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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Complex Dynamics of an Epidemic Model With Optimal Vaccination and Treatment in the Presence of Population Dispersal

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 471--497 | DOI:10.5890/DNC.2021.09.010

Manotosh Mandal$^{1,2}$, Soovoojeet Jana$^3$ , Swapan Kumar Nandi$^4$, T. K. Kar$^2$

$^{1}$ Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk -721636 , West Bengal, India

$^2$ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India

$^3$ Department of Mathematics, Ramsaday College, Amta-711401, Howrah, West Bengal, India

$^4$ Department of Mathematics, Nayabasat P.M.Sikshaniketan, Paschim Medinipur-721253, West Bengal, India

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Abstract

In this article, we describe an SEIR type epidemic model with a transport-related infection between two cities in the presence of vaccination and treatment control. The epidemiological threshold, commonly known as basic reproduction number is derived and its impact on the dynamical behaviour of the disease has been established. The optimal control problem is constructed with the objective of minimizing the effect of infection in the system and then it is solved. We compare the result of the model to a real-world problem to establish that our model can be used in some practical cases if the parameters are known.

Acknowledgments

The authors are grateful to the anonymous reviewers and Prof. Dimitri Volchenkov, Editor-in-Chief of the journal for their valuable suggestions to improve both the quality and presentation of the manuscript significantly. The work of Soovoojeet Jana is financially supported by Dept of Science \& Technology Biotechnology, Govt. of West Bengal (vide memo no. 201 (Sanc.)/ST/P/S\&T/16G-12/2018 dt 19-02-2019).

