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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Optimal Control Applied to Marine Eco-Epidemiological model with Disease in Prey Species

Journal of Applied Nonlinear Dynamics 12(4) (2023) 661--678 | DOI:10.5890/JAND.2023.12.003

S.M. Mahathy Hasan$^1$, Md. Haider Ali Biswas$^1$, Laek Sazzad Andallah$^2$

$^1$ Department of Mathematics, Jahangirnagar University, Saver, Dhaka, Bangladesh

$^2$ Mathematics Discipline, khulna University, Khulna, Bangladesh

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Abstract

In this paper, the purpose of the work is to modified of an Eco-epidemiocal model and analysis with the optimal control strategies for the infected prey. Furthermore, the dynamical behaviors, i.e. the positivity, boundedness, stability and Hopf bifurcation of the proposed model are investigated. The optimal control theory is applied to control the effect of disease on prey population and plays a vital role to eliminate disease from prey population.The results obtained suggested the optimal control is ensuring the prey-predator populations coexists in a defined habitat.

References

  1. [1]  Raid, K.N. and Kawa, A.H. (2012), The dynamics of the prey- predator model with diseae in prey, Mathematics in Computer Science, 2(4), 1052-1072.
  2. [2]  Lotka, A.J. (1924), Elements of Physical Biology, Baltimore Williams and Wilkins Co., Inc.
  3. [3]  Kermack, W.O. and McKendrick, A.G. (1927), The Evolution of Virulence in Sterilizing Pathogens, Pro-Royal Soc. London, A, 115, 700.
  4. [4]  Hethcote, H.W. (2000), The mathematics of infectious diseases, SIAM Review, 42, 599-653.
  5. [5]  Jiao, J.J., Chen, L.S., Nieto, J.J., and Angela, T. (2008), Permanence and Global attractively of the Stage-structured Predator-prey Model with Continuous Harvesting on Predator and Impulsive stocking on prey, Applied Mathematics and Mechanics (English Edition), 29(5), 653-663.
  6. [6]  Pan, S. (2013), Asymptotic spreading in a Lotka- Volterra predator-prey system, Journal of Mathematical Analysis and Applications, 407(2), 230-236.
  7. [7]  Zhou, J. and Shi, J. (2013), The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses, Journal of Mathematical Analysis and Applications, 405(2), 618-630.
  8. [8]  Gani, J. and Swift, R.J. (2013), Prey predator models with infected prey and predators, Discrete and Continuous Dynamical System, 33(11-12), 5059-5066.
  9. [9]  Jana, S. and Kar, T.K. (2013), Modeling and analysis of a prey-predator system with the disease in the prey, Nonlinear Science and Nonequilibrium and Complex Phenomena, Chaos, Solutions and Fractals, 4, 42-53.
  10. [10]  Mukhopadhyay, B. and Bhattacharyya, R. (2009), Role of predator switching in an eco-epidemiological model with disease in the prey, Ecological Modeling, 220, 931-939.
  11. [11]  Bornaa, C.S., Makinde, O.D., and Seini, I.Y. (2015), Eco-epidemiology model and Optimal Control of disease transmission between humans and animals, Communications in Mathematical Biology and Neuroscience, 8(2), 1-28.
  12. [12]  Chattopadhyay, J. (1996), Effect of toxic substances on a two-species competitive system, Ecological Modelling, 84, 287-289.
  13. [13]  Sontag, E. (1998), Mathematical Control Theory, Springer (2nd), Springer, http://doi.org/10.1007/978-3-540-69532-5-16.
  14. [14]  Okosun, K.O., Ouifki, R., and Marcus, N. (2011), Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, BioSystems, 106, 136-145.
  15. [15]  Hallam, T.G. and Clark, C.E. (1983), Effect of toxicants on populations: A qualitive approach I. Equilibrium environmental exposure, Ecological Modeling, 18, 291-304.
  16. [16]  Hadeler, K.P. and Freedman, H.I. (1989), Predator-prey populations with parasitic infection, Journal of Mathematical Biology, 27, 609-631.
  17. [17]  Freedman, H.I. and Shukla, J.B. (1991), Models for the effect of toxicant in single-species and predator-prey systems, Journal of Mathematical Biology, 30, 15-30.
