American Institute of Mathematical Sciences

Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment

 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, Jiangsu Province, China

* Corresponding author: Chengxia Lei

Received  November 2018 Revised  March 2019 Published  July 2019

Fund Project: This work was partially supported by the National Natural Science Foundation of China (No. 11801232), the Priority Academic Program Development of Jiangsu Higher Education Institution, the Natural Science Foundation of the Jiangsu Province(No. BK20180999), the Foundation of Jiangsu Normal University (17XLR008)

In the recent paper [29], a susceptible-infected-susceptible (SIS) epidemic reaction-diffusion model with a mass action infection mechanism and linear birth-death growth with no flux boundary condition was studied. It has been recognized that spontaneous infection is an important factor in disease epidemics, in addition to disease transmission [43]. In this paper, we investigate the SIS model in [29] with spontaneous infection. We establish the global boundedness and uniform persistence in the general heterogeneous environment, and derive the global stability of the unique constant endemic equilibrium in the homogeneous environment case. Moreover, we analyze the asymptotic behavior of the endemic equilibrium when the movement (migration) rate of the susceptible or infected population tends to zero. Compared to the case that there is no spontaneous infection, our study suggests that spontaneous infection can enhance persistence of infectious disease, and hence the disease becomes more threatening.

Citation: Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019173
References:
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Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565. Google Scholar [7] K. Brown, P. Dunne and R. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differential Equations, 40 (1981), 232-252. doi: 10.1016/0022-0396(81)90020-6. Google Scholar [8] R. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biology, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296. Google Scholar [9] J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mount. J. Math., 38 (2008), 1323-1334. doi: 10.1216/RMJ-2008-38-5-1323. Google Scholar [10] R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263 (2017), 2343-2373. doi: 10.1016/j.jde.2017.03.045. Google Scholar [11] R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343. doi: 10.1016/j.jde.2016.05.025. Google Scholar [12] K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946. doi: 10.1017/S0308210515000864. Google Scholar [13] W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304. doi: 10.3934/dcdsb.2013.18.1291. Google Scholar [14] Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007. Google Scholar [15] Z. Du and R. Peng, A priori $L^\infty$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439. doi: 10.1007/s00285-015-0914-z. Google Scholar [16] D. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011), 110-115. doi: 10.1016/j.mbs.2011.05.001. Google Scholar [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001. Google Scholar [18] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. Google Scholar [19] H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar [20] H. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 16 (2009), 63–89. doi: 10.1142/9789812834836_0002. Google Scholar [21] A. Hill, D. Rand, M. Nowak and N. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proceedings of the Royal Society B, 277 (2010), 3827-3835. doi: 10.1098/rspb.2010.1217. Google Scholar [22] A. Hill, D. Rand, M. Nowak and N. Christakis, Infectious disease modeling of social contagion in networks, Plos Computational Biology, 6 (2010), e1000968, 15pp. doi: 10.1371/journal.pcbi.1000968. Google Scholar [23] W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51. Google Scholar [24] M. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals,, Princeton University Press, 2008. Google Scholar [25] K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of the endemic equilibria of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56 (2017), Art. 112, 28 pp. doi: 10.1007/s00526-017-1207-8. Google Scholar [26] C. Lei, F. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517. doi: 10.3934/dcdsb.2018173. Google Scholar [27] B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25 pp. doi: 10.1007/s00033-017-0845-1. Google Scholar [28] H. Li, R. Peng and F.-B. Wang, Vary 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. Google Scholar [29] H. Li, R. Peng and Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153. doi: 10.1137/18M1167863. Google Scholar [30] H. Li, R. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur. J. Appl. Math.. doi: 10.1017/S0956792518000463. Google Scholar [31] G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406. doi: 10.1137/S003614100343651X. Google Scholar [32] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. Google Scholar [33] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. Google Scholar [34] 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 [35] S. O'Regan and J. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theoretical Econogy, 6 (2013), 333-357. Google Scholar [36] R. Peng, Qualitative analysis on a diffusive and ratio-dependent predator-prey model, IMA J. Appl. Math., 78 (2013), 566-586. doi: 10.1093/imamat/hxr066. Google Scholar [37] R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488. doi: 10.1088/0951-7715/21/7/006. Google Scholar [38] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part Ⅰ, J. Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002. Google Scholar [39] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. Google Scholar [40] 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 [41] R. Peng and F.-Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006. Google Scholar [42] H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. Google Scholar [43] Y. Tong and C. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 41 (2018), 443-460. doi: 10.1016/j.nonrwa.2017.11.002. Google Scholar [44] X. Wen, J. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2018), 715-729. doi: 10.1016/j.jmaa.2017.08.016. Google Scholar [45] Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447. doi: 10.1016/j.jde.2016.06.028. Google Scholar [46] M. Yang, G. Chen and X. Fu, A modeling SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413. Google Scholar [47] X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Can. Appl. Math. Q., 3 (1995), 473-495. Google Scholar

