July  2020, 25(7): 2433-2451. doi: 10.3934/dcdsb.2020017

Dynamical analysis of a diffusive SIRS model with general incidence rate

1. 

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

2. 

Department of Mathematics, Sichuan University, Chengdu 610064, China

3. 

Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

4. 

Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

* Corresponding author: Lan Zou, Email: lanzou@163.com

Received  May 2019 Revised  August 2019 Published  July 2020 Early access  April 2020

Fund Project: Partially supported by National Natural Science Foundation of China (No. 11671114, 11831012 and 11771168), and Natural Science Foundation of Zhejiang Province (SY20A010005)

In this paper, we propose a diffusive SIRS model with general incidence rate and spatial heterogeneity. The formula of the basic reproduction number $ \mathcal R_0 $ is given. Then the threshold dynamics, including globally attractive of the disease-free equilibrium and uniform persistence, are established in terms of $ \mathcal{R}_0 $. Special cases and numerical simulations are presented to support our main results.

Citation: Yu Yang, Lan Zou, Tonghua Zhang, Yancong Xu. Dynamical analysis of a diffusive SIRS model with general incidence rate. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2433-2451. doi: 10.3934/dcdsb.2020017
References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.  doi: 10.1016/j.mbs.2004.01.003.

[2]

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.

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[4]

Y. L. CaiX. Z. LianZ. H. Peng and W. M. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. RWA, 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.

[5]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[6]

C. CosnerD. L. DeAngelisJ. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Pop. Biol., 56 (1999), 65-75. 

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P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221. 

[8]

Q. T. GanR. Xu and P. H. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion, Nonlinear Anal. RWA, 12 (2011), 52-68.  doi: 10.1016/j.nonrwa.2010.05.035.

[9]

Z. M. GuoF.-B. Wang and X. F. 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.

[10]

H. W. HethcoteM. A. Lewis and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.  doi: 10.1007/BF00276080.

[11]

H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2.

[12]

Z. X. HuP. BiW. B. Ma and S. G. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.

[13]

G. HuangY. TakeuchiW. B. Ma and D. J. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6.

[14]

A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. 

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X. L. Lai and X. F. Zou, Repulsion effect on superinfecting virions by infected cell, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.

[17]

T. LiF. Q. ZhangH. W. Liu and Y. M. Chen, Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible, Appl. Math. Lett., 70 (2017), 52-57.  doi: 10.1016/j.aml.2017.03.005.

[18]

H. C. LiR. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.

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K. H. Li, J. M. Li and W. Wang, Epidemic reaction-diffusion systems with two types of boundary conditions, Electron. J. Differ. Equ., 2018 (2018), Paper No. 170, 21 pp.

[20]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.

[21]

Y. J. 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.

[22]

Y. T. LuoS. T. TangZ. D. Teng and L. Zhang, Global dynamics in a reaction-diffusion multi-group SIR epidemic model with nonlinear incidence, Nonlinear Anal. RWA, 50 (2019), 365-385.  doi: 10.1016/j.nonrwa.2019.05.008.

[23]

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.

[24]

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

[25]

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

[26]

J. Mena-Lorca and H. W. Hetheote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.  doi: 10.1007/BF00173264.

[27]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2000.

[28]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[29]

S. G. Ruan, Modeling the transmission dynamics and control of rabies in China, Math. Biosci., 286 (2017), 65-93.  doi: 10.1016/j.mbs.2017.02.005.

[30]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.

[31]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. TMA, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[32]

X. Y. WangD. Z. Gao and J. Wang, Influence of human behavior on cholera dynamics, Math. Biosci., 267 (2015), 41-52.  doi: 10.1016/j.mbs.2015.06.009.

[33]

X. Y. WangD. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2785-2809.  doi: 10.3934/dcdsb.2016073.

[34]

W. D. 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.

[35]

J. L. WangJ. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564.  doi: 10.1016/j.jmaa.2016.07.027.

[36]

W. WangW. B. Ma and X. L. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. RWA, 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013.