  18. [18]  Gulland, F.M.D. (1995), Impact of Infectious Diseases on Wild Animal Populations: a Review. In: Grenfell, B.T., Dobson, A. P. (Eds.), Ecology of Infectious Diseases in Natural Populations, Cambridge University Press, Cambridge, pp.20-51.
  19. [19]  Haque, M. and Venturino, E. (2006), The role of transmissible diseases in the Holling-Tanner predator-prey model, Theoretical Population Biology, 70(3), 273-288.
  20. [20]  Hugo, A., Massawe, E.S., and Oluwole, D.M. (2012), An Eco-Epidemiological Mathematical Model with Treatment and Disease Infection in both Prey and Predator population, Journal of Ecology and the Natural Environment, 4(10), 266-279.
  21. [21]  Crawford, J.D. (1991), Introduction to bifurcation theory, Reviews of Modern Physics, 63(4), 991-1037.
  22. [22]  Hasan, N., Biswas, H.A., and Uddin, S. (2019), An Ecological model for sustainable wildlife management of ecosystem based on optimal control theory, Communications in Mathematical Biology and Neuroscience, 17.
  23. [23]  Hasan, M.N., Biswas, M.H.A., and Uddin, M.S. (2020), Interactive Effects of Disease Transmission on Predator-Prey Model, Journal of Applied Non-linear Dynamics, 11(1), 33-56.
  24. [24]  Clark, C.W. (1976), Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley. Princeton. Univ. Press, New York.
  25. [25]  Hasan, N., Biswas, H.A., and Uddin, S. (2019), An ecological model for sustainable forest management of eco-system based on optimal control theory, Journal of Nepal Mathematical Society (JNMS), 2(1), 19-44.
  26. [26]  Hasan, M.N., Biswas, M.H.A., and Uddin, M.S. (2019), An ecologicall model for sustainable wildlife management of the sundarban's eco-system based on optimal control theory, Communication Biology and Neuroscience, 21(2), 251-263.
  27. [27]  Hasan, M.N., Biswas, M.H.A., and Uddin, M.S. (2019), A mathematical model for fish management in the Sundarban's ecosystem, Open Journal of Mathematical Analysis, 3(2), 42-49.
  28. [28]  Hasan, M.N., Biswas, M.H.A., and Uddin, M.S. (2020), Effect of Poaching on TigerDeer interaction Model with Ratio-Dependent functional response in the Sundarbans ecosystem, Journal of Applied Non-linear Dynamics, 9(3), 415-425.
  29. [29]  Katsukawa, T. (2004), A numerical investigation of the optimal control rule for decision-making in fisheries management, Fisheries Science, 70, 123-131.
  30. [30]  Kotani, K., Kakinaka, M., and Matsuda, H. (2008), Optimal escapement levels on renewable resource management under process uncertainty: Some implications of convex unit harvest cost, Environmental Economics and Policy Studies, 9, 107-118.
  31. [31]  Ganguli, C., Kar, T.K., and Mondal, P.K. (2017) , Optimal Harvesting of a Prey- Predator Model with Variable Carrying Capacity, International Journal of Biomathematics, 10(5), 24.
  32. [32] Lenhart, S. and Workman, J.T. (2007), Optimal Control Applied to Biological Models, CRC Mathematical and Computational Biology Series.
  33. [33]  Lenhart, S. and Workman, J. (2007), Optimal Control Applied to Biological Models, Boca Raton, Chapmal Hall/CRC.
  34. [34]  Massawe, L.N., Massawe, E.S., and Makinde, O.D. (2015), Modelling infectiology and optimal control of dengue epidemic, Applied and Computational Mathematics, 4(3), 181-191.
  35. [35]  Mpeshe, S.C., Luboobi, L.S., and Nkansah-gyekye, Y.A.W. (2014), Optimal control strategies for the dynamics of rift valey fever, Communications in Optimization Theory, pp. 1-18.
  36. [36]  Okosun, K., Makinde, O.D., and Takaidza, I. (2013), Impact of optimal control on the treatment of HIV / AIDS and screening of unaware infectives, Applied Mathematical Modelling, 37(6), 3802-3820.
  37. [37]  Tchuenche, J.M., Khamis, S.A., Agusto, F.B., and Mpeshe, S.C. (2011), Optimal control and sensitivity analysis of an influenza model with treatment and vaccination, Acta Biotheoretica, 59(1), 1-28.
  38. [38]  Pontryagin, L.S., Boltyanskii, V.G., Gamkrelize, R.V., and Mishchenko, E.F. (1967), The Mathematical Theory of Optimal Processes, Wiley, New York. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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