American Institute of Mathematical Sciences

April  2020, 19(4): 1915-1930. doi: 10.3934/cpaa.2020084

On a delayed epidemic model with non-instantaneous impulses

 1 College of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China 2 Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain 3 Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain

*Corresponding author.

Dedicated to professor Tomás Caraballo on the occasion of his 60th birthday

Received  April 2019 Revised  October 2019 Published  January 2020

We introduce a non-instantaneous pulse vaccination model. Non-instantaneous impulsive nonlinear differential equations provide an adequate biomathematical model of some medical problems. In this paper we study some basic properties such as the attractiveness of the infection-free periodic solution and the permanence of some sub-population for a vaccine model where a constant fraction of the susceptible population is vaccinated in some periodic way. Our model is a system of nonlinear differential equations with impulses.

Citation: Liang Bai, Juan J. Nieto, José M. Uzal. On a delayed epidemic model with non-instantaneous impulses. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1915-1930. doi: 10.3934/cpaa.2020084
References:
 [1] R. Agarwal, S. Hristova and D. O'Regan, Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions, J. Franklin Inst., 354 (2017), 3097-3119.  doi: 10.1016/j.jfranklin.2017.02.002. [2] R. Agarwal, S. Hristova and D. O'Regan, Non-Instantaneous Impulses in Differential Equations, Springer International Publishing, 2017. doi: 10.1007/978-3-319-66384-5. [3] L. Bai and J. J. Nieto, Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett., 73 (2017), 44-48.  doi: 10.1016/j.aml.2017.02.019. [4] L. Bai, J. J. Nieto and X. Wang, Variational approach to non-instantaneous impulsive nonlinear differential equations, J. Nonlinear Sci. Appl., 10 (2017), 2440-2448.  doi: 10.22436/jnsa.010.05.14. [5] D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66, CRC Press, 1993. [6] M. Benchohra, S. Litimein and J. J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses, Journal of Fixed Point Theory and Applications, 21 (2019), 21. doi: 10.1007/s11784-019-0660-8. [7] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag GmbH, 2012. doi: 10.1007/978-1-4614-1686-9. [8] K. L. Cooke and P. Van Den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260.  doi: 10.1007/s002850050051. [9] O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley Series in Mathematical & Computational Biology, Wiley, 2000. [10] S. Gao, L. Chen, J. J. Nieto and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037-6045.  doi: 10.1016/j.chaos.2006.04.061. [11] H. Guo, L. Chen and X. Song, Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control, Int. J. Biomath., 10 (2017), 1750093. doi: 10.1142/S1793524517500930. [12] E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2. [13] J. Jiao, S. Cai and L. Li, Impulsive vaccination and dispersal on dynamics of an SIR epidemic model with restricting infected individuals boarding transports, Physica A: Statistical Mechanics and its Applications, 449 (2016), 145 – 159. doi: 10.1016/j.physa.2015.10.055. [14] A. Khaliq and M. U. Rehman, On variational methods to non-instantaneous impulsive fractional differential equation, Applied Mathematics Letters, 83 (2018), 95-102.  doi: 10.1016/j.aml.2018.03.014. [15] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific, 1989. doi: 10.1142/0906. [16] Y. Luo, S. Gao and S. Yan, Pulse vaccination strategy in an epidemic model with two susceptible subclasses and time delay, Appl. Math., 2 (2011), 57. doi: 10.4236/am.2011.21007. [17] D. J. Nokes and J. Swinton, Vaccination in pulses: a strategy for global eradication of measles and polio?, Trends Microbiol., 5 (1997), 14-19. [18] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664. [19] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2010. doi: 10.1007/978-1-4419-7646-8. [20] R. Terzieva, Some phenomena for non-instantaneous impulsive differential equations, Int. J. Pure Appl. Math., 119 (2018), 483-490. [21] J. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., 73 (2017), 157 – 162. doi: 10.1016/j.aml.2017.04.010. [22] Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.  doi: 10.1016/S0025-5564(01)00049-9. [23] D. Yang, J. Wang and D. O'Regan, On the orbital hausdorff dependence of differential equations with non-instantaneous impulses, Comptes Rendus Mathematique, 356 (2018), 150-171.  doi: 10.1016/j.crma.2018.01.001. [24] B. Zhu and L. Liu, Periodic boundary value problems for fractional semilinear integro-differential equations with non-instantaneous impulses, Boundary Value Problems, 2018. doi: 10.1186/s13661-018-1048-1.

