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June  2020, 19(6): 3323-3340. doi: 10.3934/cpaa.2020147

Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

*Corresponding author

Received  September 2019 Revised  November 2019 Published  March 2020

Fund Project: J. Wang was supported by National Natural Science Foundation of China (No. 11871179), Natural Science Foundation of Heilongjiang Province (No. LC2018002) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, China. S. Zhu was supported by Graduate Students Innovation Research Program of Heilongjiang University (No. YJSCX2019-206HLJU)

Spatial heterogeneity and movement of population play an important role in disease spread and control in reality. This paper concerns with a spatial Susceptible-Infected-Susceptible epidemic model with spontaneous infection and logistic source, aiming to investigate the asymptotic profiles of the endemic steady state (whenever it exists) for large and small diffusion rates. We firstly establish uniform upper bound of solutions. By studying the local and global stability of the unique constant endemic equilibrium when spatial environment is homogeneous, we apply the well-known Leray-Schuauder degree index formula to confirm the existence of endemic steady state. Our theoretical results suggest that spontaneous infection and varying total population strongly enhance the persistence of disease spread in the sense that disease component of the endemic steady state will not approach zero whenever the large and small diffusion rates of the susceptible or infected population is used. This gives new insights and aspects for infectious disease modeling and control.

Citation: Siyao Zhu, Jinliang Wang. Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3323-3340. doi: 10.3934/cpaa.2020147
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]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equ., 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[3]

R. CuiK. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.

[4]

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.

[5]

J. GeK. KimZ. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.

[6]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, vol. 247, Longman Scientific & Technical Harlow, 1991.

[7]

A. Hill, D. Rand, M. Nowak and N. Christakis, Infectious Disease Modeling of Social Contagion in Networks, PLoS Comput. Biol., 6 (2010), e1000968. doi: 10.1371/journal.pcbi.1000968.

[8]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial Differ. Equ., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.

[9]

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

[10]

B. Li, H. Li and T. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), 96. doi: 10.1007/s00033-017-0845-1.

[11]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equ., 131 (1996), 400-426.  doi: 10.1006/jdeq.1996.0157.

[12]

L. Nirenberg, Topic in Nonlinear Functional Nnalysis, American Mathematical Society, Providence, RI, 1974. doi: 10.1090/cln/006.

[13]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal. Theory Meth. Appl., 71 (2009), 239-247.  doi: 10.1016/j.na.2008.10.043.

[14]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, J. Differ. Equ., 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.

[15]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[16]

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.

[17]

Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differ. Equ., 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[18]

S. Zhu and J. Wang, Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1999-2019. 

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]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equ., 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[3]

R. CuiK. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.

[4]

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.

[5]

J. GeK. KimZ. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.

[6]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, vol. 247, Longman Scientific & Technical Harlow, 1991.

[7]

A. Hill, D. Rand, M. Nowak and N. Christakis, Infectious Disease Modeling of Social Contagion in Networks, PLoS Comput. Biol., 6 (2010), e1000968. doi: 10.1371/journal.pcbi.1000968.

[8]

D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial Differ. Equ., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.

[9]

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

[10]

B. Li, H. Li and T. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), 96. doi: 10.1007/s00033-017-0845-1.

[11]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equ., 131 (1996), 400-426.  doi: 10.1006/jdeq.1996.0157.

[12]

L. Nirenberg, Topic in Nonlinear Functional Nnalysis, American Mathematical Society, Providence, RI, 1974. doi: 10.1090/cln/006.

[13]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal. Theory Meth. Appl., 71 (2009), 239-247.  doi: 10.1016/j.na.2008.10.043.

[14]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, J. Differ. Equ., 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.

[15]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[16]

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.

[17]

Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differ. Equ., 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[18]

S. Zhu and J. Wang, Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1999-2019. 

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