[37]

W. D. 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.

[38]

J. H. Wu, Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Science, 119, Springer, Berlin, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

Y. X. Wu and X. F. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differential Equations, 264 (2018), 4989-5024.  doi: 10.1016/j.jde.2017.12.027.

[40]

D. M. Xiao and S. G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.

[41]

Z. T. 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.

[42]

K. Yamazaki and X. Y. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1297-1316.  doi: 10.3934/dcdsb.2016.21.1297.

[43]

K. Yamazaki and X. Y. Wang, Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model, Math. Biosci. Eng., 14 (2017), 559-579.  doi: 10.3934/mbe.2017033.

[44]

K. Yamazaki, Global well-posedness of infectious disease models without life-time immunity: the cases of cholera and avian influenza, Math. Med. Biol., 35 (2018), 427-445.  doi: 10.1093/imammb/dqx016.

[45]

Y. Yang and D. M. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 195-211.  doi: 10.3934/dcdsb.2010.13.195.

[46]

Y. YangJ. L. Zhou and C.-H. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl., 478 (2019), 874-896.  doi: 10.1016/j.jmaa.2019.05.059.

[47]

X. J. Yu, C. F. Wu and P. X. Weng, Traveling waves for a SIRS model with nonlocal diffusion, Int. J. Biomath., 5 (2012), 1250036, 26 pp. doi: 10.1142/S1793524511001787.

[48]

T. R. Zhang and W. D. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495.  doi: 10.1016/j.jmaa.2014.04.068.

[49]

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.

[50]

T. H. ZhangT. Q. Zhang and X. Z. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1-7.  doi: 10.1016/j.aml.2016.12.007.

[51]

T.H. Zhang and H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Phys. Rev. E, 90 (2014), 052908.

[52]

J. L. ZhouY. Yang and T. H. Zhang, Global dynamics of a reaction-diffusion waterborne pathogen model with general incidence rate, J. Math. Anal. Appl., 466 (2018), 835-859.  doi: 10.1016/j.jmaa.2018.06.029.

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.  doi: 10.1016/j.mbs.2004.01.003.

[2]

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.

[3]

R. M. Anderson and R. M. May, Population biology of infectious diseases. Part Ⅰ, Nature, 280 (1979), 361-367. 

[4]

Y. L. CaiX. Z. LianZ. H. Peng and W. M. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. RWA, 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.

[5]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[6]

C. CosnerD. L. DeAngelisJ. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Pop. Biol., 56 (1999), 65-75. 

[7]

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221. 

[8]

Q. T. GanR. Xu and P. H. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion, Nonlinear Anal. RWA, 12 (2011), 52-68.  doi: 10.1016/j.nonrwa.2010.05.035.

[9]

Z. M. GuoF.-B. Wang and X. F. 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.

[10]

H. W. HethcoteM. A. Lewis and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.  doi: 10.1007/BF00276080.

[11]

H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2.

[12]

Z. X. HuP. BiW. B. Ma and S. G. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.

[13]

G. HuangY. TakeuchiW. B. Ma and D. J. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6.

[14]

A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. 

[15]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[16]

X. L. Lai and X. F. Zou, Repulsion effect on superinfecting virions by infected cell, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.

[17]

T. LiF. Q. ZhangH. W. Liu and Y. M. Chen, Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible, Appl. Math. Lett., 70 (2017), 52-57.  doi: 10.1016/j.aml.2017.03.005.

[18]

H. C. LiR. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.

[19]

K. H. Li, J. M. Li and W. Wang, Epidemic reaction-diffusion systems with two types of boundary conditions, Electron. J. Differ. Equ., 2018 (2018), Paper No. 170, 21 pp.

[20]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.

[21]

Y. J. 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.

[22]

Y. T. LuoS. T. TangZ. D. Teng and L. Zhang, Global dynamics in a reaction-diffusion multi-group SIR epidemic model with nonlinear incidence, Nonlinear Anal. RWA, 50 (2019), 365-385.  doi: 10.1016/j.nonrwa.2019.05.008.

