\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Dynamics and asymptotic profiles on an age-structured SIS epidemic model with random diffusion and advection

  • *Corresponding author: Shi-Ke Hu

    *Corresponding author: Shi-Ke Hu 

This work is supported by the National Natural Science Foundation of China (No.12171039) and Science and Technology Plan Foundation of Gansu Province of China (No.21JR7RA216)

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • In this paper, we introduce an age-structured SIS (susceptible-infectious-susceptible) epidemic model with random diffusion and advection, in which birth and transmission rates depend on individuals. First, the well-posedness of this model was obtained. Next, we established the existence and uniqueness of the nontrivial nonnegative steady state, and derived the basic reproduction number which is also a threshold for the existence of nontrivial nonnegative steady states. Then, we studied the local stability of the nontrivial nonnegative steady state. In particular, we determined the upper and lower bounds of the principal eigenvalue to better obtain the local stability. Finally, we investigated the asymptotic profiles of the principal eigenvalue and the nontrivial nonnegative steady state with respect to diffusion rate and advection rate, respectively.

    Mathematics Subject Classification: Primary: 35L60, 47D06, 47H07, 92D25.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. 1, Monographs in Mathematics, vol. 89, Birkhauser Boston Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.
    [2] H. Brezis and H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Vol. 2, Springer, New York, 2011.
    [3] S. N. BusenbergM. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM Journal on Mathematical Analysis, 22 (1991), 1065-1080.  doi: 10.1137/0522069.
    [4] R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction–diffusion–advection SIS epidemic model with mass action infection mechanism, Calculus of Variations and Partial Differential Equations, 60 (2021), Paper No. 184, 38 pp. doi: 10.1007/s00526-021-01992-w.
    [5] R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, Journal of Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.
    [6] K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, Journal of Mathematical Biology, 41 (2000), 1-23.  doi: 10.1007/s002850000025.
    [7] D. Daners and P. Koch-Medina, Abstract Evolution Equations, Periodic Problems and Applications, Longmon Scientific & Technical, Hariow. New York, 1992.
    [8] G. Di Blasio, Mathematical analysis for an epidemic model with spatial and age structure, Journal of Evolution Equations, 10 (2010), 929-953.  doi: 10.1007/s00028-010-0077-8.
    [9] O. DiekmannJ. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio ${R}_{0}$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.
    [10] Z.-K. GuoH.-F. Huo and H. Xiang, Optimal control of TB transmission based on an age structured HIV-TB co-infection model, Journal of the Franklin Institute, 359 (2022), 4116-4137.  doi: 10.1016/j.jfranklin.2022.04.005.
    [11] Z.-K. GuoH.-F. Huo and H. Xiang, Analysis of an age-structured model for HIV-TB co-infection, Discrete & Continuous Dynamical Systems-Series B, 27 (2022), 199-228.  doi: 10.3934/dcdsb.2021037.
    [12] Z.-K. GuoH. Xiang and H.-F. Huo, Analysis of an age-structured tuberculosis model with treatment and relapse, Journal of Mathematical Biology, 82 (2021), 1-37.  doi: 10.1007/s00285-021-01595-1.
    [13] B. Guo and W. Chan, On the semigroup for age dependent population dynamics with spatial diffusion, Journal of Mathematical Analysis and Applications, 184 (1994), 190-199.  doi: 10.1006/jmaa.1994.1193.
    [14] D. He, L. Lin, Y. Artzy-Randrup, H. Demirhan, B. J. Cowling and L. Stone, Resolving the enigma of Iquitos and Manaus: A modeling analysis of multiple COVID-19 epidemic waves in two Amazonian cities, Proceedings of the National Academy of Sciences, 120 (2023), e2211422120. doi: 10.1073/pnas.2211422120.
    [15] H. J. Heijmans, The dynamical hehaviour of the age-size-distribution of a cell population, The Dynamics of Physiologically Structured Populations (Amsterdam, 1983), Lecture Notes in Biomath., Springer-Verlag, Berlin, 68 (1986), 185-202. doi: 10.1007/978-3-662-13159-6_5.
    [16] J. Huo and R. Yuan, Mathematical analysis of autonomous and non-autonomous age-structured reaction-diffusion-advection population model, Mathematical Methods in the Applied Sciences, 46 (2023), 2667-2696.  doi: 10.1002/mma.8668.