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References:
 [1] N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equation, Commun. Partial Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113. Google Scholar [2] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profiles of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309. doi: 10.1137/060672522. Google Scholar [3] L. Allen, B. Bolker, Y. Lou and A. 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. Google Scholar [4] F. Altarelli, A. Braunstein, L. Dall'Asta, J. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Physical Review X, 4 (2014), 021024. doi: 10.1103/PhysRevX.4.021024. Google Scholar [5] R. Anderson and R. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367. doi: 10.1007/978-3-642-68635-1. Google Scholar [6] H. Brezis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565. Google Scholar [7] K. Brown, P. Dunne and R. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differential Equations, 40 (1981), 232-252. doi: 10.1016/0022-0396(81)90020-6. Google Scholar [8] R. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biology, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296. Google Scholar [9] J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mount. J. Math., 38 (2008), 1323-1334. doi: 10.1216/RMJ-2008-38-5-1323. Google Scholar [10] R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263 (2017), 2343-2373. doi: 10.1016/j.jde.2017.03.045. Google Scholar [11] R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343. doi: 10.1016/j.jde.2016.05.025. Google Scholar [12] K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946. doi: 10.1017/S0308210515000864. Google Scholar [13] W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304. doi: 10.3934/dcdsb.2013.18.1291. Google Scholar [14] Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956. doi: 10.1016/j.jde.2008.11.007. Google Scholar [15] Z. Du and R. Peng, A priori $L^\infty$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439. doi: 10.1007/s00285-015-0914-z. Google Scholar [16] D. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011), 110-115. doi: 10.1016/j.mbs.2011.05.001. Google Scholar [17] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001. Google Scholar [18] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. Google Scholar [19] H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar [20] H. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 16 (2009), 63–89. doi: 10.1142/9789812834836_0002. Google Scholar [21] A. Hill, D. Rand, M. Nowak and N. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proceedings of the Royal Society B, 277 (2010), 3827-3835. doi: 10.1098/rspb.2010.1217. Google Scholar [22] A. Hill, D. Rand, M. Nowak and N. Christakis, Infectious disease modeling of social contagion in networks, Plos Computational Biology, 6 (2010), e1000968, 15pp. doi: 10.1371/journal.pcbi.1000968. Google Scholar [23] W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51. Google Scholar [24] M. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals,, Princeton University Press, 2008. Google Scholar [25] K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of the endemic equilibria of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56 (2017), Art. 112, 28 pp. doi: 10.1007/s00526-017-1207-8. Google Scholar [26] C. Lei, F. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517. doi: 10.3934/dcdsb.2018173. Google Scholar [27] B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25 pp. doi: 10.1007/s00033-017-0845-1. Google Scholar [28] H. Li, R. Peng and F.-B. Wang, Vary 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. Google Scholar [29] H. Li, R. Peng and Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153. doi: 10.1137/18M1167863. Google Scholar [30] H. Li, R. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur. J. Appl. Math.. doi: 10.1017/S0956792518000463. Google Scholar [31] G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406. doi: 10.1137/S003614100343651X. Google Scholar [32] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. Google Scholar [33] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. Google Scholar [34] 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 [35] S. O'Regan and J. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theoretical Econogy, 6 (2013), 333-357. Google Scholar [36] R. Peng, Qualitative analysis on a diffusive and ratio-dependent predator-prey model, IMA J. Appl. Math., 78 (2013), 566-586. doi: 10.1093/imamat/hxr066. Google Scholar [37] R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488. doi: 10.1088/0951-7715/21/7/006. Google Scholar [38] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part Ⅰ, J. Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002. Google Scholar [39] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. Google Scholar [40] 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 [41] R. Peng and F.-Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006. Google Scholar [42] H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. Google Scholar [43] Y. Tong and C. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal. Real World Appl., 41 (2018), 443-460. doi: 10.1016/j.nonrwa.2017.11.002. Google Scholar [44] X. Wen, J. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2018), 715-729. doi: 10.1016/j.jmaa.2017.08.016. Google Scholar [45] Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447. doi: 10.1016/j.jde.2016.06.028. Google Scholar [46] M. Yang, G. Chen and X. Fu, A modeling SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413. Google Scholar [47] X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Can. Appl. Math. Q., 3 (1995), 473-495. Google Scholar
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