A., Noise in ecosystems: a short review, Math Biosci Eng, 1, 185-211 (2004) · Zbl 1086.92055 |
| [5] | Tornatore, E.; Buccellato, S. M.; Vetro, P., Stability of a stochastic SIR system, Physica A, 354, 111-126 (2005) |
| [6] | Ji, C.; Jiang, D.; Shi, N., Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390, 1747-1762 (2011) |
| [7] | Meng, X.; Deng, F., Stability of stochastic switched epidemic systems with discrete or distributed time delay, J Syst Eng Electron, 25, 660-670 (2014) |
| [8] | Zhou, Y.; Zhang, W.; Yuan, S., Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl Math Comput, 244, 118-131 (2014) · Zbl 1335.92109 |
| [9] | Yang, Q.; Mao, X., Stochastic dynamics of SIRS epidemic models with random perturbation, Math Biosci Eng, 11, 1003-1025 (2014) · Zbl 1306.92064 |
| [10] | Schurz, H.; Tosun, K., Stochastic asymptotic stability of SIR model with variable diffusion rates, J Dyn Diff Equ, 27, 69-82 (2015) · Zbl 1312.60077 |
| [11] | 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 |
| [12] | Du, N. H.; Nhu, N. N., Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Appl Math Lett, 64, 223-230 (2017) · Zbl 1354.92090 |
| [13] | Witbooi, P. J., Stability of a stochastic model of an SIR epidemic with vaccination, Acta Biotheor, 65, 151-165 (2017) |
| [14] | Xie, X.; Ma, L.; Xu, J., Asymptotic behavior and stability of stochastic SIR model with variable diffusion rates, Appl Math, 8, 1031-1044 (2017) |
| [15] | Dieu, N. T., Asymptotic properties of a stochastic SIR epidemic model with Beddington-DeAngelis incidence rate, J Dyn Diff Equ, 30, 93-106 (2018) · Zbl 1387.34076 |
| [16] | Han, P.; Chang, Z.; Meng, X., Asymptotic dynamics of a stochastic SIR epidemic system affected by mixed nonlinear incidence rates, Complexity, 8596371 (2020) · Zbl 1435.92073 |
| [17] | Bena, I., Dichotomous Markov noise: exact results for out-of equilibrium systems, Int J Mod Phys B, 20, 2825-2888 (2006) · Zbl 1107.82357 |
| [18] | 031120-11 |
| [19] | Jin, Y.; Wang, H., Noise-induced dynamics in a Josephson junction driven by trichotomous noises, Chaos Solitons Fractals, 133, 109633 (2020) · Zbl 1483.82008 |
| [20] | d’Onofrio, A.; Caravagna, G.; de Franciscis, S., Bounded noise induced first-order phase transitions in a baseline non-spatial model of gene transcription, Physica A, 492, 2056-2068 (2018) · Zbl 1514.82078 |
| [21] | Siewe Siewe, M.; Fokou Kenfack, W.; Kofane, T. C., Probabilistic response of an electromagnetic transducer with nonlinear magnetic coupling under bounded noise excitation, Chaos Solitons Fractals, 124, 26-35 (2019) |
| [22] | Wang, C. J.; Lin, Q. F.; Yao, Y. G.; Yang, K. L.; Tian, M. Y.; Wang, Y., Dynamics of a stochastic system driven by cross-correlated sine-wiener bounded noises, Nonlinear Dyn, 95, 1941-1956 (2019) · Zbl 1432.60065 |
| [23] | Cheng, G.; Liu, W.; Gui, R.; Yao, Y., Sine-wiener bounded noise-induced logical stochastic resonance in a two-well potential system, Chaos Solitons Fractals, 131, 109514 (2020) · Zbl 1495.82024 |
| [24] | d’Onofrio A., editor. Bounded noises in physics, biology, and engineering. New York: Birkhäuser; 2013. · Zbl 1276.60002 |
| [25] | Chichigina, O.; Valenti, D.; Spagnolo, A., A simple noise model with memory for biological systems, Fluct Noise Lett, 5, 243-250 (2005) |
| [26] | Chichigina, O.; Dubkov, A. A.; Valenti, D.; Spagnolo, A., Stability in a system subject to noise with regulated periodicity, Phys Rev E, 84, 021134 (2011) |
