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doi: 10.3934/dcdsb.2022116
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Spatial dynamics of an epidemic model in time almost periodic and space periodic media

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

* Corresponding author: Bin-Guo Wang

Received  March 2022 Revised  May 2022 Early access June 2022

Fund Project: Research supported by NSF of China 11501269 and Natural Science Foundation of Gansu Province of China (grant number: 21JR7RA535)

This paper is devoted to the study of a reaction-diffusion-advection epidemic model in time almost periodic and space periodic media. First, we obtain a threshold result on the global stability of either zero or the positive time almost periodic solution in terms of the basic reproduction ratio $ \mathcal{R}_0 $. Second, we prove the existence of spreading speeds in the partially spatially homogeneous case and the general case. At last, we use numerical simulations to investigate the influence of model parameters on spreading speeds.

Citation: Ming-Zhen Xin, Bin-Guo Wang. Spatial dynamics of an epidemic model in time almost periodic and space periodic media. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022116
References:
[1]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

S. AltizerA. HosseiniP. HudsonM. Rohani and P. Rohani, Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006), 467-484. 

[3]

Y. Atsushi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. Ⅱ., Funkcial. Ekvac., 33 (1990), 139-150. 

[4]

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

[5]

X. Bao, Spreading speeds for two species competition systems in time almost periodic and space periodic media, Acta Appl. Math., 171 (2021), Paper No. 11, 28 pp. doi: 10.1007/s10440-020-00376-0.

[6]

X. BaoW.-T. LiW. Shen and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), 3048-3091.  doi: 10.1016/j.jde.2018.05.003.

[7]

V. Capasso and R. E. Wilson, Analysis of reaction-diffusion system modeling man environment man epidemics, SIAM J. Appl. Math., 57 (1997), 327-346.  doi: 10.1137/S0036139995284681.

[8]

T. CaraballoJ. A. LangaR. Obaya and A. M. Sanz, Global and cocycle attractors for non-autonomous reaction–diffusion equations. The case of null upper Lyapunov exponent, J. Differential Equations, 265 (2018), 3914-3951.  doi: 10.1016/j.jde.2018.05.023.

[9]

C. Cheng and Z. Zheng, Analysis of a reaction-diffusion system about West Nile virus with free boundaries in the almost periodic heterogeneous environment, Z. Angew. Math. Phys., 73 (2022), Paper No. 84, 27 pp. doi: 10.1007/s00033-022-01729-5.

[10]

C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989.

[11]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[12]

L.-J. Du, W.-T. Li and W. Shen, Propagation phenomena for time-space periodic monotone semiflows and applications to cooperative systems in multi-dimensional media, J. Funct. Anal., 282 (2022), Paper No. 109415, 59 pp. doi: 10.1016/j.jfa.2022.109415.

[13]

J. FangX. Lai and F.-B. Wang, Spatial dynamics of a dengue transmission model in time-space periodic environment, J. Differ. Equation., 269 (2020), 149-175.  doi: 10.1016/j.jde.2020.04.034.

[14]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[15]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.

[16]

M. I. Friedlin, On wavefront propagation in periodic media, stochastic analysis and applications, in: Adv. Probab. Related Topics, vol. 7, Dekker, New York, 1984, 147–166.

[17]

J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Sov. Math., Dokl., 20 (1979), 1282-1286. 

[18]

F. Hamel and L. Roques, Persistence and propagation in periodic reaction-diffusion models, Tamkang J. Math., 45 (2014), 217-228.  doi: 10.5556/j.tkjm.45.2014.1656.

[19]

M. W. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.

[20]

J. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J. Appl. Dyn. Syst., 8 (2009), 790-821.  doi: 10.1137/080723259.

[21]

V. HutsonW. Shen and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679.  doi: 10.1090/S0002-9939-00-05808-1.

[22]

W. Hutter and F. Räbiger, Spectral mapping theorems for evolution semigroups on spaces of almost perioidc functions, Quaest. Math., 26 (2003), 191-211.  doi: 10.2989/16073600309486054.

[23]

X. Liang and X.-Q Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[24]

L. Maniar and R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, Evolution Equations, 299–318, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003.

[25]

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

[26]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynam. Differential Equations, 25 (2013), 1201-1231.  doi: 10.1007/s10884-013-9337-y.

[27]

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.

[28]

L. QiangB.-G. Wang and Z.-C. Wang, A reaction-diffusion epidemic model with incubation period in almost periodic environments, European J. Appl. Math., 32 (2021), 1153-1176.  doi: 10.1017/S0956792520000303.

[29]

L. QiangB.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differential Equations, 269 (2020), 4440-4476.  doi: 10.1016/j.jde.2020.03.027.