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References:
 [1] R. Agarwal, S. Hristova and D. O'Regan, Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions, J. Franklin Inst., 354 (2017), 3097-3119.  doi: 10.1016/j.jfranklin.2017.02.002. [2] R. Agarwal, S. Hristova and D. O'Regan, Non-Instantaneous Impulses in Differential Equations, Springer International Publishing, 2017. doi: 10.1007/978-3-319-66384-5. [3] L. Bai and J. J. Nieto, Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett., 73 (2017), 44-48.  doi: 10.1016/j.aml.2017.02.019. [4] L. Bai, J. J. Nieto and X. Wang, Variational approach to non-instantaneous impulsive nonlinear differential equations, J. Nonlinear Sci. Appl., 10 (2017), 2440-2448.  doi: 10.22436/jnsa.010.05.14. [5] D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66, CRC Press, 1993. [6] M. Benchohra, S. Litimein and J. J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses, Journal of Fixed Point Theory and Applications, 21 (2019), 21. doi: 10.1007/s11784-019-0660-8. [7] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag GmbH, 2012. doi: 10.1007/978-1-4614-1686-9. [8] K. L. Cooke and P. Van Den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260.  doi: 10.1007/s002850050051. [9] O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley Series in Mathematical & Computational Biology, Wiley, 2000. [10] S. Gao, L. Chen, J. J. Nieto and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037-6045.  doi: 10.1016/j.chaos.2006.04.061. [11] H. Guo, L. Chen and X. Song, Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control, Int. J. Biomath., 10 (2017), 1750093. doi: 10.1142/S1793524517500930. [12] E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2. [13] J. Jiao, S. Cai and L. Li, Impulsive vaccination and dispersal on dynamics of an SIR epidemic model with restricting infected individuals boarding transports, Physica A: Statistical Mechanics and its Applications, 449 (2016), 145 – 159. doi: 10.1016/j.physa.2015.10.055. [14] A. Khaliq and M. U. Rehman, On variational methods to non-instantaneous impulsive fractional differential equation, Applied Mathematics Letters, 83 (2018), 95-102.  doi: 10.1016/j.aml.2018.03.014. [15] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific, 1989. doi: 10.1142/0906. [16] Y. Luo, S. Gao and S. Yan, Pulse vaccination strategy in an epidemic model with two susceptible subclasses and time delay, Appl. Math., 2 (2011), 57. doi: 10.4236/am.2011.21007. [17] D. J. Nokes and J. Swinton, Vaccination in pulses: a strategy for global eradication of measles and polio?, Trends Microbiol., 5 (1997), 14-19. [18] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664. [19] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2010. doi: 10.1007/978-1-4419-7646-8. [20] R. Terzieva, Some phenomena for non-instantaneous impulsive differential equations, Int. J. Pure Appl. Math., 119 (2018), 483-490. [21] J. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., 73 (2017), 157 – 162. doi: 10.1016/j.aml.2017.04.010. [22] Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.  doi: 10.1016/S0025-5564(01)00049-9. [23] D. Yang, J. Wang and D. O'Regan, On the orbital hausdorff dependence of differential equations with non-instantaneous impulses, Comptes Rendus Mathematique, 356 (2018), 150-171.  doi: 10.1016/j.crma.2018.01.001. [24] B. Zhu and L. Liu, Periodic boundary value problems for fractional semilinear integro-differential equations with non-instantaneous impulses, Boundary Value Problems, 2018. doi: 10.1186/s13661-018-1048-1.
Simulation with $R^*<1$, infection-free solution
Simulation with $R_*>1$, permanence of infected population
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