References

  1. [1]  Diekmann, O. and Heesterbeek, J.A.P. (1999), Mathematical Epidemiology of Infectious Diseases, { Model Building, Analyis and Interpretation}, Wiley, New York.
  2. [2]  Hethcote, H.W. (2000), The mathematics of infectious diseases, { SIAM Rev.}, 42(4), 599-653.
  3. [3]  Keeling, M.J. and Rohani, P. (2008), { Modelling Infectious Diseases in Humans and Animals}, Princeton University Press, Princeton, New Jersey.
  4. [4]  Chen, Y., Yan, M., and Xiang, Z. (2014), Transmission dynamics of a two-city SIR epidemic model with transport related infection, { J Appl Math}, 2014, Article ID 764278.
  5. [5]  Jana, S., Nandi, S.K., and Kar, T.K. (2016), Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment, { Acta biotheoretica}, 64, 65-84.
  6. [6]  Nandi, S.K., Jana, S., Mandal, M., and Kar, T.K. (2018), Analysis of a fuzzy epidemic model with saturated treatment and disease transmission,{ Int. J. of Biomathematics}, 11(1), 1850002.
  7. [7]  Nandi, S.K., Jana, S., Mandal, M., and Kar, T.K. (2019), Complex Dynamics and Optimal Treatment of an Epidemic Model with Two Infectious Diseases, { Int. J. Appl. Comput. Math}, 5(29), doi: https://doi.org/10.1007/s40819-019-0613-3.
  8. [8]  Wang, W. and Mulone, G. (2003), Threshold of disease transmission in a patch environment, { J. Math. Anal. Appl.}, 285, 321-335.
  9. [9]  Wang, W. and Zhao, X.Q. (2004), An epidemic model in a patchy environment, { Math. Biosci.}, 190, 97-112.
  10. [10]  Wang, W. and Zhao, X.Q. (2005), An age-structured epidemic model in a patchy environment,{ SIAM J. Appl. Math.}, 65, 1597-1614.
  11. [11]  Arino, J. and Driessche, P.V. (2003), A multi-city epidemic model, { Math. Popul. Stud}, 10, 175-193.
  12. [12]  Wan, H. and Cui, J. (2007), An SEIS epidemic model with transport related infection, { J. Theor. Biol.}, 247 507-524.
  13. [13]  Wang, L. and Yang, W. (2011), Global dynamics of a two-patch SIS model with infection during transport, { Applied Mathematics and Computation}, 217, 8458-8467.
  14. [14]  Denphedtnong, A., Chinviriyasit, S., and Chinviriyasit, W. (2013), On the dynamics of SEIRS epidemic model with transport related infection, { Math Biosci}, 245, 188-205.
  15. [15]  Jana, S., Haldar, P., and Kar, T.K. (2016), Optimal control and stability analysis of an epidemic model with population dispersal, { Chaos, Solitons and Fractals}, 83, 67-81.
  16. [16]  Nandi, S.K., Jana, S., Mandal, M., and Kar, T.K. (2018), Mathematical Analysis of an Epidemic System in the Presence of Optimal Control and Population Dispersal, { Biophysical Reviews and Letters}, 13(1), 1-17.
  17. [17]  Hethcote, H.W. (1978), An immunization model for a heterogeneous population, { Theor. Populat. Biol.}, 14, 338.
  18. [18]  Makinde, O.D. (2007), Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy, { Appl. Math. Comput.}, 184, 842-848.
  19. [19]  Thomasey, D.H. and Martcheva, M.(2008), Serotype replacement of vertically transmitted diseases through perfect vaccination, { J. Biol. sys.}, 16(2), 255-277.
  20. [20]  Zaman, G., Kang, Y.H., and Jung, I.H. (2008), Stability analysis and optimal vaccination of an SIR epidemic model, { BioSystems}, 93, 240-249.
  21. [21]  Buonomo, B., dOnofrio, A., and Lacitignola, D. (2008), Global stability of an SIR epidemic model with information dependent vaccination, { Math. Biosci}, 216, 9-16.
  22. [22]  Buonomo, B. and Lacitignola, D. (2011), On the backward bifurcation of a vaccination model with nonlinear incidence, { Nonlinear Anal. Model. Control}, 16(1), 30-46.
  23. [23]  Zhou, Y., Yang, K., Zhou, K,. and Liang, Y. (2014), Optimal vaccination policies for an SIR model with limited resources, { Acta Biotheor}, 62, 171-181.
  24. [24]  Jung, E., Lenhart, S., and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, { Discrete Contin. Dyn. Syst. Ser.}, 2(4), 473-482.
  25. [25]  Wang, W. (2006), Backward bifurcation of an epidemic model with treatment, { Mathematical Biosciences}, 201, 58-71.
  26. [26]  Cai, L., Li, X., Ghosh, M., and Guo, B. (2009), Stability analysis of an HIV/AIDS epidemic model with treatment, { J. Comput. Appl. Math.}, 229, 313-323.
  27. [27]  Eckalbar, J.C. and Eckalbar, W. L.(2011), Dynamics of an epidemic model with quadratic treatment, { Nonlinear Anal Real World Appl}, 12, No. 1, 320-332.
  28. [28]  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.
  29. [29]  Kar, T. K. and Jana, S. (2013), A theoretical study on mathematical modeling of an infectious disease with application of optimal control, { Biosystems}, 111(1), 37-50.
  30. [30]  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 Biotheor}, 59(1), 1-28.
  31. [31]  Laarabi, H., Abta, A., and Hattaf, K. (2015), Optimal control of a delayed SIRS epidemic model with vaccination and treatment, { Acta Biotheor}, 63(2), 87-97.
  32. [32]  Mishra, A.K., Sharma, A., and Shukla, J. B. (2015), Stability analysis and optimal control of an epidemic model with awareness programs by media, { BioSystems}, 138, 53-62.
  33. [33]  Kar, T.K., Nandi, S.K., Jana, S., and Mandal, M. (2019), Stability and bifurcation analysis of an epidemic model with the effect of media, { Chaos, Solitons $\&$ Fractals}, 120, 188-199.
  34. [34]  Kar, T.K., Jana, S., and Ghorai, A. (2013), Effect of isolation in an infectious disease, { Int. J. Ecol. Econ. Stat.}, 29(2), 87-106.
  35. [35]  Jana, S., Haldar, P., and Kar, T.K. (2016), Mathematical analysis of an epidemic model with isolation and optimal controls, { International Journal of Computer Mathematics}, 94(2017), 1318-1336.
  36. [36]  Kar, T. K. and Jana, S.(2013), Application of three controls optimally in a vector-borne disease---a mathematical study, { Commun Nonlinear Sci Numer Simulat}, 18, 2868-2884.
  37. [37]  Allen, L.J.S. (2007), { An introduction to mathematical biology}, Pearson education limited, USA.
  38. [38]  Diekmann, O., Metz, J.A.J., and Heesterbeek, J.A.P. (1990), On the definition on the computation of the basic reproduction number ratio $R_0$ in models for infectious diseases in heterogenious population, { Jr. Math. Biol.}, 28, 365-382.
  39. [39]  Fulford, G.R., Roberts, M.G., and Heesterbeek, J.A.P. (2002), The meta population dynamics of an infectious disease: tuberculosis in possums, { Jr. Theor. Biol.}, 61, 15-29.
  40. [40]  Cui, J., Takeuchi, Y., and Saito, Y. (2006), Spreading disease with transport related infection, { Jr. Theo. Biol.}, 239, 376-90.
  41. [41]  Joshi, H.R. (2002), Optimal control of an HIV immunology model, { Optim. Con. Appl. Methods}, 23, 199-213.
  42. [42]  Kar, T.K. and Batabyal, A. (2011), Stability and optimal control of a sir epidemic model with vaccination, { Biosystems}, 104, 127-135.
  43. [43]  Lukes, D.L. (1982), { Differential equations:Classical to controlled, Mathematics in science and engineering}, 162, Academic press, New York.
  44. [44]  Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F.(1962), { The mathematical theory of optimal processes}, Wiley, New York.
  45. [45]  Rodrigues, P., Silva, C.J., and Torres, D. F. M. (2014), Cost effectiveness analysis of optimal control measures for tuberculosis, { Bull Math Biol}, 76(10), 2627-2645.
  46. [46]  Agusto, F.B. (2013), Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, { BioSystems} 113, 155-164.
  47. [47]  Okosun, K.O., Ouifki, R., Marcus, N. (2013), Optimal control strategies and cost-effectiveness analysis of malaria model, { BioSystems}, 111, 83-101.
  48. [48]  Liu, X. and Takeuchi, Y. (2006), Spread of disease with transport related infection and entry screening, { Jr. Theor. Bio.}, 242, 517-528.
  49. [49]  Murray, J.D. (2002), { Mathematical Biology}, Berlin, Springer.
  50. [50]  Buck, C., Llopis, A., Najera, E., and Terris, M. (1998), `The Challenge of Epidemiology: Issues and Selected Readings,{ Scientific Publication, Pan American Health Organization}, 505.
  51. [51]  Alchon, S. (2003), { A Pest in the land: New world epidemic in a global perspective}, University of New Mexico Press.
  52. [52]  Liu, J. and Zhou, Y. (2009), Global stability of a SIRS epidemic model with transport related infection, { Chaos solitons Fractals}, 40, 145-158.
  53. [53]  World Health Organization. { Ebola Situation Report-29 April 2015}, http://apps.who.int/ebola/current-situation/ebola-situation-report-29-april-2015, 2015. Accessed: 01/09/2016.
  54. [54]  Zhang, F.F., Jin, Z., and Sun, G.Q. (2010), Biffurcation analysis of a delayed epidemic model, { Appl Math Compu}, 216, 753-767.

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