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Periodic and almost periodic solutions for a non-autonomous respiratory disease model with a lag effect

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Abstract

This paper studies a kind of non-autonomous respiratory disease model with a lag effect. First of all, the permanence and extinction of the system are discussed by using the comparison principle and some differential inequality techniques. Second, it assumes that all coefficients of the system are periodic. The existence of positive periodic solutions of the system is proven, based on the continuation theorem in coincidence with the degree theory of Mawhin and Gaines. In the meantime, the global attractivity of positive periodic solutions of the system is obtained by constructing an appropriate Lyapunov functional and using the Razumikin theorem. In addition, the existence and uniform asymptotic stability of almost periodic solutions of the system are analyzed by assuming that all parameters in the model are almost periodic in time. Finally, the theoretical derivation is verified by a numerical simulation.

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References

  1. Zhu S, Lian X, Wei L, et al, PM2.5 forecasting using SVR with PSOGSA algorithm based on CEEMD, GRNN and GCA considering meteorological factors. Atmos Environ, 2018, 183(5): 20–32

    Article  Google Scholar 

  2. Jo E J, Lee W S, Jo H Y, et al, Effects of particulate matter on respiratory disease and the impact of meteorological factors in Busan. Korea, Resp Med, 2017, 124: 79–87

    Article  Google Scholar 

  3. Weber S A, Insaf T Z, Hall E S, et al, Assessing the impact of fine particulate matter (PM2.5) on respiratory-cardiovascular chronic diseases in the New York City Metropolitan area using Hierarchical Bayesian Model estimates. Environ Res, 2016, 151: 399–409

    Article  Google Scholar 

  4. Li Y, Ma Z, Zheng C, et al, Ambient temperature enhanced acute cardiovascular-respiratory mortality effects of PM2.5 in Beijing, China. Int J Biometeorol, 2015, 59(12): 1761–1770

    Article  Google Scholar 

  5. Tang S, Yan Q, Shi W, et al, Measuring the impact of air pollution on respiratory infection risk in China. Environ Pollut, 2018, 232: 477–486

    Article  Google Scholar 

  6. Kuniya T, Nakata Y, Permanence and extinction for a nonautonomous SEIRS epidemic model. Appl Math Comput, 2012, 218(18): 9321–9331

    MathSciNet  MATH  Google Scholar 

  7. Lu C, Ding X, Persistence and extinction in general non-autonomous logistic model with delays and stochastic perturbation. Appl Math Comput, 2014, 229: 1–15

    MathSciNet  MATH  Google Scholar 

  8. Shi C L, Li Z, Chen F, The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays. Nonl Anal: Real World Appl, 2007, 8(5): 1536–1550

    Article  MathSciNet  MATH  Google Scholar 

  9. Teng Z, Liu Y, Zhang L, Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality. Nonl Anal, 2008, 69(8): 2599–2614

    Article  MathSciNet  MATH  Google Scholar 

  10. Martcheva M, A non-autonomous multi-strain SIS epidemic model. J Biol Dyn, 2009, 3(2/3): 235–251

    Article  MathSciNet  MATH  Google Scholar 

  11. Qi H, Zhang S, Meng X, et al, Periodic solution and ergodic stationary distribution of two stochastic SIQS epidemic systems. Physica A: Stat Mech Appl, 2018, 508: 223–241

    Article  MathSciNet  Google Scholar 

  12. Du Z, Feng Z, Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays. J Comput Appl Math, 2014, 258: 87–98

    Article  MathSciNet  MATH  Google Scholar 

  13. Arenas A J, Gilberto G P, Lucas J, Periodic solutions of nonautonomous differential systems modeling obesity population. Chaos, Solit Fract, 2009, 42(2): 1234–1244

    Article  MathSciNet  MATH  Google Scholar 

  14. Yuan S L, Ma Z E, Jin Z, Persistence and periodic solution on a nonautonomous SIS model with delays. Acta Math Appl Sin-E, 2003, 19(1): 167–176

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhou Y, Ma Z, The periodic solutions for time dependent age-structured population models. Acta Math Sci, 2000, 20B(2): 155–161

    Article  MathSciNet  MATH  Google Scholar 

  16. Gakkhar S, Singh B, Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters. Chaos, Sol Frac, 2006, 27(5): 1239–1255

    Article  MathSciNet  MATH  Google Scholar 

  17. Moussaoui A, Bouguima S M, Seasonal influences on a prey-predator model. J Appl Math Comput, 2016, 50(1/2): 39–57

    Article  MathSciNet  MATH  Google Scholar 

  18. Doveri F, Scheffer M, Rinaldi S, et al, Seasonality and chaos in a plankton fish model. Theor Popul Biol, 1993, 43(2): 159–183

