Mathematical Analysis for a Zika Virus Dynamics in a Seasonal Environment

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DOI: https://doi.org/10.28924/2291-8639-22-2024-71
Published: Feb 22, 2024
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Fatema Ahmed Al Najim, Mathematical Analysis for a Zika Virus Dynamics in a Seasonal Environment, Int. J. Anal. Appl., 22 (2024), 71.

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Fatema Ahmed Al Najim

Abstract

We propose a mathematical model for the Zika virus (ZIKV) spread under the influence of a seasonal environment. The basic reproduction number R0 was calculated for both cases, the fixed and seasonal environment permitting the characterisation of the extinction and the persistence of the disease for both cases. We proved that the virus-free steady state is globally asymptotically stable if R0≤1, while the disease will be persist if R0>1. Finally, extensive numerical simulations are given to confirm the theoretical findings.

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References

  1. T. Magalhaes, B.D. Foy, E.T.A. Marques, G.D. Ebel, J. Weger-Lucarelli, Mosquito-Borne and Sexual Transmission of Zika Virus: Recent Developments and Future Directions, Virus Res. 254 (2018), 1–9. https://doi.org/10.1016/j.virusres.2017.07.011.
  2. L.R. Petersen, D.J. Jamieson, A.M. Powers, M.A. Honein, Zika Virus, N. Engl. J. Med. 374 (2016), 1552–1563. https://doi.org/10.1056/nejmra1602113.
  3. G.W.A. Dick, S.F. Kitchen, A.J. Haddow, Zika Virus (I). Isolations and Serological Specificity, Trans. R. Soc. Trop. Med. Hyg. 46 (1952), 509–520. https://doi.org/10.1016/0035-9203(52)90042-4.
  4. P.S. Mead, N.K. Duggal, S.A. Hook, M. Delorey, M. Fischer, et al. Zika Virus Shedding in Semen of Symptomatic Infected Men, N. Engl. J. Med. 378 (2018), 1377–1385. https://doi.org/10.1056/nejmoa1711038.
  5. N.C. Grassly, C. Fraser, Mathematical Models of Infectious Disease Transmission, Nat. Rev. Microbiol. 6 (2008), 477–487. https://doi.org/10.1038/nrmicro1845.
  6. M.E. Hajji, A.H. Albargi, AMathematical Investigation of an "SVEIR" EpidemicModel for theMeasles Transmission, Math. Biosci. Eng. 19 (2022), 2853–2875. https://doi.org/10.3934/mbe.2022131.
  7. A.A. Alsolami, M. El Hajji, Mathematical Analysis of a Bacterial Competition in a Continuous Reactor in the Presence of a Virus, Mathematics. 11 (2023), 883. https://doi.org/10.3390/math11040883.
  8. A.H. Albargi, M. El Hajji, Bacterial Competition in the Presence of a Virus in a Chemostat, Mathematics. 11 (2023), 3530. https://doi.org/10.3390/math11163530.
  9. F. Brauer, C. Castillo-Chavez, A. Mubayi, S. Towers, Some models for epidemics of vector-transmitted diseases, Infect. Dis. Model. 1 (2016), 79–87. https://doi.org/10.1016/j.idm.2016.08.001.
  10. Y. Nakata, T. Kuniya, Global Dynamics of a Class of SEIRS Epidemic Models in a Periodic Environment, J. Math. Anal. Appl. 363 (2010), 230–237. https://doi.org/10.1016/j.jmaa.2009.08.027.
  11. S.K. Sasmal, I. Ghosh, A. Huppert, J. Chattopadhyay, Modeling the Spread of Zika Virus in a Stage-Structured Population: Effect of Sexual Transmission, Bull. Math. Biol. 80 (2018), 3038–3067. https://doi.org/10.1007/s11538-018-0510-7.
  12. A. Alshehri, M. El Hajji, Mathematical Study for Zika Virus Transmission With General Incidence Rate, AIMS Math. 7 (2022), 7117–7142. https://doi.org/10.3934/math.2022397.
  13. D. Xiao, Dynamics and Bifurcations on a Class of Population Model With Seasonal Constant-Yield Harvesting, Discr. Contin. Dyn. Syst. - Ser. B. 21 (2015), 699–719. https://doi.org/10.3934/dcdsb.2016.21.699.
  14. N. Bacaër, S. Guernaoui, The Epidemic Threshold of Vector-Borne Diseases With Seasonality, J. Math. Biol. 53 (2006), 421–436. https://doi.org/10.1007/s00285-006-0015-0.
  15. N. Bacaër, Approximation of the Basic Reproduction Number R0 for Vector-Borne Diseases with a Periodic Vector Population, Bull. Math. Biol. 69 (2007), 1067–1091. https://doi.org/10.1007/s11538-006-9166-9.
