Structured parametric epidemic models

Abstract

A stage-structured model for a theoretical epidemic process that incorporates immature, susceptible and infectious individuals in independent stages is formulated. In this analysis, an input interpreted as a birth function is considered. The structural identifiability is studied using the Markov parameters. Then, the unknown parameters are uniquely determined by the output structure corresponding to an observation of infection. Two different birth functions are considered: the linear case and the Beverton-Holt type to analyse the structured epidemic model. Some conditions on the parameters to obtain non-zero disease-free equilibrium points are given. The identifiability of the parameters allows us to determine uniquely the basic reproduction number Script capital R-0 and the stability of the model in the equilibrium is studied using Script capital R-0 in terms of the model parameters.This work has been partially supported by MTM2010-18228. The authors wish to express their thanks to the reviewers for helpful comments and suggestions.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2014). Structured parametric epidemic models. International Journal of Computer Mathematics. 91(2):188-197. https://doi.org/10.1080/00207160.2013.800864188197912Allen, L. J. S., & Thrasher, D. B. (1998). The effects of vaccination in an age-dependent model for varicella and herpes zoster. IEEE Transactions on Automatic Control, 43(6), 779-789. doi:10.1109/9.679018Ben-Zvi, A., McLellan, P. J., & McAuley, K. B. (2004). Identifiability of Linear Time-Invariant Differential-Algebraic Systems. 2. The Differential-Algebraic Approach. Industrial & Engineering Chemistry Research, 43(5), 1251-1259. doi:10.1021/ie030534jBoyadjiev, C., & Dimitrova, E. (2005). An iterative method for model parameter identification. Computers & Chemical Engineering, 29(5), 941-948. doi:10.1016/j.compchemeng.2004.08.036Cantó, B., Coll, C., & Sánchez, E. (2011). Identifiability for a Class of Discretized Linear Partial Differential Algebraic Equations. Mathematical Problems in Engineering, 2011, 1-12. doi:10.1155/2011/510519Cao, H., & Zhou, Y. (2012). The discrete age-structured SEIT model with application to tuberculosis transmission in China. Mathematical and Computer Modelling, 55(3-4), 385-395. doi:10.1016/j.mcm.2011.08.017Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28(4). doi:10.1007/bf00178324Dion, J.-M., Commault, C., & van der Woude, J. (2003). Generic properties and control of linear structured systems: a survey. Automatica, 39(7), 1125-1144. doi:10.1016/s0005-1098(03)00104-3Emmert, K. E., & Allen, L. J. S. (2004). Population Persistence and Extinction in a Discrete-time, Stage-structured Epidemic Model. Journal of Difference Equations and Applications, 10(13-15), 1177-1199. doi:10.1080/10236190410001654151Farina, L., & Rinaldi, S. (2000). Positive Linear Systems. doi:10.1002/9781118033029Van den Hof, J. M. (1998). Structural identifiability of linear compartmental systems. IEEE Transactions on Automatic Control, 43(6), 800-818. doi:10.1109/9.679020T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.Li, C.-K., & Schneider, H. (2002). Applications of Perron-Frobenius theory to population dynamics. Journal of Mathematical Biology, 44(5), 450-462. doi:10.1007/s002850100132Li, X., & Wang, W. (2005). A discrete epidemic model with stage structure☆. Chaos, Solitons & Fractals, 26(3), 947-958. doi:10.1016/j.chaos.2005.01.063Ma, J., & Earn, D. J. D. (2006). Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease. Bulletin of Mathematical Biology, 68(3), 679-702. doi:10.1007/s11538-005-9047-7Wang, W., & Zhao, X.-Q. (2004). An epidemic model in a patchy environment. Mathematical Biosciences, 190(1), 97-112. doi:10.1016/j.mbs.2002.11.00