    Article  MATH  Google Scholar 

  19. Wang C, Wang S, Li L, Periodic solution and almost periodic solution of a nonmonotone reaction-diffusion system with time delay. Acta Math Sci, 2010, 30B(2): 517–524

    MathSciNet  MATH  Google Scholar 

  20. Mahieddine K, Tatar N E, Existence and global attractivity of a periodic solution to a nonautonomous dispersal system with delays. Appl Math Model, 2007, 31(4): 780–793

    Article  MATH  Google Scholar 

  21. Wang Q, Dai B, Three periodic solutions of nonlinear neutral functional differential equations. Nonl Anal: Real World Appl, 2008, 9(3): 977–984

    Article  MathSciNet  MATH  Google Scholar 

  22. Xia Y, Chen F, Chen A, et al, Existence and global attractivity of an almost periodic ecological model. Appl Math Comput, 2004, 157(2): 449–475

    MathSciNet  MATH  Google Scholar 

  23. Xie Y, Li X, Almost periodic solutions of single population model with hereditary effects. Appl Math Comput, 2008, 203(2): 690–697

    MathSciNet  MATH  Google Scholar 

  24. Fan M, Wang K, Jiang D, Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments. Math Biosic, 1999, 160(1): 47–61

    Article  MathSciNet  MATH  Google Scholar 

  25. Chen X X, Almost periodic solutions of nonlinear delay population equation with feedback control. Nonl Anal: Real World Appl, 2007, 8(1): 62–72

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen X X, Chen F D. Almost-periodic solutions of a delay population equation with feedback control, Nonl Anal: Real World Appl, 2006, 7: 559–571

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang R, Wang L, Almost periodic solutions for cellular neural networks with distributed delays. Acta Math Sci, 2011, 31B(2): 422–429

    MathSciNet  MATH  Google Scholar 

  28. Menouer M A, Moussaoui A, Ait Dads E H, Existence and global asymptotic stability of positive almost periodic solution for a predator-prey system in an artificial lake. Chaos, Solit Frac, 2017, 103: 271–278

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang T, Gan X, Almost periodic solutions for a discrete fishing model with feedback control and time delays. Commun Nonlinear Sci Numer Simulat, 2014, 19(1): 150–163

    Article  MathSciNet  MATH  Google Scholar 

  30. Huang P, Li X, Liu B, Almost periodic solutions for an asymmetric oscillation. J Differ Equations, 2017, 263(12): 8916–8946

    Article  MathSciNet  MATH  Google Scholar 

  31. Shi L, Feng X L, Qi L X, et al, Modeling and predicting the influence of PM2.5 on children’s respiratory diseases. Int J Bifurc Chaos, 2020, 30(15): 2050235

    Article  MATH  Google Scholar 

  32. Phosri A, Ueda K, Phung V, et al, Effects of ambient air pollution on daily hospital admissions for respiratory and cardiovascular diseases in Bangkok, Thailand. Sci Total Environ, 2019, 651: 1144–1153

    Article  Google Scholar 

  33. Nhung N, Schindler C, Dien T M, et al, Acute effects of ambient air pollution on lower respiratory infections in Hanoi children: An eight-year time series study. Environ Int, 2018, 110: 139–148

    Article  Google Scholar 

  34. He S, Tang S, Xiao Y, et al, Stochastic modelling of air pollution impacts on respiratory infection risk. Bull Math Biol, 2018, 80: 3127–3153

    Article  MathSciNet  MATH  Google Scholar 

  35. Cairncross E K, John J, Zunckel M, A novel air pollution index based on the relative risk of daily mortality associated with short-term exposure to common air pollutants. Atmos Environ, 2007, 41(38): 8442–8454

    Article  Google Scholar 

  36. Takeuchi Y, Beretta E, Ma W B, Global asymptotic properties of a SIR epidemic model with nite incubation time. Nonl Anal: TMA, 2000, 42(6): 931–947

    Article  MATH  Google Scholar 

  37. Ma W B, Song M, Takeuchi Y, Global stability of an SIR epidemic model with time delays. Appl Math Lett, 2004, 17(10): 1141–1145

    Article  MathSciNet  MATH  Google Scholar 

  38. Mccluskey C C, Complete global stability for an SIR epidemic model with delay-Distributed or discrete. Nonl Anal: Real World Appl, 2010, 11(1): 55–59

    Article  MathSciNet  MATH  Google Scholar 

  39. Thieme H R, Uniform persistence and permanence for non-autonomous semiflows in population biology. Math Biosci, 2000, 166(2): 173–201

    Article  MathSciNet  MATH  Google Scholar 

  40. Abdurhaman X, Teng Z D, On the persistence and extinction for a non-autonomous SIRS epidemic model. Int J Biomath, 2006, 21(2): 167–176

    MathSciNet  MATH  Google Scholar 

  41. Tian B, Qiu Y, Chen N, Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay. Appl Math Comput, 2009, 215(2): 779–790

