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January  2022, 27(1): 257-275. doi: 10.3934/dcdsb.2021040

Random perturbations of an eco-epidemiological model

Lopo F. de Jesus 1, , César M. Silva 2, and  Helder Vilarinho 2,,

1. 

Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, and Instituto Superior de Ciências da Educação, Rua Sarmento Rodrigues, Lubango, Angola
2. 

Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, 6201-001, Covilhã, Portugal

* Corresponding author: Helder Vilarinho

Received  September 2020 Revised  December 2020 Published  January 2022 Early access  January 2021

Fund Project: L. F. de Jesus, C. M. Silva and H. Vilarinho were partially supported by FCT through CMA-UBI (project UIDB/MAT/00212/2020). L. F. de Jesus was also supported by INAGBE

We consider random perturbations of a general eco-epidemiological model. We prove the existence of a global random attractor, the persistence of susceptibles preys and provide conditions for the simultaneous extinction of infectives and predators. We also discuss the dynamics of the corresponding random epidemiological $ SI $ and predator-prey models. We obtain for this cases a global random attractor, prove the prevalence of susceptibles/preys and provide conditions for the extinctions of infectives/predators.

Keywords: Eco-epidemiological model, random attractor, random dynamical systems, random differential equations.
Mathematics Subject Classification: Primary: 34F05, 60H25, 37A50.
Citation: Lopo F. de Jesus, César M. Silva, Helder Vilarinho. Random perturbations of an eco-epidemiological model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 257-275. doi: 10.3934/dcdsb.2021040
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Anal., 17 (2013), 511–528.

[3]

T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynam., 84 (2016), 115-126.  doi: 10.1007/s11071-015-2238-3.

[4]

T. Caraballo, R. Colucci, J. López-de-la-Cruz and A. Rapaport, A way to model stochastic perturbations in population dynamics models with bounded realizations, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 239-257.  doi: 10.1016/j.cnsns.2019.04.019.

[5]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, SpringerBriefs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.

[6]

T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.  doi: 10.3934/cpaa.2017007.

[7]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, Vol. 580. Springer-Verlag, Berlin-New York, 1977.

[8]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Math, Vol. 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[9]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., 176 (1999), 57-72.  doi: 10.1007/BF02505989.

[10]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[12]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.

[13]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.

[14]

H. Crauel, Random Probability Measures on Polish Spaces Stochastics Monographs, V.11, London, 2002.

[15] J. W. Cholewa and T. Dloko, Global Attractors in the Abstract Parabolics Problems, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.
[16]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[17]

M. Garrione and C. Rebelo, Persistence in seasonally varying predator-prey systems via the basic reproduction number, Nonlinear Anal. Real World Appl., 30 (2016), 73-98.  doi: 10.1016/j.nonrwa.2015.11.007.

[18]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Probability Theory and Stochastic Modelling, 85, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.

[19]

L. F. de Jesus, C. M. Silva and H. Vilarinho, An Eco-epidemiological model with general functional response of predator to prey, preprint.

[20]

L. F. de Jesus, C. M. Silva and H. Vilarinho, Periodic orbits for periodic eco-epidemiological systems with infected prey, Electron. J. Qual. Theory Differ. Equ., 54 (2020), 1-20.  doi: 10.14232/ejqtde.2020.1.54.

[21]

Y. Lu, X. Wang and S. Liu, A non-autonomous predator-prey model with infected prey, Discrete Contin. Dyn. Syst. B, 23 (2018), 3817-3836.  doi: 10.3934/dcdsb.2018082.

[22]

C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949.  doi: 10.1007/s00285-011-0440-6.

[23]

C. M. Silva, Existence of Periodic Solutions for Eco-Epidemic Model with Disease in the Prey, J. Math. Anal. Appl., 453 (2017), 383-397.  doi: 10.1016/j.jmaa.2017.03.074.

[24]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.

[25]

X. Niu, T. Zhang and Z. Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Appl. Math. Model., 35 (2011), 457-470.  doi: 10.1016/j.apm.2010.07.010.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Anal., 17 (2013), 511–528.

[3]

T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynam., 84 (2016), 115-126.  doi: 10.1007/s11071-015-2238-3.

[4]

T. Caraballo, R. Colucci, J. López-de-la-Cruz and A. Rapaport, A way to model stochastic perturbations in population dynamics models with bounded realizations, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 239-257.  doi: 10.1016/j.cnsns.2019.04.019.

[5]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, SpringerBriefs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.

[6]

T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.  doi: 10.3934/cpaa.2017007.

[7]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, Vol. 580. Springer-Verlag, Berlin-New York, 1977.

[8]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Math, Vol. 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[9]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., 176 (1999), 57-72.  doi: 10.1007/BF02505989.

[10]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[12]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.

[13]

H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718.  doi: 10.1016/j.jde.2018.03.011.

[14]

H. Crauel, Random Probability Measures on Polish Spaces Stochastics Monographs, V.11, London, 2002.

[15] J. W. Cholewa and T. Dloko, Global Attractors in the Abstract Parabolics Problems, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.
[16]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[17]

M. Garrione and C. Rebelo, Persistence in seasonally varying predator-prey systems via the basic reproduction number, Nonlinear Anal. Real World Appl., 30 (2016), 73-98.  doi: 10.1016/j.nonrwa.2015.11.007.

[18]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Probability Theory and Stochastic Modelling, 85, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.

[19]

L. F. de Jesus, C. M. Silva and H. Vilarinho, An Eco-epidemiological model with general functional response of predator to prey, preprint.

[20]

L. F. de Jesus, C. M. Silva and H. Vilarinho, Periodic orbits for periodic eco-epidemiological systems with infected prey, Electron. J. Qual. Theory Differ. Equ., 54 (2020), 1-20.  doi: 10.14232/ejqtde.2020.1.54.

[21]

Y. Lu, X. Wang and S. Liu, A non-autonomous predator-prey model with infected prey, Discrete Contin. Dyn. Syst. B, 23 (2018), 3817-3836.  doi: 10.3934/dcdsb.2018082.

[22]

C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949.  doi: 10.1007/s00285-011-0440-6.

[23]

C. M. Silva, Existence of Periodic Solutions for Eco-Epidemic Model with Disease in the Prey, J. Math. Anal. Appl., 453 (2017), 383-397.  doi: 10.1016/j.jmaa.2017.03.074.

[24]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.

[25]

X. Niu, T. Zhang and Z. Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Appl. Math. Model., 35 (2011), 457-470.  doi: 10.1016/j.apm.2010.07.010.

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