## Modeling the dynamics of an epidemic under vaccination in two interacting populations.(English)Zbl 1251.93088

Summary: We present a model for an SIR epidemic in a population consisting of two components-locals and migrants. We identify three equilibrium points and we analyse the stability of the disease free equilibrium. Then we apply optimal control theory to find an optimal vaccination strategy for this 2-group population in a very simple form. Finally we support our analysis by numerical simulation using the fourth order Runge-Kutta method.

### MSC:

 93C95 Application models in control theory 92D25 Population dynamics (general) 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 37N25 Dynamical systems in biology
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### References:

 [1] K. Chalvet-Monfray, M. Artzrouni, J. P. Gouteux, P. Auger, and P. Sabatier, “A two-patch model of Gambian sleeping sickness: Application to vector control strategies in a village and plantations,” Acta Biotheoretica, vol. 46, no. 3, pp. 207-222, 1998. [2] J. Tumwiine, J. Y. T. Mugisha, and L. S. Luboobi, “A host-vector model for malaria with infective immigrants,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 139-149, 2010. · Zbl 1176.92045 [3] Y. Zhou, K. Khan, Z. Feng, and J. Wu, “Projection of tuberculosis incidence with increasing immigration trends,” Journal of Theoretical Biology, vol. 254, no. 2, pp. 215-228, 2008. · Zbl 1400.92567 [4] Z. W. Jia, G. Y. Tang, Z. Jin et al., “Modeling the impact of immigration on the epidemiology of tuberculosis,” Theoretical Population Biology, vol. 73, no. 3, pp. 437-448, 2008. · Zbl 1210.92042 [5] R. Naresh, A. Tripathi, and D. Sharma, “Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infectives,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 880-892, 2009. · Zbl 1165.34377 [6] M. De La Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination,” Informatica, vol. 22, no. 3, pp. 339-370, 2011. · Zbl 1263.92029 [7] Y. Li, L. Chen, and K. Wang, “Permanence for a delayed nonautonomous SIR epidemic model with density-dependent birth rate,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 350892, 10 pages, 2011. · Zbl 1229.92069 [8] A. Kaddar, A. Abta, and H. T. Alaoui, “A comparison of delayed SIR and SEIR epidemic models,” Nonlinear Analysis: Modelling and Control, vol. 16, no. 2, pp. 181-190, 2011. · Zbl 1322.92073 [9] 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. · Zbl 1271.93015 [10] G. Zaman, Y. Han Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR epidemic model,” BioSystems, vol. 93, no. 3, pp. 240-249, 2008. [11] C. Piccolo III and L. Billings, “The effect of vaccinations in an immigrant model,” Mathematical and Computer Modelling, vol. 42, no. 3-4, pp. 291-299, 2005. · Zbl 1080.92056 [12] J. Yu, D. Jiang, and N. Shi, “Global stability of two-group SIR model with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 235-244, 2009. · Zbl 1184.34064 [13] Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson, and Y. L. Danon, “Pulse mass measles vaccination across age cohorts,” Proceedings of the National Academy of Sciences of the United States of America, vol. 90, no. 24, pp. 11698-11702, 1993. [14] L. Acedo, J.-A. Moraño, 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. · Zbl 1205.92044 [15] J. M. Tchuenche, S. A. Khamis, F. B. Agusto, and S. C. Mpeshe, “Optimal control and sensitivity analysis of an influenza model with treatment and vaccination,” Acta Biotheoretica, vol. 59, no. 1, pp. 1-28, 2011. [16] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series, Chapman & Hall/CRC, London, UK, 2007. · Zbl 1291.92010
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