×
Zakary, Omar; Bidah, Sara; Rachik, Mostafa; Elmouki, Ilias

Cell and patch vicinity travel restrictions in a multi-regions SI discrete epidemic control model. (English) Zbl 1467.92236

Int. J. Adv. Appl. Math. Mech. 6, No. 2, 30-41 (2018).
Summary: Some epidemics can spread in geographical large scales and in unprecedented durations if the authorities concerned do not hurry to save susceptible populations. In such cases, mathematical modelers are required to take into account the spatial aspect of the propagation of an epidemic. Infected travelers either exiting the main source of epidemic or coming from other locations that have been partially affected later, and aiming to enter safer regions, represent the population category that should be hastily controlled. Based on these assumptions, and with the consideration of the factor of their mobility, we devise a discrete-time susceptible-infected (SI) model which describes the spatial spread of an epidemic emerging in regions that are connected by any kind of movement. Furthermore, we introduce into SI discrete equations, controls variables which restrict movements of infected people for avoiding any contact with the susceptible class. Explicitly, we present regions by \(M^2\) cells assembled in one grid, and we seek the optimal values of controls using a discrete version of Pontryagin’s maximum principle. In a first case, we aim to control only one cell by restricting movements of infected people coming from its neighbors, and in a second case, we aim to control a group of cells or patches, based on the same logic of inter-interventions and which is related to the vicinity travel restrictions optimal control strategy. In a first time, the patch is supposed to be located in the opposite side of the epidemic source, and in a second time, in the interior of the domain of study in order to verify the effectiveness of such control policies when the number of regions in the vicinity set, is either small or important.

MSC:

92D30 Epidemiology
49J15 Existence theories for optimal control problems involving ordinary differential equations

Keywords:

spatial spread of epidemic; discrete-time model; SI epidemic model; multi-regions; optimal control; travel restrictions; vicinity; cell; patch
PDF BibTeX XML Cite
\textit{O. Zakary} et al., Int. J. Adv. Appl. Math. Mech. 6, No. 2, 30--41 (2018; Zbl 1467.92236)
Full Text: Link
  • logo of hbz
  • logo of WorldCat

References:

[1] Kermack, W. O., McKendrick, A. G. (1927). A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A. 115 (772): 700-721. doi:10.1098/rspa.1927.0118. · JFM 53.0517.01
[2] Zakary, O., Rachik, M., & Elmouki, I. (2016). On the impact of awareness programs in HIV/AIDS prevention: an SIR model with optimal control. Int. J. Comput. Appl, 133(9), 1-6. · Zbl 1418.92065
[3] Zakary, O., Rachik, M., & Elmouki, I. (2017). How much time is sufficient for benefiting of awareness programs in epidemics prevention? A free final time optimal control approach. Int. J. Adv. Appl. Math. and Mech, 4(4), 26-40. · Zbl 1397.92698
[4] Li, J., Teng, Z., Wang, G., Zhang, L., & Hu, C. (2017). Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment. Chaos, Solitons & Fractals, 99, 63-71. · Zbl 1373.92129
[5] Zaman, G., Kang, Y. H., Cho, G., & Jung, I. H. (2017). Optimal strategy of vaccination & treatment in an SIR epidemic model. Mathematics and Computers in Simulation, 136, 63-77. · Zbl 07313807
[6] Mpeshe, S. C., Nyerere, N., & Sanga, S. (2017). Modeling approach to investigate the dynamics of Zika virus fever: A neglected disease in Africa. Int. J. Adv. Appl. Math. and Mech, 4(3), 14-21. · Zbl 1382.92244
[7] Kandhway, K., & Kuri, J. (2014). How to run a campaign: Optimal control of SIS and SIR information epidemics. Applied Mathematics and Computation, 231, 79-92. · Zbl 1410.92124
[8] Safan, M., & Rihan, F. A. (2014). Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation. Mathematics and Computers in Simulation, 96, 195-206. · Zbl 07312549
[9] Wu, Q., Lou, Y., & Zhu, W. (2016). Epidemic outbreak for an SIS model in multiplex networks with immunization. Mathematical biosciences, 277, 38-46. · Zbl 1358.92099
[10] Liu, B., Duan, Y., & Luan, S. (2012). Dynamics of an SI epidemic model with external effects in a polluted environment. Nonlinear Analysis: Real World Applications, 13(1), 27-38. · Zbl 1238.37040
[11] Tang, S. (2011). A modified SI epidemic model for combating virus spread in wireless sensor networks. International Journal of Wireless Information Networks, 18(4), 319-326.
[12] Romero-L, M., & Gallego, L. (2017). Analysis of voltage sags propagation in distribution grids using a SI epidemic model. In Power Electronics and Power Quality Applications (PEPQA), 2017 IEEE Workshop on (pp. 1-6). IEEE.
[13] Zakary, O., Rachik, M., & Elmouki, I. (2017). On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach. International Journal of Dynamics and Control, 5(3), 917-930. · Zbl 1360.92123
[14] Zakary, O., Rachik, M., & Elmouki, I. (2017). A new analysis of infection dynamics: multi-regions discrete epidemic model with an extended optimal control approach. International Journal of Dynamics and Control, 5(4), 10101019. · Zbl 1360.92123
[15] Zakary, O., Larrache, A., Rachik, M., & Elmouki, I. (2016). Effect of awareness programs and travel-blocking operations in the control of HIV/AIDS outbreaks: a multi-domains SIR model. Advances in Difference Equations, 2016(1), 169. · Zbl 1418.92065
[16] Zakary, O., Rachik, M., & Elmouki, I. (2017). A multi-regional epidemic model for controlling the spread of Ebola: awareness, treatment, and travel-blocking optimal control approaches. Mathematical Methods in the Applied Sciences, 40(4), 1265-1279. · Zbl 1360.92123
[17] Zakary, O., Rachik, M., & Elmouki, I. (2017). A new epidemic modeling approach: Multi-regions discrete-time model with travel-blocking vicinity optimal control strategy. Infectious Disease Modelling, 2(3), 304-322. · Zbl 1422.92178
[18] Abouelkheir, I., Rachik, M., Zakary, O., & Elmouki, I. (2017). A multi-regions SIS discrete influenza pandemic model with a travel-blocking vicinity optimal control approach on cells. American Journal of Computational and Applied Mathematics, 7(2), 37-45.
[19] Abouelkheir, I., El Kihal, F., Rachik, M., Zakary, O., & Elmouki, I. (2017). A multi-regions SIRS discrete epidemic model with a travel-blocking vicinity optimal control approach on cells. Br. J. Math. Comput. Sci, 20(4), 1-16. · Zbl 1382.92237
[20] El Kihal, F., Rachik, M., Zakary, O., & Elmouki, I. (2017). A multi-regions SEIRS discrete epidemic model with a travel-blocking vicinity optimal control approach on cells. Int. J. Adv. Appl. Math. Mech, 4(3), 60-71. · Zbl 1382.92237
[21] El Kihal, F., Abouelkheir, I., Rachik, M., & Elmouki, I. Optimal Control and Computational Method for the Resolution of Isoperimetric Problem in a Discrete-Time SIRS System. Mathematical and Computational Applications, 23(4), 52. (2018).
[22] Zakary, O., Rachik, M., Elmouki, I., & Lazaiz, S. (2017). A multi-regions discrete-time epidemic model with a travel-blocking vicinity optimal control approach on patches. Advances in Difference Equations, 2017(1), 120. · Zbl 1422.92178
[23] Sethi, S. P., & Thompson, G. L. (2000). What is optimal control theory? (pp. 1-22). Springer US.
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.