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A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon

1 School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, P.R. China
2 Department of Earth, Ocean, and Atmospheric Science and Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, United States
3 Department of Scientific Computing, Florida State University, Tallahassee, FL 32306-4120, United States

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This paper focuses on numerical approximation of the basic reproduction number $\mathcal{R}_0$ , which is the threshold defined by the spectral radius of the next-generation operator in epidemiology. Generally speaking, $\mathcal{R}_0$ cannot be explicitly calculated for most age-structured epidemic systems. In this paper, for a deterministic age-structured epidemic system and its stochastic version, we discretize a linear operator produced by the infective population with a theta scheme in a finite horizon, which transforms the abstract problem into the problem of solving the positive dominant eigenvalue of the next-generation matrix. This leads to a corresponding threshold $\mathcal{R}_0$,n . Using the spectral approximation theory, we obtain that $\mathcal{R}_0$,n → $\mathcal{R}_0$ as n → +∞. Some numerical simulations are provided to certify the theoretical results.
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Citation: Wenjuan Guo, Ming Ye, Xining Li, Anke Meyer-Baese, Qimin Zhang. A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon. Mathematical Biosciences and Engineering, 2019, 16(5): 4107-4121. doi: 10.3934/mbe.2019204

References

• 1. X. Zhang, D. Jiang, A. Alsaedi, et al., Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Appl. Math. Lett., 59 (2016), 87–93.
• 2. P. Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000), 525–540.
• 3. W. Guo, Y. Cai, Q. Zhang, et al., Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage, Physica A, 492 (2018), 2220–2236.
• 4. J. Pan, A. Gray, D. Greenhalgh, et al., Parameter estimation for the stochastic SIS epidemic model, J. Stat. Inference Stoch. Process, 17 (2014), 75–98.
• 5. Y. Cai, Y. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240.
• 6. S. Busenberg, M. Iannelli and H. Thieme, Global behavior of an age-structured epidemic model, Siam J. Math. Anal., 22 (1991), 1065–1080.
• 7. B. Cao, H. Huo and H. Xiang, Global stability of an age-structure epidemic model with imperfect vaccination and relapse, Physica A, 486 (2017), 638–655.
• 8. H. Inaba, Age-structured population dynamics in demography and epidemiology, Springer, Sin- gapore, 2017.
• 9. T.Kuniya, Globalstabilityanalysiswithadiscretizationapproachforanage-structuredmultigroup SIR epidemic model, Nonlinear Anal. Real World Appl., 12 (2011), 2640–2655.
• 10. K. Toshikazu, Numerical approximation of the basic reproduction number for a class of age- structured epidemic models, Appl. Math. Lett., 73 (2017), 106–112.
• 11. H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411–434.
• 12. O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio R 0 , in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382.
• 13. T. Kuniya and R. Oizumi, Existence result for an age-structured SIS epidemic model with spatial diffusion, Nonlinear Anal. Real World Appl., 23 (2015), 196–208.
• 14. N. Bacaër, Approximation of the basic reproduction number R 0 for Vector-Borne diseases with a periodic vector population, B. Math. Biol., 69 (2007), 1067–1091.
• 15. Z. Xu, F. Wu and C. Huang, Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 278 (2015), 258–277.
• 16. X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 238 (2013), 14–28.
• 17. F. Chatelin, The spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators, Siam Rev., 23 (1981), 495–522.
• 18. A. Berman and R. Plemmons, Nonnegative matrices in the mathematical sciences, Academic press, New York, 1979.
• 19. K. Ito and F. Kappel, The Trotter-Kato theorem and approximation of PDEs, Math. Comput., 67 (1998), 21–44.
• 20. B. Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149–153.
• 21. M. Krein, Linear operators leaving invariant acone in a Banach space, Amer. Math. Soc. Transl.,10 (1962), 3–95.
• 22. W. Guo, Q. Zhang, X. Li, et al., Dynamic behavior of a stochastic SIRS epidemic model with media coverage, Math. Method. Appl. Sci., 41 (2018), 5506-5525.
• 23. C. Mills, J. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904–906.