    [17] M. IannelliF. Milner and A. Pugliese, Analytical and numerical results for the age-structured SIS epidemic model with mixed inter-intracohort transmission, SIAM Journal on Mathematical Analysis, 23 (1992), 662-688.  doi: 10.1137/0523034.
    [18] M. IannelliM.-Y. Kim and E.-J. Park, Asymptotic behavior for an SIS epidemic model and its approximation, Nonlinear Analysis, 35 (1999), 797-814.  doi: 10.1016/S0362-546X(97)00597-X.
    [19] H. Inaba, Threshold and stability results for an age-structured epidemic model, Journal of Mathematical Biology, 28 (1990), 411-434.  doi: 10.1007/BF00178326.
    [20] H. Inaba, On a pandemic threshold theorem of the early Kermack–Mckendrick model with individual heterogeneity, Mathematical Population Studies, 21 (2014), 95-111.  doi: 10.1080/08898480.2014.891905.
    [21] H. Kang and S. Ruan, Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion,, Journal of Mathematical Biology, 83 (2021), Paper No. 5, 30 pp. doi: 10.1007/s00285-021-01634-x.
    [22] H. Kang and S. Ruan, Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions, Journal of Differential Equations, 278 (2021), 430-462.  doi: 10.1016/j.jde.2021.01.004.
    [23] H. KangS. Ruan and X. Yu, Age-structured population dynamics with nonlocal diffusion, Journal of Dynamics and Differential Equations, 34 (2022), 789-823.  doi: 10.1007/s10884-020-09860-5.
    [24] M. Kim, Global dynamics of approximate solutions to an age-structured epidemic model with diffusion, Advances in Computational Mathematics, 25 (2006), 451-474.  doi: 10.1007/s10444-004-7639-7.
    [25] T. Kuniya and H. Inaba, Endemic threshold results for an age-structured SIS epidemic model with periodic parameters, Journal of Mathematical Analysis and Applications, 402 (2013), 477-492.  doi: 10.1016/j.jmaa.2013.01.044.
    [26] T. KuniyaH. Inaba and J. Yang, Global behavior of SIS epidemic models with age structure and spatial heterogeneity, Japan Journal of Industrial and Applied Mathematics, 35 (2018), 669-706.  doi: 10.1007/s13160-018-0300-5.
    [27] T. Kuniya and R. Oizumi, Existence result for an age-structured SIS epidemic model with spatial diffusion, Nonlinear Analysis: Real World Applications, 23 (2015), 196-208.  doi: 10.1016/j.nonrwa.2014.10.006.
    [28] M. Langlais and S. Busenberg, Global behaviour in age structured SIS models with seasonal periodicities and vertical transmission, Journal of Mathematical Analysis and Applications, 213 (1997), 511-533.  doi: 10.1006/jmaa.1997.5554.
    [29] Z. LiuP. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, Journal of Differential Equations, 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.
    [30] F. LutscherE. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Review, 47 (2005), 749-772.  doi: 10.1137/050636152.
    [31] F. LutscherM. A. Lewis and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bulletin of Mathematical Biology, 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.
    [32] P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer. New York, 2018. doi: 10.1007/978-3-030-01506-0.
    [33] P. Magal and C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM Journal on Applied Mathematics, 73 (2013), 1058-1095.  doi: 10.1137/120882056.
    [34] J. Simon, Compact sets in the space ${L}^{p}(0, {T}; {B})$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.
    [35] L. StoneD. HeS. Lehnstaedt and Y. Artzy-Randrup, Extraordinary curtailment of massive typhus epidemic in the Warsaw Ghetto, Science Advances, 6 (2020), 0927.  doi: 10.1126/sciadv.abc0927.
    [36] H. Triebel, Interpolation Theory, Function Space, Differential Operators, 2nd edn, Johann Ambrosius Barth, Heidelberg, 1995.
    [37] C. Walker, Some results based on maximal regularity regarding population models with age and spatial structure, Journal of Elliptic and Parabolic Equations, 4 (2018), 69-105.  doi: 10.1007/s41808-018-0010-9.
    [38] C. Walker and J. Zehetbauer, The principle of linearized stability in age-structured diffusive populations, Journal of Differential Equations, 341 (2022), 620-656.  doi: 10.1016/j.jde.2022.09.025.
    [39] G. F. Webb, A recovery-relapse epidemic model with spatial diffusion, Journal of Mathematical Biology, 14 (1982), 177-194.  doi: 10.1007/BF01832843.
    [40] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker. New York, 1985.
    [41] L. XueS. JingK. ZhangR. Milne and H. Wang, Infectivity versus fatality of SARS-CoV-2 mutations and influenza, International Journal of Infectious Diseases, 121 (2022), 195-202.  doi: 10.1016/j.ijid.2022.05.031.
  • 加载中
SHARE

Article Metrics

HTML views(1133) PDF downloads(308) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return