  16. N. Bacaër, R. Ouifki, Growth Rate and Basic Reproduction Number for Population Models With a Simple Periodic Factor, Math. Biosci. 210 (2007), 647–658. https://doi.org/10.1016/j.mbs.2007.07.005.
  17. J. Ma, Z. Ma, Epidemic Threshold Conditions for Seasonally Forced SEIR Models, Math. Biosci. Eng. 3 (2006), 161–172. https://doi.org/10.3934/mbe.2006.3.161.
  18. T. Zhang, Z. Teng, On a Nonautonomous SEIRS Model in Epidemiology, Bull. Math. Biol. 69 (2007), 2537–2559. https://doi.org/10.1007/s11538-007-9231-z.
  19. S. Guerrero-Flores, O. Osuna, C. Vargas-de-Leon, Periodic Solutions for Seasonal Siqrs Models With Nonlinear Infection Terms, Electron. J. Diff. Equ. 2019 (2019), 92.
  20. W. Wang, X.Q. Zhao, Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments, J. Dyn. Diff. Equ. 20 (2008), 699–717. https://doi.org/10.1007/s10884-008-9111-8.
  21. M. El Hajji, D.M. Alshaikh, N.A. Almuallem, Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination, Mathematics. 11 (2023), 2350. https://doi.org/10.3390/math11102350.
  22. M. El Hajji, Periodic Solutions for an "SVIQR" Epidemic Model in a Seasonal Environment with General Incidence Rate, Int. J. Biomath. In Press.
  23. B.S. Alshammari, D.S. Mashat, F.O. Mallawi, Mathematical and Numerical Investigations for a Cholera Dynamics With a Seasonal Environment, Int. J. Anal. Appl. 21 (2023), 127. https://doi.org/10.28924/2291-8639-21-2023-127.
  24. M. El Hajji, R.M. Alnjrani, Periodic Trajectories for HIV Dynamics in a Seasonal Environment With a General Incidence Rate, Int. J. Anal. Appl. 21 (2023), 96. https://doi.org/10.28924/2291-8639-21-2023-96.
  25. M. El Hajji, N.S. Alharbi, M.H. Alharbi, Mathematical Modeling for a CHIKV Transmission Under the Influence of Periodic Environment, Int. J. Anal. Appl. 22 (2024), 6. https://doi.org/10.28924/2291-8639-22-2024-6.
  26. M. El Hajji, Periodic Solutions for Chikungunya Virus Dynamics in a Seasonal Environment With a General Incidence Rate, AIMS Math. 8 (2023), 24888–24913. https://doi.org/10.3934/math.20231269.
  27. J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 2nd ed., Biomathematics Texts, Vol. 18, 1980.
  28. A.K. Chakraborty, M.A. Haque, M.A. Islam, Mathematical Modelling and Analysis of Dengue Transmission in Bangladesh With Saturated Incidence Rate and Constant Treatment, Commun. Biomath. Sci. 3 (2020), 101–113.
  29. N.K. Goswami, B. Shanmukha, Stability and Optimal Control Analysis of Zika Virus With Saturated Incidence Rate, Malaya J. Mat. 8 (2020), 331–342. https://doi.org/10.26637/mjm0802/0004.
  30. E. Bonyah, K.O. Okosun, Mathematical modeling of Zika virus, Asian Pac. J. Trop. Dis. 6 (2016), 673–679. https://doi.org/10.1016/s2222-1808(16)61108-8.
  31. O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations, J. Math. Biol. 28 (1990), 365–382. https://doi.org/10.1007/bf00178324.
  32. P. van den Driessche, J. Watmough, Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission, Math. Biosci. 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6.
  33. J. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.
  34. M. El Hajji, A.Y. Al-Subhi, M.H. Alharbi, Mathematical Investigation for Two-Bacteria Competition in Presence of a Pathogen With Leachate Recirculation, Int. J. Anal. Appl. 22 (2024), 45. https://doi.org/10.28924/2291-8639-22-2024-45.
  35. A.H. Albargi, M.E. Hajji, Mathematical Analysis of a Two-Tiered Microbial Food-Web Model for the Anaerobic Digestion Process, Math. Biosci. Eng. 20 (2023), 6591–6611. https://doi.org/10.3934/mbe.2023283.
  36. M. El Hajji, Mathematical Modeling for Anaerobic Digestion Under the Influence of Leachate Recirculation, AIMS Math. 8 (2023), 30287–30312. https://doi.org/10.3934/math.20231547.
  37. M. El Hajji, Influence of the Presence of a Pathogen and Leachate Recirculation on a Bacterial Competition, Int. J. Biomath. In Press.
  38. F. Zhang, X.Q. Zhao, A Periodic Epidemic Model in a Patchy Environment, J. Math. Anal. Appl. 325 (2007), 496–516. https://doi.org/10.1016/j.jmaa.2006.01.085.
  39. X.Q. Zhao, Dynamical Systems in Population Biology, Springer, 2003. https://doi.org/10.1007/978-3-319-56433-3.