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January  2020, 25(1): 81-98. doi: 10.3934/dcdsb.2019173

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  January 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
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.

[2]

L. AllenB. BolkerY. 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.

[3]

L. AllenB. BolkerY. 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.

[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.

[5]

R. Anderson and R. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367.  doi: 10.1007/978-3-642-68635-1.

[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.

[7]

K. BrownP. 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.

[8]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biology, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296.

[9]

J. CuiX. 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.

[10]

R. CuiK.-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.

[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.

[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.

[13]

W. DingW. 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.

[14]

Y. DuR. 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.

[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.

[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.

[17]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001.

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[19]

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

[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.

[21]

A. HillD. RandM. 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.

[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.

[23]

W. HuangM. 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.

[24] M. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals,, Princeton University Press, 2008. 
[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.

[26]

C. LeiF. 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.

[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.

[28]

H. LiR. 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.

[29]

H. LiR. 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.

[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.

[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.

[32]

C.-S. LinW.-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.

[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.

[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.

[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. 

[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.

[37]

R. PengJ. 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.

[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.

[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.

[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.

[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.

[42]

H. ShiZ. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. 

[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.

[44]

X. WenJ. 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.

[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.

[46]

M. YangG. 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. 

[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. 

show all references

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.

[2]

L. AllenB. BolkerY. 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.

[3]

L. AllenB. BolkerY. 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.

[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.

[5]

R. Anderson and R. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367.  doi: 10.1007/978-3-642-68635-1.

[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.

[7]

K. BrownP. 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.

[8]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biology, Wiley, Chichester, UK, 2003. doi: 10.1002/0470871296.

[9]

J. CuiX. 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.

[10]

R. CuiK.-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.

[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.

[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.

[13]

W. DingW. 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.

[14]

Y. DuR. 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.

[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.

[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.

[17]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001.

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[19]

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

[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.

[21]

A. HillD. RandM. 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.

[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.

[23]

W. HuangM. 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.

[24] M. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals,, Princeton University Press, 2008. 
[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.

[26]

C. LeiF. 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.

[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.

[28]

H. LiR. 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.

[29]

H. LiR. 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.

[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.

[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.

[32]

C.-S. LinW.-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.

[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.

[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.

[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. 

[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.

[37]

R. PengJ. 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.

[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.

[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.

[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.

[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.

[42]

H. ShiZ. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. 

[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.

[44]

X. WenJ. 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.

[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.

[46]

M. YangG. 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. 

[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. 

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