[30]

G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971.

[31]

W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.

[32]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), no. 647. doi: 10.1090/memo/0647.

[33]

W. Shen and Y. Yi, Convergence in almost periodic Fisher and Kolmogorov models, J. Math. Biol., 37 (1998), 84-102.  doi: 10.1007/s002850050121.

[34]

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.

[35]

C. C. Travis and G. F. Webb, Existence and stablitiy for partial functional differnetial equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3.

[36]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[37]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, American Mathematical Society, Providence, RI, 1994. doi: 10.1090/mmono/140.

[38]

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.

[39]

J.-B. Wang and W.-T. Li, Pulsating waves and entire solutions for a spatially periodic nonlocal dispersal system with a quiescent stage, Sci. China Math., 62 (2019), 2505-2526.  doi: 10.1007/s11425-019-1588-1.

[40]

J.-B. WangW.-T. Li and J.-W. Sun, Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 849-880.  doi: 10.1017/S0308210518000045.

[41]

D. WattsD. BurkeB. HarrisonR. Whitmire and A. Nisalak, Effect of temperature on the vector effciency of Aedes aegypti for dengue 2 virus, Am. J. Trop. Hyg., 36 (1987), 143-152. 

[42]

H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.

[43]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic hahitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[44]

C. WuD. Xiao and X.-Q. Zhao, Spreading speed of a partially degenerate reaction-diffusion system in a periodic hahitat, J. Differential Equations, 255 (2013), 3983-4011.  doi: 10.1016/j.jde.2013.07.058.

[45]

S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. Real World Appl., 13 (2012), 1991-2005.  doi: 10.1016/j.nonrwa.2011.12.020.

[46]

S.-L. Wu and C.-H. Hsu, Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062.  doi: 10.1090/tran/6526.

[47]

S.-L. WuC.-H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105.  doi: 10.1016/j.jde.2014.10.009.

[48]

A. Yagi, Abstract Evolution Equations and their Applications, Springer, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

[49]

M. ZhaoW.-T. Li and Y. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620.  doi: 10.3934/cpaa.2020208.

[50]

M. ZhaoW.-T. Li and W. Ni, Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 981-999.  doi: 10.3934/dcdsb.2019199.

[51]

M. ZhaoY. ZhangW.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.

[52]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2nd edition, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-56433-3.

[53]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509.  doi: 10.1016/S0022-0396(02)00054-2.

show all references

References:
[1]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

S. AltizerA. HosseiniP. HudsonM. Rohani and P. Rohani, Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006), 467-484. 

[3]

Y. Atsushi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. Ⅱ., Funkcial. Ekvac., 33 (1990), 139-150. 

[4]

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

[5]

X. Bao, Spreading speeds for two species competition systems in time almost periodic and space periodic media, Acta Appl. Math., 171 (2021), Paper No. 11, 28 pp. doi: 10.1007/s10440-020-00376-0.

[6]

X. BaoW.-T. LiW. Shen and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), 3048-3091.  doi: 10.1016/j.jde.2018.05.003.

[7]

V. Capasso and R. E. Wilson, Analysis of reaction-diffusion system modeling man environment man epidemics, SIAM J. Appl. Math., 57 (1997), 327-346.  doi: 10.1137/S0036139995284681.

[8]

T. CaraballoJ. A. LangaR. Obaya and A. M. Sanz, Global and cocycle attractors for non-autonomous reaction–diffusion equations. The case of null upper Lyapunov exponent, J. Differential Equations, 265 (2018), 3914-3951.  doi: 10.1016/j.jde.2018.05.023.

[9]

C. Cheng and Z. Zheng, Analysis of a reaction-diffusion system about West Nile virus with free boundaries in the almost periodic heterogeneous environment, Z. Angew. Math. Phys., 73 (2022), Paper No. 84, 27 pp. doi: 10.1007/s00033-022-01729-5.

[10]

C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989.

[11]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[12]

L.-J. Du, W.-T. Li and W. Shen, Propagation phenomena for time-space periodic monotone semiflows and applications to cooperative systems in multi-dimensional media, J. Funct. Anal., 282 (2022), Paper No. 109415, 59 pp. doi: 10.1016/j.jfa.2022.109415.

[13]

J. FangX. Lai and F.-B. Wang, Spatial dynamics of a dengue transmission model in time-space periodic environment, J. Differ. Equation., 269 (2020), 149-175.  doi: 10.1016/j.jde.2020.04.034.

[14]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[15]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.

[16]

M. I. Friedlin, On wavefront propagation in periodic media, stochastic analysis and applications, in: Adv. Probab. Related Topics, vol. 7, Dekker, New York, 1984, 147–166.

