Main Article Content
Seasonal epidemic model, respiratory syncytial virus, stochastic differential equation, Itˆo’s formula, positive solutions, brownian motion.
In this paper we investigate the positivity and boundedness of the solution of a stochastic seasonal epidemic model for the respiratory syncytial virus (RSV ). The stochasticity in the model is due to fluctuating physical and social environments and is introduced by perturbing the transmission parameter of the seasonal disease. We show the existence and uniqueness of the positive solution of the stochastic seasonal epidemic model which is required in the modeling of populations since all populations must be positive from a biological point of view. In addition, the positivity and boundedness of solutions is important to other nonlinear models that arise in sciences and engineering. Numerical simulations of the stochastic model are performed using the Milstein numerical scheme and are included to support our analytic results.
 H. Hethcote, “Mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2005.
 S. Gao, Z. Teng, J. J. Nieto, and A. Torres, “Analysis of an SIR epidemic model with pulse vaccination and distributed time delay,” Journal of Biomedicine and Biotechnology, vol. 2007, p. 10, 2007.
 O. Diallo and Y. Koné, “Melnikov analysis of chaos in a general epidemiological model,” Nonlinear Analysis: Real World Applications, vol. 8, pp. 20–26, 2007.
 M. Keeling, P. Rohani, and B. Grenfell, “Seasonally forced disease dynamics explored as switching between attractors,” Physica D: Nonlinear Phenomena, vol. 148, no. 3-4, pp. 317–335, 2001.
 A. Weber, M. Weber, and P. Milligan, “Modeling epidemics caused by respiratory syncytial virus (RSV),” Math. Biosc., vol. 172, pp. 95–113, 2001.
 D. Greenhalgh and I. A. Moneim, “SIRS epidemic model and simulations using different types of seasonal contact rate,” Systems Analysis Modelling Simulation, vol. 43(5), pp. 573–600, 2003.
 D. J. D. Earn, P. Rohani, B. M. Bolker, J. Earn, and B. T. Grenfell, “A simple model for complex dynamical transitions in epidemics,” Science, vol. 287, pp. 667–670, 2000.
 T. Zhang and Z. Teng, “Permanence and extinction for a nonautonomous SIRS epidemic model with time delay,” Applied Mathematical Modelling, vol. 33, pp. 1058–1071, 2009.
 Z. Teng, Y. Liu, and L. Zhang, “Persistence and extinction of disease in nonautonomous SIRS epidemic models with disease-induced mortality,” Nonlinear Analysis, vol. 69, pp. 2599–2614, 2008.
 T. Zhang and Z. Teng, “On a nonautonomous SEIRS model in epidemiology,” Bulletin of mathematical biology, vol. 69, no. 8, pp. 2537–2559, 2007.
 T. Zhang, Z. Teng, and S. Gao, “Threshold conditions for a non-autonomous epidemic model with vaccination,” Applicable Analysis, vol. 87, no. 2, pp.181–199, 2008.
 T. Saha and M. Bandyopadhyay, “Effect of randomy fluctuating environment on Autotroph-Herbivore model system,” International Journal of Mathematics and Mathematical, vol. 68, pp. 3703–3716, 2004.
 Y. Ding, M. Xu, and L. Hu, “Asymptotic behavior and stability of a stochastic model for AIDS transmission,” Appl. Math. Comput., vol. 204, pp. 99–108, 2008.
 B. Spagnolo, M. Cirone, A. L. Barbera, and F. de Pasquale, “Noiseinduced effects in population dynamics,” Journal of Physics: Condensed Matter, vol. 14, no. 9, pp. 2247–2255, 2002. [Online]. Available: http://stacks.iop.org/0953-8984/14/2247
 M. Bandyopadhyay, “Effect of environmental fluctuation on a detritus based ecosystem,” Journal of Applied Mathematics and Computing, vol. 26, no. 1-2, pp. 433–450, 2008.
 L. Jódar, R. J. Villanueva, and A. Arenas, “Modeling the spread of seasonal epidemiological diseases: theory and applications,” Mathematical and Computer Modelling, vol. 48, pp. 548–557, 2008.
 N. Dalal, D. Greenhalgh, and X. Mao, “A stochastic model of AIDS and condom use,” J. Math. Anal. Appl., vol. 325, pp. 36–53, 2007.
 A. Bahar and X. Mao, “Stochastic delay Lotka-Volterra model,” J. Math. Anal. Appl., vol. 292, pp. 364–380, 2004.
 R. R. Sarkar, “A stochastic model for autotroph-herbivore system with nutrient reclycing,” Journal of Theoretical Biology, vol. 1, no. 3-4, pp. 429–440, 2004.
 T. Saha and M. Bandyopadhyay, “Dynamical analysis of a delayed ratiodependent prey-predator model within fluctuating environment,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 458–478, 2008.
 H. C. Tuckwell and L. Toubiana, “Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density dependence,” Biosystems,, vol. 90, no. 2, pp. 546–559, 0 2007.