[23]

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.

[24]

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

[25]

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

[26]

J. Mena-Lorca and H. W. Hetheote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.  doi: 10.1007/BF00173264.

[27]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2000.

[28]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[29]

S. G. Ruan, Modeling the transmission dynamics and control of rabies in China, Math. Biosci., 286 (2017), 65-93.  doi: 10.1016/j.mbs.2017.02.005.

[30]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.

[31]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. TMA, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[32]

X. Y. WangD. Z. Gao and J. Wang, Influence of human behavior on cholera dynamics, Math. Biosci., 267 (2015), 41-52.  doi: 10.1016/j.mbs.2015.06.009.

[33]

X. Y. WangD. Posny and J. Wang, A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2785-2809.  doi: 10.3934/dcdsb.2016073.

[34]

W. D. 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.

[35]

J. L. WangJ. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564.  doi: 10.1016/j.jmaa.2016.07.027.

[36]

W. WangW. B. Ma and X. L. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. RWA, 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013.

[37]

W. D. 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.

[38]

J. H. Wu, Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Science, 119, Springer, Berlin, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

Y. X. Wu and X. F. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differential Equations, 264 (2018), 4989-5024.  doi: 10.1016/j.jde.2017.12.027.

[40]

D. M. Xiao and S. G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.

[41]

Z. T. 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.

[42]

K. Yamazaki and X. Y. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1297-1316.  doi: 10.3934/dcdsb.2016.21.1297.

[43]

K. Yamazaki and X. Y. Wang, Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model, Math. Biosci. Eng., 14 (2017), 559-579.  doi: 10.3934/mbe.2017033.

[44]

K. Yamazaki, Global well-posedness of infectious disease models without life-time immunity: the cases of cholera and avian influenza, Math. Med. Biol., 35 (2018), 427-445.  doi: 10.1093/imammb/dqx016.

[45]

Y. Yang and D. M. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 195-211.  doi: 10.3934/dcdsb.2010.13.195.

[46]

Y. YangJ. L. Zhou and C.-H. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl., 478 (2019), 874-896.  doi: 10.1016/j.jmaa.2019.05.059.

[47]

X. J. Yu, C. F. Wu and P. X. Weng, Traveling waves for a SIRS model with nonlocal diffusion, Int. J. Biomath., 5 (2012), 1250036, 26 pp. doi: 10.1142/S1793524511001787.

[48]

T. R. Zhang and W. D. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495.  doi: 10.1016/j.jmaa.2014.04.068.

[49]

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.

[50]

T. H. ZhangT. Q. Zhang and X. Z. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1-7.  doi: 10.1016/j.aml.2016.12.007.

[51]

T.H. Zhang and H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Phys. Rev. E, 90 (2014), 052908.

[52]

J. L. ZhouY. Yang and T. H. Zhang, Global dynamics of a reaction-diffusion waterborne pathogen model with general incidence rate, J. Math. Anal. Appl., 466 (2018), 835-859.  doi: 10.1016/j.jmaa.2018.06.029.

Figure 1.  Variation of populations of system (15) with $ D = 1 $, $ \alpha_2 = 0 $ and other parameters in (16)
Figure 2.  Variation of populations of system (15) with $ D = 1 $, $ \alpha_2 = 0.1 $ and other parameters in (16)
Figure 3.  The relationship of $ \mathcal R_0 $ and $ \gamma_1 $
Figure 4.  The relationship of $ \mathcal R_0 $ and $ \alpha_2 $
Figure 5.  When $ \alpha_2 = 0 $ and $ D = 10^{-5} $, the evolution of the infective individuals $ I(x, t) $ with parameters in (16) and $ \mathcal R_0\approx 1.9375 $
Figure 6.  The relation between $ \mathcal R_0 $ and $ c $ in $ \beta(x) $
Figure 7.  When $ \alpha_2 = 0 $ and $ D = 10^{6} $, the evolution of the infective individuals $ I(x, t) $ with parameters in (16) and $ \mathcal R_0\approx 1.0373 $
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