[17]

J. Gärtner and M. I. Freidlin, The propagation of concentration waves in periodic and random media, Sov. Math., Dokl., 20 (1979), 1282-1286. 

[18]

F. Hamel and L. Roques, Persistence and propagation in periodic reaction-diffusion models, Tamkang J. Math., 45 (2014), 217-228.  doi: 10.5556/j.tkjm.45.2014.1656.

[19]

M. W. HirschH. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), 107-131.  doi: 10.1023/A:1009044515567.

[20]

J. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J. Appl. Dyn. Syst., 8 (2009), 790-821.  doi: 10.1137/080723259.

[21]

V. HutsonW. Shen and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679.  doi: 10.1090/S0002-9939-00-05808-1.

[22]

W. Hutter and F. Räbiger, Spectral mapping theorems for evolution semigroups on spaces of almost perioidc functions, Quaest. Math., 26 (2003), 191-211.  doi: 10.2989/16073600309486054.

[23]

X. Liang and X.-Q Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[24]

L. Maniar and R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, Evolution Equations, 299–318, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003.

[25]

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

[26]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynam. Differential Equations, 25 (2013), 1201-1231.  doi: 10.1007/s10884-013-9337-y.

[27]

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.

[28]

L. QiangB.-G. Wang and Z.-C. Wang, A reaction-diffusion epidemic model with incubation period in almost periodic environments, European J. Appl. Math., 32 (2021), 1153-1176.  doi: 10.1017/S0956792520000303.

[29]

L. QiangB.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differential Equations, 269 (2020), 4440-4476.  doi: 10.1016/j.jde.2020.03.027.

[30]

G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971.

[31]

W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.

[32]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), no. 647. doi: 10.1090/memo/0647.

[33]

W. Shen and Y. Yi, Convergence in almost periodic Fisher and Kolmogorov models, J. Math. Biol., 37 (1998), 84-102.  doi: 10.1007/s002850050121.

[34]

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.

[35]

C. C. Travis and G. F. Webb, Existence and stablitiy for partial functional differnetial equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3.

[36]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[37]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, American Mathematical Society, Providence, RI, 1994. doi: 10.1090/mmono/140.

[38]

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.

[39]

J.-B. Wang and W.-T. Li, Pulsating waves and entire solutions for a spatially periodic nonlocal dispersal system with a quiescent stage, Sci. China Math., 62 (2019), 2505-2526.  doi: 10.1007/s11425-019-1588-1.

[40]

J.-B. WangW.-T. Li and J.-W. Sun, Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 849-880.  doi: 10.1017/S0308210518000045.

[41]

D. WattsD. BurkeB. HarrisonR. Whitmire and A. Nisalak, Effect of temperature on the vector effciency of Aedes aegypti for dengue 2 virus, Am. J. Trop. Hyg., 36 (1987), 143-152. 

[42]

H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.

[43]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic hahitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[44]

C. WuD. Xiao and X.-Q. Zhao, Spreading speed of a partially degenerate reaction-diffusion system in a periodic hahitat, J. Differential Equations, 255 (2013), 3983-4011.  doi: 10.1016/j.jde.2013.07.058.

[45]

S.-L. Wu, Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics, Nonlinear Anal. Real World Appl., 13 (2012), 1991-2005.  doi: 10.1016/j.nonrwa.2011.12.020.

[46]

S.-L. Wu and C.-H. Hsu, Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062.  doi: 10.1090/tran/6526.

[47]

S.-L. WuC.-H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105.  doi: 10.1016/j.jde.2014.10.009.

[48]

A. Yagi, Abstract Evolution Equations and their Applications, Springer, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

[49]

M. ZhaoW.-T. Li and Y. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pure Appl. Anal., 19 (2020), 4599-4620.  doi: 10.3934/cpaa.2020208.

[50]

M. ZhaoW.-T. Li and W. Ni, Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 981-999.  doi: 10.3934/dcdsb.2019199.

[51]

M. ZhaoY. ZhangW.-T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differential Equations, 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.

[52]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2nd edition, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-56433-3.

[53]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509.  doi: 10.1016/S0022-0396(02)00054-2.

Figure 1.  The evolution of $ u_1 $ and $ u_2 $
Figure 2.  The relationship between $ c_-^* $, $ c_+^* $ and $ k_{d_1}, k_{d_2}, k_q, k_p $
Figure 3.  The evolution of $ u_1 $ and $ u_2 $
Figure 4.  The relationship between $ c_-^* $, $ c_+^* $ and $ k_{d_1}, k_{d_2}, k_q, k_p $
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