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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

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

[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. Google Scholar

[42]

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

[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. Google Scholar

[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. Google Scholar

[45]

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

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[55]

B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dynam. Differential Equations, 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7. Google Scholar

[56]

L. WangZ. Liu and X. Zhang, Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence, Appl. Math. Comput., 284 (2016), 47-65. doi: 10.1016/j.amc.2016.02.058. Google Scholar

[57]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar

[58]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. Google Scholar

[59]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction–diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942. Google Scholar

[60]

W. Wang and X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170. doi: 10.1137/140981769. Google Scholar

[61]

J. Wu, Spatial structure: Partial differential equations models, Mathematical Epidemiology, Springer, Berlin, 1945 (2008), 191-203. doi: 10.1007/978-3-540-78911-6_8. Google Scholar

[62]

D. Xu and X.-Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl., 311 (2005), 417-438. doi: 10.1016/j.jmaa.2005.02.062. Google Scholar

[63]

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. Google Scholar

[64]

L. Zhang and J.-W. Sun, Global stability of a nonlocal epidemic model with delay, Taiwanese J. Math., 20 (2016), 577-587. doi: 10.11650/tjm.20.2016.6291. Google Scholar

[65]

L. Zhang and Z. -C. Wang, A time-periodic reaction-diffusion epidemic model with infection period, Z. Angew. Math. Phys. , 67 (2016), Art. 117, 14 pp. doi: 10.1007/s00033-016-0711-6. Google Scholar

[66]

L. ZhangZ.-C. Wang and Y. Zhang, Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission, Comput. Math. Appl., 72 (2016), 202-215. doi: 10.1016/j.camwa.2016.04.046. Google Scholar

[67]

L. ZhangZ.-C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period, J. Differential Equations, 258 (2015), 3011-3036. doi: 10.1016/j.jde.2014.12.032. Google Scholar

[68]

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. Google Scholar

[69]

X. -Q. Zhao, Dynamical System in Population Biology, Spring-Verlag, New York. 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

[70]

X.-Q. Zhao, Global dynamics of a reaction and diffusion model for Lyme disease, J. Math. Biol., 65 (2012), 787-808. doi: 10.1007/s00285-011-0482-9. Google Scholar

[71]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyman. Differential Equations, 29 (2017), 67-82. doi: 10.1007/s10884-015-9425-2. Google Scholar

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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

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

[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. Google Scholar

[42]

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

[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. Google Scholar

[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. Google Scholar

[45]

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

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[55]

B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dynam. Differential Equations, 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7. Google Scholar

[56]

L. WangZ. Liu and X. Zhang, Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence, Appl. Math. Comput., 284 (2016), 47-65. doi: 10.1016/j.amc.2016.02.058. Google Scholar

[57]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. Google Scholar

[58]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. Google Scholar

[59]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction–diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942. Google Scholar

[60]

W. Wang and X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170. doi: 10.1137/140981769. Google Scholar

[61]

J. Wu, Spatial structure: Partial differential equations models, Mathematical Epidemiology, Springer, Berlin, 1945 (2008), 191-203. doi: 10.1007/978-3-540-78911-6_8. Google Scholar

[62]

D. Xu and X.-Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl., 311 (2005), 417-438. doi: 10.1016/j.jmaa.2005.02.062. Google Scholar

[63]

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. Google Scholar

[64]

L. Zhang and J.-W. Sun, Global stability of a nonlocal epidemic model with delay, Taiwanese J. Math., 20 (2016), 577-587. doi: 10.11650/tjm.20.2016.6291. Google Scholar

[65]

L. Zhang and Z. -C. Wang, A time-periodic reaction-diffusion epidemic model with infection period, Z. Angew. Math. Phys. , 67 (2016), Art. 117, 14 pp. doi: 10.1007/s00033-016-0711-6. Google Scholar

[66]

L. ZhangZ.-C. Wang and Y. Zhang, Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission, Comput. Math. Appl., 72 (2016), 202-215. doi: 10.1016/j.camwa.2016.04.046. Google Scholar

[67]

L. ZhangZ.-C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period, J. Differential Equations, 258 (2015), 3011-3036. doi: 10.1016/j.jde.2014.12.032. Google Scholar

[68]

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. Google Scholar

[69]

X. -Q. Zhao, Dynamical System in Population Biology, Spring-Verlag, New York. 2003. doi: 10.1007/978-0-387-21761-1. Google Scholar

[70]

X.-Q. Zhao, Global dynamics of a reaction and diffusion model for Lyme disease, J. Math. Biol., 65 (2012), 787-808. doi: 10.1007/s00285-011-0482-9. Google Scholar

[71]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyman. Differential Equations, 29 (2017), 67-82. doi: 10.1007/s10884-015-9425-2. Google Scholar

[1]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169

[2]

Zhenguo Bai. Threshold dynamics of a periodic SIR model with delay in an infected compartment. Mathematical Biosciences & Engineering, 2015, 12 (3) : 555-564. doi: 10.3934/mbe.2015.12.555

[3]

Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064

[4]

Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata. Dynamics of an ultra-discrete SIR epidemic model with time delay. Mathematical Biosciences & Engineering, 2018, 15 (3) : 653-666. doi: 10.3934/mbe.2018029

[5]

Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1375-1393. doi: 10.3934/mbe.2014.11.1375

[6]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191

[7]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039

[8]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483

[9]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083

[10]

Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091

[11]

Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure & Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005

[12]

Xinli Hu. Threshold dynamics for a Tuberculosis model with seasonality. Mathematical Biosciences & Engineering, 2012, 9 (1) : 111-122. doi: 10.3934/mbe.2012.9.111

[13]

Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074

[14]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

[15]

Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19

[16]

Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2479-2500. doi: 10.3934/dcdsb.2017127

[17]

Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128

[18]

Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801

[19]

Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277

[20]

Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model with seasonal fluctuation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 1-14. doi: 10.3934/dcdsb.2011.15.1

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (29)
  • HTML views (112)
  • Cited by (0)

Other articles
by authors

[Back to Top]