    MathSciNet  MATH  Google Scholar 

  42. Alzabut J O, Stamov G T, Sermutlu E, Positive almost periodic solutions for a delay logarithmic population model. Math Comput Model, 2011, 53(1/2): 161–167

    Article  MathSciNet  MATH  Google Scholar 

  43. Xu C, Li P, Guo Y, Global asymptotical stability of almost periodic solutions for a non-autonomous competing model with time-varying delays and feedback controls. J Biol Dynam, 2019, 13(1): 407–421

    Article  MathSciNet  MATH  Google Scholar 

  44. Song X, Chen L, Optimal harvesting and stability for a two-species competitive system with stage structure. Math Biosci, 2001, 170(2): 173–186

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen F, Li Z, Chen X, et al, Dynamic behaviors of a delay differential equation model of plankton allelopathy. J Comput Appl Math, 2007, 206(2): 733–754

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang T L, Teng Z D. On a nonautonomous SEIRS model in epidemiology, Bull. Math Biol, 2007, 69: 2537–2559

    Article  MathSciNet  MATH  Google Scholar 

  47. Liu Z, Chen L, On positive periodic solutions of a nonautonomous neutral delay-species competitive system. Nonl Anal-Theor, 2008, 68(6): 1409–1420

    Article  MathSciNet  MATH  Google Scholar 

  48. Lou X, Cui B, Existence and global attractivity of almost periodic solutions for neural field with time delay. Appl Math Comput, 2008, 200(1): 465–472

    MathSciNet  MATH  Google Scholar 

  49. Zhang T, Almost periodic oscillations in a generalized Mackey-Glass model of respiratory dynamics with several delays. Int J Biomath, 2014, 7(3): 1450029

    Article  MathSciNet  MATH  Google Scholar 

  50. Kuang Y, MA Z, Delay differential equations with applications in population dynamics. Math Comput Simul, 1993, 35(5): 452–453

    Article  Google Scholar 

  51. Chen F, Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model. Appl Math Comput, 2005, 171(2): 760–770

    MathSciNet  MATH  Google Scholar 

  52. Zhang T, Xiong L, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative. Appl Math Lett, 2020, 101: 106072

    Article  MathSciNet  MATH  Google Scholar 

  53. Duan X, Wei G, Yang H, Positive solutions and infinitely many solutions for a weakly coupled system. Acta Math Sci, 2020, 40B(5): 1585–1601

    Article  MathSciNet  Google Scholar 

  54. Geng J, Xia Y, Almost periodic solutions of a nonlinear ecological model. Commun Nonlinear Sci Numer Simulat, 2011, 16: 2575–2597

    Article  MathSciNet  MATH  Google Scholar 

  55. Abbas S, Sen M, Banerjee M, Almost periodic solution of a non-autonomous model of phytoplankton allelopathy. Nonlinear Dynam, 2012, 67(1): 203–214

    Article  MathSciNet  MATH  Google Scholar 

  56. Yuan H, Time-periodic isentropic supersonic euler flows in one-dimensional ducts driving by periodic boundary conditions. Acta Math Sci, 2019, 39B(2): 403–412

    Article  MathSciNet  Google Scholar 

  57. Wang C, Li L, Zhang Q, et al, Dynamical behaviour of a Lotka-Volterra competitive-competitive- cooperative model with feedback controls and time delays. J Biol Dynam, 2019, 13(1): 43–68

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang T, Yang L, Xu L, Stage-structured control on a class of predator-prey system in almost periodic environment. Int J Control, 2020, 93(6): 1442–1460

    Article  MathSciNet  MATH  Google Scholar 

  59. Bohner M, Stamov G Tr, Stamova I M, Almost periodic solutions of Cohen-Grossberg neural networks with time-varying delay and variable impulsive perturbations. Commun Nonlinear Sci Numer Simulat, 2020, 80: 1–14

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, China

    Lei Shi, Longxing Qi & Sulan Zhai

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  1. Lei Shi
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  2. Longxing Qi
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  3. Sulan Zhai
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Correspondence to Longxing Qi.

Additional information

This research was supported by the National Natural Science Foundation of China (11401002, 11771001), the Natural Science Foundation of Anhui Province (2008085MA02), the Natural Science Fund for Colleges and Universities in Anhui Province (KJ2018A0029), the Teaching Research Project of Anhui University (ZLTS2016065), the Quality engineering project of colleges and universities in Anhui Province (2020jyxm0103), and the Science Foundation of Anhui Province Universities (KJ2019A005).

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Shi, L., Qi, L. & Zhai, S. Periodic and almost periodic solutions for a non-autonomous respiratory disease model with a lag effect. Acta Math Sci 42, 187–211 (2022). https://doi.org/10.1007/s10473-022-0110-3

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  • DOI: https://doi.org/10.1007/s10473-022-0110-3

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