 N. Dalal, D. Greenhalgh, and X. Mao, “A stochastic model for internal HIV dynamics,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1084–1101, 2008.
 D. Jiang, J. Yu, C. Ji, and N. Shi, “Asymptotic behavior of global positive solution to a stochastic SIR model,” Mathematical and Computer Modelling, vol. 54, pp. 221–232, 2011.
 H. Andersson and T. Britton, Stochastic epidemic models and their statistical analysis. Springer, 2000.
 E. Renshaw, Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge, 1991.
 D. Rand and H. Wilson, “Chaotic stochasticity: A ubiquitous source of unpredictability
in epidemics,” Proceedings: Biological Sciences, vol. 246, pp. 179–184, 1991.
 J. C. Cortés, L. Jódar, and L. Villafuerte, “A random Euler method for solving differential equations with uncertainties,” pp. 944–948, 2008.
 G. González-Parra, J. F. Querales, and D. Aranda, “Predicción de la epidemia del virus respiratorio sincitial en bogotá dc utilizando variables climatológicas,” Biomédica, vol. 36, no. 3, 2016.
 A. B. Hogan, K. Glass, H. C. Moore, and R. S. Anderssen, “Age structures in mathematical models for infectious diseases, with a case study of respiratory syncytial virus,” in Applications+ Practical Conceptualization+ Mathematics=fruitful Innovation. Springer, 2016, pp. 105–116.
 D. F. Aranda-Lozano, G. C. González-Parra, and J. Querales, “Modelling respiratory syncytial virus (RSV) transmission children aged less than five years-old,” Revista de Salud Pública, vol. 15, no. 4, pp. 689–700, 2013.
 A. J. Arenas, G. González-Parra, and J.-A. Morano, “Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia, Spain,” BioSystems, vol. 96, pp. 206–212, 2009.
 P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1995.
 G. González-Parra and H. M. Dobrovolny, “Assessing uncertainty in A2 respiratory syncytial virus viral dynamics,” Computational and mathematical methods in medicine, vol. 2015, 2015.
 G. Chowell, C. Ammon, N. Hengartner, and J. Hyman, “Transmission dynamics of the great influenza pandemic of 1918 in geneva, switzerland: Assessing the effects of hypothetical interventions,” Journal of Theoretical Biology, vol. 241, no. 2, pp. 193–204, 2006.
 Z. Liu, “Dynamics of positive solutions to sir and seir epidemic models with saturated incidence rates,” Nonlinear Analysis: Real World Applications, vol. 14, pp. 1286–1299, 2013.
 G. Chen, T. Li, and C. Liu, “Lyapunov exponent of a stochastic sirs model,” C. R. Acad. Sci. Paris, Ser. I, vol. 351, pp. 33–35, 2013.
 A. Lahrouz, L. Omari, and D. Kiouach, “Global analysis of a deterministic and stochastic nonlinear sirs epidemic model,” Nonlinear Analysis: Modelling and Control, vol. 16, no. 1, pp. 59–76, 2011.
 T. Gard, Introduction to Stochastic Differential Equations. Marcel Dekker, New York, 1988.
 X. Mao, Stochastic Differential Equations and Applications. Horwood Publishing, Chichester, 1997.
 C. Hall, “Respiratory syncytial virus, in: R.D. Feigin, J.D. Cherry (eds.),” Textbook of Pediatric Infectious Diseases, Saunders, Philadelphia, PA., pp. 1633–1640, 1992.
 J. Morano-Fernandez, “Mathematical modelling of respiratory syncytial virus spread in the spanish region of valencia. preventive applications,” Ph.D. dissertation, Institute of Mathematics Multidisciplinary, Department of Applied Mathematics, Valencia Polytechnic University, Sept 2010.
 A. J. A. Tawil, “Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of valencian,” Ph.D. dissertation, Institute of Mathematics Multidisciplinary, Department of Applied Mathematics, Valencia Polytechnic University, May 2009.
 L. Acedo, J. Díez-Domingo, J.-A. Morano, and R.-J. Villanueva, “Mathematical modelling of respiratory syncytial virus (rsv): vaccination strategies and budget applications. epidemiology and infection,” Epidemiology and Infection, vol. 138, pp. 853–860, 2010.
 L. Acedo, J.-A. Morano, and J. Díez-Domingo, “Cost analysis of a vaccination strategy for respiratory syncytial virus (rsv) in a network model,” Mathematical and Computer Modelling, vol. 52, no. 7-8, pp. 1016 – 1022, 2010.
 L. Acedo, J.-A. Morano, R.-J. Villanueva, J. Villanueva-Oller, and J. Díez- Domingo, “Using random networks to study the dynamics of respiratory syncytial virus (RSV) in the spanish region of Valencia,” Mathematical and Computer Modelling, vol. 54, no. 7 - 8, pp. 1650 – 1654, 2011.