October 2017, 14(5&6): 1535-1563. doi: 10.3934/mbe.2017080

Threshold dynamics of a time periodic and two–group epidemic model with distributed delay

a. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

b. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Z.-C. Wang

Received  May 14, 2016 Accepted  December 31, 2016 Published  May 2017

In this paper, a time periodic and two–group reaction–diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number $R_0$ for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of $R_0$, that is, the disease is uniformly persistent if $R_0 > 1$, while the disease goes to extinction if $R_0 < 1$. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.

Citation: Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
References:
[1]

S. AltizerA. DobsonP. HosseiniP. HudsonM. Pascual and P. Rohani, Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.

[2]

R. M. Anderson, Discussion: the Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol., 53 (1991), 3-32. doi: 10.1007/BF02464422.

[3]

R. M. Anderson and R. May, Infectious Diseases of Humanns: Dynamics and Control, Oxford University Press, Oxford, 1991.

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N. BacaërD. Ait and H. El, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762. doi: 10.1007/s00285-010-0354-8.

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E. BerettaT. HaraW. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.

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B. BonziA. A. FallA. Iggidr and G. Sallet, Stability of differential susceptibility and infectivity epidemic models, J. Math. Biol., 62 (2011), 39-64. doi: 10.1007/s00285-010-0327-y.

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F. Brauer, Compartmental models in epidemiology, Mathematical Epidemiology, Springer, 56 (2008), 19-79. doi: 10.1007/978-3-540-78911-6_2.

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L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach space, Arch. Math., 56 (1991), 49-57. doi: 10.1007/BF01190081.

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L. CaiM. Martcheva and X.-Z. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, J. Biol. Dyn., 7 (2013), 47-67. doi: 10.1080/17513758.2013.772253.

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D. Dancer and P. Koch Medina, Abstract ecolution equations, Periodic problem and applications, Longman, Harlow, UK, 1992.

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O. DiekmannJ. 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. doi: 10.1007/BF00178324.

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W. E. FitzgibbonC. B. Martin and J. J. Morgan, A diffusive epidemic model with criss–cross dynamics, J. Math. Anal. Appl., 184 (1994), 399-414. doi: 10.1006/jmaa.1994.1209.

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H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z.

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Y. Jin and X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction–diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516. doi: 10.1137/070709761.

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Y. Lou and X.-Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8.

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Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with deasonal developmental durations, J Nonlinear Sci., 27 (2017), 573-603. doi: 10.1007/s00332-016-9344-3.

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P. Magal and C. McCluskey, Two–group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.

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P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

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M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

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R. Martain and H. L. Smith, Abstract functional differential equations and reaction–diffusion system, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

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C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

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C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.05.003.

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R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020.

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Y. TakeuchiW. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947. doi: 10.1016/S0362-546X(99)00138-8.

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H. R. Thieme, Mathematics in population biology, Princeton University Press, Princeton, NJ, 2003.

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H. R. Thieme, Spectral bound and reproduction number for infinite–dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.

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H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801. doi: 10.1016/j.jde.2011.01.007.

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P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

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B.-G. WangW.-T. Li and Z.-C. Wang, A reaction–diffusion SIS epidemic model in an almost periodic environment, Z. Angew. Math. Phys., 66 (2015), 3085-3108. doi: 10.1007/s00033-015-0585-z.

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Z. Xu and X.-Q. Zhao, A vector–bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2615-2634. doi: 10.3934/dcdsb.2012.17.2615.

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Y. Zhang and X.-Q. Zhao, A reaction–diffusion Lyme disease model with seasonality, SIAM J. Appl. Math., 73 (2013), 2077-2099. doi: 10.1137/120875454.

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show all references

References:
[1]

S. AltizerA. DobsonP. HosseiniP. HudsonM. Pascual and P. Rohani, Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.

[2]

R. M. Anderson, Discussion: the Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol., 53 (1991), 3-32. doi: 10.1007/BF02464422.

[3]

R. M. Anderson and R. May, Infectious Diseases of Humanns: Dynamics and Control, Oxford University Press, Oxford, 1991.

[4]

N. BacaërD. Ait and H. El, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762. doi: 10.1007/s00285-010-0354-8.

[5]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector–borne disease with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0.

[6]

E. BerettaT. HaraW. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.

[7]

B. BonziA. A. FallA. Iggidr and G. Sallet, Stability of differential susceptibility and infectivity epidemic models, J. Math. Biol., 62 (2011), 39-64. doi: 10.1007/s00285-010-0327-y.

[8]

F. Brauer, Compartmental models in epidemiology, Mathematical Epidemiology, Springer, 56 (2008), 19-79. doi: 10.1007/978-3-540-78911-6_2.

[9]

L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach space, Arch. Math., 56 (1991), 49-57. doi: 10.1007/BF01190081.

[10]

L. CaiM. Martcheva and X.-Z. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, J. Biol. Dyn., 7 (2013), 47-67. doi: 10.1080/17513758.2013.772253.

