# American Institute of Mathematical Sciences

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. Allen, B. M. Bolker, Y. 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. Cai, X. Z. Lian, Z. 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. Cosner, D. L. DeAngelis, J. 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. Gan, R. 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. Guo, F.-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. Hethcote, M. 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. Hu, P. Bi, W. 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. Huang, Y. Takeuchi, W. 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. Li, F. Q. Zhang, H. 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. Li, R. 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. Liu, S. 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. Luo, S. T. Tang, Z. 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. Wang, D. 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. Wang, D. 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. Wang, J. 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. Wang, W. 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. Yang, J. 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. Zhang, Z.-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. Zhang, T. 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. Zhou, Y. 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.

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##### 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. Allen, B. M. Bolker, Y. 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. Cai, X. Z. Lian, Z. 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. Cosner, D. L. DeAngelis, J. 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. Gan, R. 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. Guo, F.-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. Hethcote, M. 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. Hu, P. Bi, W. 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. Huang, Y. Takeuchi, W. 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. Li, F. Q. Zhang, H. 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. Li, R. 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. Liu, S. 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. Luo, S. T. Tang, Z. 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. Wang, D. 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. Wang, D. 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. Wang, J. 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. Wang, W. 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. Yang, J. 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. Zhang, Z.-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. Zhang, T. 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. Zhou, Y. 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.
Variation of populations of system (15) with $D = 1$, $\alpha_2 = 0$ and other parameters in (16)
Variation of populations of system (15) with $D = 1$, $\alpha_2 = 0.1$ and other parameters in (16)
The relationship of $\mathcal R_0$ and $\gamma_1$
The relationship of $\mathcal R_0$ and $\alpha_2$
When $\alpha_2 = 0$ and $D = 10^{-5}$, the evolution of the infective individuals $I(x, t)$ with parameters in (16) and $\mathcal R_0\approx 1.9375$
The relation between $\mathcal R_0$ and $c$ in $\beta(x)$
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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