Global Dynamics and Optimal Control of Multi-Age Structured Vector Disease Model with Vaccination, Relapse and General Incidence

• Autores: Sheng-Fu Wang, Lin-Fei Nie
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 1, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Vaccine effectiveness, disease recovery and recurrence are important issues that must be faced in the prevention and control of vector-borne infectious diseases. We develop, in this paper, a dynamical model of vector disease with multi-age-structure to describe the transmission of parasites (or bacteria) between vectors and hosts, where vaccination, relapse and general incidence are introduced to study how these factors influence the spread and control of disease. First, the accurate formulation of the basic reproduction number is gained, which determines the existence and local asymptotic stability of the disease-free and endemic steady states. Further, by utilizing the fluctuation theorem and the method of Lyapunov function, we verify that the disease-free steady state is globally asymptotically stable if the basic reproduction number is less than one. In addition, we also prove that the endemic steady state of this model without relapse is globally asymptotically stable if the basic reproduction number is greater than one.

    Moreover, the optimal control problem for our model is formulated and analyzed.

    Finally, some numerical simulations are conducted to explain these analytical results.

• Referencias bibliográficas
  • 1. Malaria, https://www.who.int/news-room/fact-sheets/detail/malaria [30 September 2021]
  • 2. Ross, R.: The prevention of malaria, 2nd edn. Murray, London (1911)
  • 3. Dietz, K.: Models for parasitic disease control. Bull. Inst. Internat. Statist. 46, 531–544 (1975)
  • 4. Macdonald, G.: The epidemiology and control of malaria. Oxford University Press, London (1957)
  • 5. Niger, A.M., Gumel, A.B.: Mathematical analysis of the role of repeated exposure on malaria transmission dynamics. Differ. Equat. Dyn....
  • 6. Chitnis, N., Cushing, J.M., Hyman, J.M.: Bifurcation analysis of a mathematical model for malaria transmission. SIAM J. Appl. Math. 67(1),...
  • 7. Osman, M.A., Li, J.H: Analysis of a vector-bias malaria transmission model with application to Mexico, Sudan and Democratic Republic of...
  • 8. Zheng, T.T., Nie, L.F., Teng, Z.D., Luo, Y.T.: Competitive exclusion in a multi-strain malaria transmission model with incubation period....
  • 9. Martcheva, M.: An introduction to mathematical epidemiology. Springer, New York (2015)
  • 10. Duan, X.C., Yuan, S.L., Li, X.Z.: Global stability of an SVIR model with age of vaccination. Appl. Math. Comput. 226, 528–540 (2014)
  • 11. Yang, J.Y., Chen, Y.M., Xu, F.: Effect of infection age on an SIS epidemic model on complex networks. J. Math. Biol. 73, 1227–1249 (2016)
  • 12. Yang, J.Y., Xu, R., Li, J.X.: Threshold dynamics of an age-space structured brucellosis disease model with Neumann boundary condition....
  • 13. Hathout, F.Z., Touaoula, T.M., Djilali, S.: Mathematical analysis of a triple age dependent epidemiological model with including a protection...
  • 14. Wang, S.F., Nie, L.F.: Global dynamics for a vector-borne disease model with class-age-dependent vaccination, latency and general incidence...
  • 15. Wang, X., Chen, Y.M., Liu, S.Q.: Dynamics of an age-structured host-vector model for malaria transmission. Math. Methods Appl. Sci. 41,...
  • 16. Liu, L.L., Wang, J.L., Liu, X.N.: Global stability of an SEIR epidemic model with age-dependent latency and relapse. Nonlinear Anal. Real...
  • 17. Magal, P.: Compact attractors for time-periodic age-structured population models. Electron. J. Differ. Equ. 65, 1–35 (2001)
  • 18. Yang, J.Y., Modnak, C., Wang, J.: Dynamical analysis and optimal control simulation for an agestructured cholera transmission model. J....
