On a safety set for an epidemic model with a bounded population
Abstract
Given a class of non-linear SIRS epidemic model, we analyse some useful conditions on the model parameters to determine a safety set for the containment of an epidemic. In addition, once that set is determined, we find control actions so that the epidemic remains within the security set with infection rates below an allowed amount. More specifically, for every initial state in a certain safety set of the state space there exists an adequate control policy maintaining the state of the system in such safety set. Sufficient conditions for the existence of a solution under a feedback are derived in terms of linear inequalities on the input vectors at the vertices of a polytope.
Keyword : epidemiological process, discrete-time non-linear system, positivity, stability, control feedback
How to Cite
Coll, C., Romero-Vivó, S., & Sánchez, E. (2022). On a safety set for an epidemic model with a bounded population. Mathematical Modelling and Analysis, 27(2), 263–281. https://doi.org/10.3846/mma.2022.14586
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L.J.S. Allen and P. Van den Driessche. The basic reproduction number in some discrete-time epidemic models. Journal of difference equations and applications, 14(10-11):1127–1147, 2008. https://doi.org/10.1080/10236190802332308
M. Althoff, O. Stursberg and M. Buss. Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In 2008 47th IEEE Conference on Decision and Control, pp. 4042–4048, Cancun, Mexico, December 2–6, 2008. IEEE. https://doi.org/10.1109/CDC.2008.4738704
Z. Bartosiewicz. Local positive reachability of nonlinear continuoustime systems. IEEE Trans Autom Control, 61(12):4217–4221, 2015. https://doi.org/10.1109/TAC.2015.2511921
Z. Bartosiewicz. Positive reachability of discrete-time nonlinear systems. In IEEE Conference on Control Applications (CCA), pp. 1203–1208, Buenos Aires, Argentina, September 19–22, 2016. IEEE. https://doi.org/10.1109/CCA.2016.7587970
A. Berman and R.J. Plemmons. Nonnegative matrices in the mathematical sciences, volume 9. Siam, Philadelphia, 1994.
C.P. Bhunu and W. Garira. A two strain tuberculosis transmission model with therapy and quarantine. Mathematical Modelling and Analysis, 14(3):291–312, 2009. https://doi.org/10.3846/1392-6292.2009.14.291-312
J. Calatayud, J.C. Cortés, M. Jornet and R.J. Villanueva. Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC. Mathematical Methods in the Applied Sciences, 41(18):9618–9627, 2018. https://doi.org/10.1002/mma.5315
B. Cantó, C. Coll and E. Sánchez. Structured parametric epidemic models. Int. J. Comp. Math., 91(2):188–197, 2014. https://doi.org/10.1080/00207160.2013.800864
B. Cantó, C. Coll and E. Sánchez. Estimation of parameters in a structured SIR model. Advances in Difference Equations, 2017(1):1–13, 2017. https://doi.org/10.1186/s13662-017-1078-5
V. Capasso. Mathematical structures of epidemic systems, volume 88. Springer, 1993. https://doi.org/10.1007/978-3-540-70514-7
T. Dang, C. Le Guernic and O. Maler. Computing reachable states for nonlinear biological models. In International Conference on Computational Methods in Systems Biology (CMSB 2009), pp. 126–141, Bologna, Italy, 31 August - 1 September, 2009. Springer. https://doi.org/10.1007/978-3-642-03845-7_9
A. Delabouglise, A. James, J.F. Valarcher, S. Hagglund, D. Raboisson and J. Rushton. Linking disease epidemiology and livestock productivity: The case of bovine respiratory disease in France. PLoS ONE, 12(12):1–23, 2017. https://doi.org/10.1371/journal.pone.0189090
B. Grünbaum. Convex polytopes, volume 221. Springer Science & Business Media, 2013.
A.B. Gumel, C. Castillo-Chavez, R.E. Mickens and D.P. Clemence. Mathematical studies on human disease dynamics: emerging paradigms and challenges, volume 410. Amer. Math. Soc., Boston, MA, USA, 2006.
L.C.G.J.M. Habets and J.H. Van Schuppen. A control problem for affine dynamical systems on a full-dimensional polytope. Automat, 40(1):21–35, 2004. https://doi.org/10.1016/j.automatica.2003.08.001
T. Kaczorek. Locally positive nonlinear systems. In European Control Conference (ECC), pp. 1792–1797, Cambridge, United Kingdom, September 1–4, 2003. IEEE. https://doi.org/10.23919/ECC.2003.7085225
T. Kaczorek. Positivity and stability of discrete-time and continuous-time nonlinear systems. In 16th International Conference on Computational Problems of Electrical Engineering (CPEE), pp. 59–61, Lviv, Ukraine, September 2–5, 2015. IEEE. https://doi.org/10.1109/CPEE.2015.7333337
X. Ma, Y. Zhou and H. Cao. Global stability of the endemic equilibrium of a discrete SIR epidemic model. Adv. Diff. Eq., 2013(1):42, 2013. https://doi.org/10.1186/1687-1847-2013-42
R.M. Mitchell, R.H. Whitlock, S.M. Stehman, A. Benedictys, P.P. Chapagain, Y.T. Grohn and Y.H. Schukken. Simulation modeling to evaluate the persistence of Mycobacterium avium subsp. paratuberculosis (MAP) on commercial dairy farms in the United States. Prevetive Veterinary Medicine, 83(3–4):360–380, 2008. https://doi.org/10.1016/j.prevetmed.2007.09.006
M. Naim, F. Lahmidi and A. Namir. Controllability and observability analysis of nonlinear positive discrete systems. Discrete Dyn. in Nat. and Soc., 2018, 2018. https://doi.org/10.1155/2018/3279290
S.V. Rakovic, E.C. Kerrigan, D.Q. Mayne and J. Lygeros. Reachability analysis of discrete-time systems with disturbances. IEEE Trans. Autom. Control, 51(4):546–561, 2006. https://doi.org/10.1109/TAC.2006.872835
E. Renshaw. Modelling biological populations in space and time, volume 11. Cambridge University Press, 1993.