| [27] | Morrisson, J. A.; McKenna, J., Analysis of some stochastic ordinary differential equations, SIAM-AMS Proc, 6, 97-162 (1973) · Zbl 0275.34061 |
| [28] | Khasminskii, R., Stochastic stability of differential equations (2012), Springer: Springer New York · Zbl 1259.60058 |
| [29] | Buckwar, E.; Kelly, C., Asymptotic and transient mean-square properties of stochastic systems arising in ecology, fluid dynamics, and system control, SIAM J Appl Math, 74, 411-433 (2014) · Zbl 1297.93172 |
| [30] | Sauga, A.; Laas, K.; Mankin, R., Memory-induced sign reversals of the spatial cross-correlation for particles in viscoelastic shear flows, Chaos Solitons Fractals, 81, 443-450 (2015) · Zbl 1355.60054 |
| [31] | Floris, C., Mean square stability of a second-order parametric linear system excited by a colored gaussian noise, J Sound Vib, 336, 82-95 (2015) |
| [32] | Wu, J.; Li, X.; Liu, X., On the pth moment stability of the binary airfoil induced by bounded noise, Chaos Solitons Fractals, 98, 109-120 (2017) · Zbl 1372.74059 |
| [33] | Willems, J. L., Moment stability of linear white noise and coloured noise systems, (Clarkson, B. L., Stochastic problems in dynamics (1977), Pitman: Pitman London), 67-87 · Zbl 0374.93042 |
| [34] | Shapiro, V. E.; Loginov, V. M., “Formulae of differentiation” and their use for solving stochastic equations, Physica A, 91, 563-574 (1978) |
| [35] | Cameron, R. H.; Martin, W. T., Transformations of wiener integrals under translations, Ann Math, 45, 386-396 (1944) · Zbl 0063.00696 |
| [36] | Bobryk, R. V., Closure methods and asymptotic expansions for linear stochastic systems, J Math Anal Appl, 329, 703-711 (2007) · Zbl 1108.60051 |
| [37] | Bobryk, R. V., Stochastic stability of coupled oscillators, Appl Math Comp, 198, 544-550 (2008) · Zbl 1134.70011 |
| [38] | Bobryk, R. V.; Chrzeszczyk, A., Stability regions for Mathieu equation with imperfect periodicity, Phys Lett A, 373, 3532-3535 (2009) · Zbl 1233.70007 |
| [39] | Bobryk, R. V., Stochastic multiresonance in oscillators induced by bounded noise, Commun Nonlinear Sci Numer Simul, 93, 105460 (2020) · Zbl 1457.34097 |
| [40] | Kozin, F., On relations between moment properties and almost sure Lyapunov stability for linear stochastic systems, J Math Anal Appl, 10, 342-351 (1965) · Zbl 0131.16501 |
| [41] | Feng, X.; Loparo, K. A.; Ji, Y.; Chizeck, H. J., Stochastic stability properties of jump linear systems, IEEE Trans Automat Control, 37, 38-53 (1992) · Zbl 0747.93079 |
| [42] | Arnold, L.; Kliemann, W., Qualitative theory of stochastic systems, (Bharucha-Reid, A. T., Probabilistic analysis and related topics, vol. 3 (1983), Academic Press: Academic Press New York), 1-79 · Zbl 0536.60070 |
| [43] | Cao, B.; Shan, M.; Zhang, Q.; Wang, W., A stochastic SIS epidemic model with vaccination, Physica A, 486, 127-143 (2017) · Zbl 1499.92097 |
| [44] | Zhang, Y.; Li, Y.; Zhang, Q.; Li, A., Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501, 178-187 (2018) · Zbl 1514.92177 |
| [45] | Wang, Z.; Zhang, Q.; Li, X., Markovian switching for near-optimal control of a stochastic SIV epidemic model, Math Biosci Eng, 16, 1348-1375 (2019) · Zbl 1497.92313 |
| [46] | Liu, R.; Wei, F., Persistence and extinction for an age-structured stochastic SVIR epidemic model with generalized nonlinear incidence rate, Physica A, 513, 572-587 (2019) · Zbl 1514.92150 |
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