zbMATH — the first resource for mathematics

Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching. (English) Zbl 1400.92484
Summary: We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible-infectious-recovered-susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence. We then explain how the results can be generalised to a finite-state MC. The results for the SIR (Susceptible-Infectious-Removed) model with Markovian Switching (MS) are a special case. Numerical simulations are produced to confirm our theoretical results.

92D30 Epidemiology
34F05 Ordinary differential equations and systems with randomness
60J28 Applications of continuous-time Markov processes on discrete state spaces
Full Text: DOI
[1] Du, N. H.; Sam, V. H., Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl., 324, 1, 82-97 (2006) · Zbl 1107.92038
[2] Gray, A.; Greenhalgh, D.; Mao, X.; Pan, J., The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394, 2, 496-516 (2012) · Zbl 1271.92030
[3] Mao, X., Stochastic Differential Equations and Applications (2008), Horwood: Horwood Chichester, UK
[4] Mao, X.; Sabanis, S.; Renshaw, E., Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287, 1, 141-156 (2003) · Zbl 1048.92027
[5] Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97, 1, 95-110 (2004) · Zbl 1058.60046
[6] Du, N. H.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170, 399-422 (2004) · Zbl 1089.34047
[7] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 9, 2, 249-256 (1978)
[8] Takeuchi, Y., Global Dynamical Properties of Lotka-Volterra Systems (1996), World Scientific · Zbl 0844.34006
[9] Takeuchi, Y.; Du, N. H.; Hieu, N. T.; Sato, K., Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323, 938-957 (2006) · Zbl 1113.34042
[10] Li, X.; Jiang, D.; Mao, X., Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232, 427-448 (2009) · Zbl 1173.60020
[11] Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115, 700-721 (1927) · JFM 53.0517.01
[12] Hethcote, H. W., Qualitative analyses of communicable disease models, Math. Biosci., 28, 335-356 (1976) · Zbl 0326.92017
[13] Tornatore, E.; Buccellato, S. M.; Vetro, P., Stability of a stochastic SIR system, Physica A, 354, 111-126 (2005)
[14] Lu, Q., Stability of SIRS system with random perturbations, Physica A, 388, 18, 3,677-3,686 (2009)
[15] 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, 1, 248-271 (2012) · Zbl 1231.92058
[16] Zhao, Y.; Jiang, D., The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett., 34, 90-93 (2014) · Zbl 1314.92174
[17] Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71, 3, 876-902 (2011) · Zbl 1263.34068
[18] O’Regan, S. N.; Kelly, T. C.; Korobeinikov, A.; O’Callaghan, M. J.A.; Pokrovskii, A. V., Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23, 4, 446-448 (2010) · Zbl 1193.34102
[19] Korobeinikov, A., Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68, 615-626 (2006) · Zbl 1334.92410
[20] Vargas-de-León, C., On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos Solitons Fractals, 44, 12, 1,106-1,110 (2011)
[21] Liu, X.; Stechlinski, P., Pulse and constant control schemes for epidemic models with seasonality, Nonlinear Anal. RWA, 12, 2, 931-946 (2011) · Zbl 1203.92058
[22] Nasell, I., Stochastic models of some endemic infections, Math. Biosci., 179, 1-19 (2002) · Zbl 0991.92026
[23] Chen, G.; Li, T., Stability of stochastic delayed SIR model, Stoch. Dyn., 9, 2, 231-252 (2009) · Zbl 1176.93079
[24] Shrestha, M.; Scarpino, S. V.; Moore, C., A message-passing approach for recurrent-state epidemic models on networks, Phys. Rev. E, 92, 2, 0220821 (2015)
[25] Hethcote, H. W.; Yorke, J. A., (Gonorrhea Transmission Dynamics and Control. Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, vol. 56 (1984), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0542.92026
[26] Lamb, K. E.; Greenhalgh, D.; Robertson, C., A simple mathematical model for genetic effects in pneumococcal carriage and transmission, J. Comput. Appl. Math., 235, 7, 1,812-1,818 (2011)
[27] Lipsitch, M., Vaccination against colonising bacteria with multiple serotypes, Proc. Natl. Acad. Sci. USA, 94, 6,571-6,576 (1997)
[28] Wei, Q.; Xiong, Z.; Wang, F., Dynamics of a stochastic SIR model under regime switching, J. Inf. Comput. Sci., 10, 2,727-2,734 (2013)
[29] Anderson, R. M.; May, R. M., Infectious Diseases of Humans (1992), Oxford University Press: Oxford University Press Oxford
[30] Allen, L. J.S., An introduction to stochastic epidemic models, (Brauer, F.; van den Driessche, P.; Wu, J., Mathematical Epidemiology. Mathematical Epidemiology, Lecture Notes in Biomathematics, Mathematical Biosciences Subseries, vol. 1,945 (2008), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1206.92022
[31] Bailey, N. T.J., The Mathematical Theory of Infectious Diseases and its Applications (1975), Griffin: Griffin London · Zbl 0334.92024
[32] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325, 36-53 (2007) · Zbl 1101.92037
[33] Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases, Vol. 146 (2000), Wiley: Wiley Chichester
[34] Hethcote, H. W.; van den Driessche, P., An SIS epidemic model with variable population size and a delay, J. Math. Biol., 34, 177-194 (1995) · Zbl 0836.92022
[35] Korobeinikov, A.; Wake, G. C., Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15, 955-960 (2002) · Zbl 1022.34044
[36] Lahrouz, A.; Settati, A., Asymptotic properties of switching diffusion epidemic model with varying population size, Appl. Math. Comput., 219, 11,134-11,148 (2013)
[37] Nasell, I., (Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model. Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model, Lecture Notes in Mathematics, vol. 2,022 (2011), Springer-Verlag) · Zbl 1320.92019
[38] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033
[39] Keeling, M. J.; Rohani, P., Modelling Infectious Diseases in Humans and Animals (2008), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 1279.92038
[40] Anderson, W. J., Continuous-Time Markov Chains (1991), Springer-Verlag: Springer-Verlag Berlin-Heidelberg
[41] Butler, G. J.; Waltman, P., Persistence in dynamical systems, J. Differential Equations, 63, 255-263 (1986) · Zbl 0603.58033
[42] Waltman, P., A brief survey of persistence in dynamical systems, (Delay Differential Equations and Dynamical Systems. Delay Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, vol. 1,475 (1991), Springer Berlin, Heidelberg), 31-40 · Zbl 0756.34054
[43] Tang, J. W., The effect of environmental parameters on the survival of airborne infectious agents, J. R. Soc. Interface, 6, S737-S746 (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.