[11]

D. Dancer and P. Koch Medina, Abstract ecolution equations, Periodic problem and applications, Longman, Harlow, UK, 1992.

[12]

O. DiekmannJ. 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. doi: 10.1007/BF00178324.

[13]

W. E. FitzgibbonM. LanglaisM. E. Parrott and G. F. Webb, A diffusive system with age dependency modeling FIV, Nonlinear Anal., 25 (1995), 975-989. doi: 10.1016/0362-546X(95)00092-A.

[14]

W. E. FitzgibbonC. B. Martin and J. J. Morgan, A diffusive epidemic model with criss–cross dynamics, J. Math. Anal. Appl., 184 (1994), 399-414. doi: 10.1006/jmaa.1994.1209.

[15]

W. E. FitzgibbonM. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics, Math. Biosci., 128 (1995), 131-155. doi: 10.1016/0025-5564(94)00070-G.

[16]

D. Gao and S. Ruan, Malaria models with spatial effects, John Wiley & Sons. (in press)

[17]

I. GudeljK. A. J. White and N. F. Britton, The effects of spatial movement and group interactions on disease dynamics of social animals, Bull. Math. Biol., 66 (2004), 91-108. doi: 10.1016/S0092-8240(03)00075-2.

[18]

Z. GuoF.-B. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non–local infections, J. Math. Biol., 65 (2012), 1387-1410. doi: 10.1007/s00285-011-0500-y.

[19]

P. Hess, Periodic–Parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.

[20]

H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.

[21]

W. HuangK. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple–group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854. doi: 10.1137/0152047.

[22]

G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691. doi: 10.1016/j.aml.2013.01.010.

[23]

J. M. Hyman and J. Li, Differential susceptibility epidemic models, J. Math. Biol., 50 (2005), 626-644. doi: 10.1007/s00285-004-0301-7.

[24]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z.

[25]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction–diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516. doi: 10.1137/070709761.

[26]

T. Kato, Peturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelerg, 1976.

[27]

J. Li and X. Zou, Generalization of the Kermack–McKendrick SIR model to a patchy environment for a disease with latency, Math. Model. Nat. Phenom., 4 (2009), 92-118. doi: 10.1051/mmnp/20094205.

[28]

J. Li and X. Zou, Dynamics of an epidemic model with non–local infections for diseases with latency over a patchy environment, J. Math. Biol., 60 (2010), 645-686. doi: 10.1007/s00285-009-0280-9.

[29]

M. LiZ. Shuai and C. Wang, Global stability of multi–group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017.

[30]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.

[31]

Y. Lou and X.-Q. Zhao, Threshold dynamics in a time–delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186. doi: 10.3934/dcdsb.2009.12.169.

[32]

Y. Lou and X.-Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8.

[33]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with deasonal developmental durations, J Nonlinear Sci., 27 (2017), 573-603. doi: 10.1007/s00332-016-9344-3.

[34]

P. Magal and C. McCluskey, Two–group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.

[35]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[36]

M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[37]

R. Martain and H. L. Smith, Abstract functional differential equations and reaction–diffusion system, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[38]

C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[39]

C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.05.003.

[40]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[41]

R. Peng and X.-Q. Zhao, A reaction–diffusion SIS epidemic model in a time–periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451.

[42]

B. Perthame, Parabolic Equations in Biology, Springer, Cham, 2015. doi: 10.1007/978-3-319-19500-1.

[43]

L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, Mathematical Surveys and Monographs, 102. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/102.

[44]

R. Ross, An application of the theory of probabilities to the study of a priori pathometry: Ⅰ, Proc. R. Soc. Lond., 92 (1916), 204-230. doi: 10.1098/rspa.1916.0007.

[45]

S. Ruan, Spatial−temporal dynamics in nonlocal epidemiological models, Mathematics for Life Science and Medicine, Springer−Verlag, Berlin, (2007), 99–122.

[46]

S. Ruan and J. Wu, Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts, Chapman & Hall/CRC, Boca Raton, FL, (2009), 293–316.

[47]

H. L. Smith, Monotone Dynamical System: An Introduction to the Theorey of Competitive and Cooperative Systems, Math. Surveys and Monogr. vol 41, American Mathematical Society, Providence, 1995.

[48]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020.

[49]

Y. TakeuchiW. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947. doi: 10.1016/S0362-546X(99)00138-8.

[50]

H. R. Thieme, Mathematics in population biology, Princeton University Press, Princeton, NJ, 2003.

[51]

H. R. Thieme, Spectral bound and reproduction number for infinite–dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.

[52]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801. doi: 10.1016/j.jde.2011.01.007.

[53]

P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[54]

B.-G. WangW.-T. Li and Z.-C. Wang, A reaction–diffusion SIS epidemic model in an almost periodic environment, Z. Angew. Math. Phys., 66 (2015), 3085-3108. doi: 10.1007/s00033-015-0585-z.

[55]

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