  • 19. Dang, Y.X., Qiu, Z.P., Li, X.Z., Martcheva, M.: Global dynamics of a vector-host epidemic model with age of infection. Math. Biosci. Eng....
  • 20. Duan, X.C., Cheng, H.H., Martcheva, M., Yuan, S.L.: Dynamics of an age structured heroin transmission model with imperfect vaccination....
  • 21. Tumwiine, J., Mugisha, J.Y.T., Luboobi, L.S.: A mathematical model for the dynamics of malaria in a human host and mosquito vector with...
  • 22. Zhang, F.M., Qiu, Z.P., Huang, A.J., Zhao, X.: Optimal control and cost-effectiveness analysis of a Huanglongbing model with comprehensive...
  • 23. Mohammed-Awel, J., Numfor, E., Zhao, R.J., Lenhart, S.: A new mathematical model studying imperfect vaccination: optimal control analysis....
  • 24. Jan, R., Xiao, Y.N.: Effect of partial immunity on transmission dynamics of dengue disease with optimal control. Math. Methods Appl. Sci....
  • 25. Tang, B., Xiao, Y.N., Tang, S.Y., Wu, J.H.: Modelling weekly vector control against dengue in the Guangdong Province of China. J. Theor....
  • 26. Jia, P.Q., Yang, J.Y., Li, X.Z.: Optimal control and cost-effective analysis of an age-structured emerging infectious disease model. Infect....
  • 27. Roop-O, P., Chinviriyasit,W., Chinviriyasit, S.: The effect of incidence function in backward bifurcation for malaria model with temporary...
  • 28. Kokomo, E., Emvudu, Y.: Mathematical analysis and numerical simulation of an age-structured model of cholera with vaccination and demographic...
  • 29. Yang, Y., Xu, Y.C.: Global stability of a diffusive and delayed virus dynamics model with BeddingtonDeAngelis incidence function and CTL...
  • 30. Tadmon, C., Foko, S., Rendall, A.D.: Global stability analysis of a delay cell-population model of hepatitis B infection with humoral...
  • 31. Kumar, A.: Nilam: Dynamic behavior of an SIR epidemic model along with time delay; CrowleyMartin type incidence rate and Holling type...
  • 32. Liu, W.M., Levin, S.A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol....
  • 33. Lu, M., Huang, J.C., Ruan, S.G., Yu, P.: Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized nonmonotone...
  • 34. Hale, J.: Theory of functional differential equations. Springer-Verlag, New York (1971)
  • 35. Hirsch, W.M., Hanish, H., Gabriel, J.P.: Differential equation models of some parasitic infections: methods for the study of asymptotic...
  • 36. Iannelli, M.: Mathematical theory of age-structured population dynamics. Giardini Editori E Stampatori, Pisa (1995)
  • 37. Thieme, H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166(2), 173–201 (2000)
  • 38. Castillo-Chavez, C., Thieme, H.R.: Asymptotically autonomous epidemic models. In: Mathematical Population Dynamics: Analysis of Heterogeneity,...
  • 39. Thieme, H.R.: Convergence results and a Poincaré-Bendixson trichotomy for asymptotically alltonomous differential equations. J. Math....
  • 40. Kang, Y.H.: Identification problem of two operators for nonlinear systems in Banach spaces. Nonlinear Anal. 70, 1443–1458 (2009)
  • 41. Fister, K.R., Gaff, H., Lenhart, S., Numfor, E., Schaefer, E., Wang, J.: Optimal control of vaccination in an age-structured cholera model....
  • 42. Lenhart, S., Workman, J.T.: Optimal control applied to biological models. Chapman & Hall/Crc, London (2007)
  • 43. Smith, H.L., Thieme, H.R.: Dynamical systems and population persistence. Amer. Math. Soc, Providence, RI (2011)
  • 44. Wang, X., Zhang, Y., Song, X.Y.: An age-structured epidemic model with waning immunity and general nonlinear incidence rate. Int. J. Biomath....
  • 45. Wang, X., Chen, Y., Liu, S.: Global dynamics of a vector-borne disease model with infection ages and general incidence rates. Comp. Appl....
  • 46. Chitnis, N., Hyman, J.M., Cushing, J.M.: Determining important parameters in the spread of malaria through the sensitivity analysis of...
  • 47. Ngonghala, C.N., Mohammed-Awel, J., Zhao, R.J., Prosper, O.: Interplay between insecticide-treated bed-nets and mosquito demography: implications...