May  2020, 25(5): 1999-2019. doi: 10.3934/dcdsb.2020013

Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment

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

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

In this paper, we investigate a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. We first prove that the solution of the model is bounded when the susceptible and infected individuals have same or distinct dispersal rates. The global stability of the constant endemic equilibrium is proved by constructing suitable Lyapunov functionals when all parameters are positive constants. We employ the topological degree argument to show the existence of positive steady state. Most importantly, we have also investigated the asymptotic profiles of the positive steady state as the dispersal rate of susceptible or infected individuals tends to zero or infinity. Our result reveals that a linear source and spontaneous infection can significantly enhance disease persistence no matter what dispersal rate of the susceptible or infected population is small or large, which leads to the situation that when total population number allows to vary, disease becomes more difficult to control.

Citation: Siyao Zhu, Jinliang Wang. Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1999-2019. doi: 10.3934/dcdsb.2020013
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, Disc. Cont. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.

[3]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[4]

R. H. 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.

[5]

Z. J. 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.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[7]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2006), 599-653.  doi: 10.1137/S0036144500371907.

[8]

A. L. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Infectious disease modeling of social contagion in networks, PLoS Comput. Biol., 6 (2010), e1000968, 15 pp. doi: 10.1371/journal.pcbi.1000968.

[9]

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

[10]

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.

[11]

B. Li, H. C. Li and Y. C. 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.

[12]

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.

[13]

Y. J. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Bio., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[14]

L. Nirenberg, Topic in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.

[15]

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.

[16]

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.

[17]

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, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[18]

R. PengJ. P. Shi and M. X. 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.

[19]

Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Analysis: RWA, 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.

[20]

H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8 (1975), 295-310. 

[21]

Y. X. Wu and X. F. 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.

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, Disc. Cont. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.

[3]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[4]

R. H. 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.

[5]

Z. J. 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.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[7]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2006), 599-653.  doi: 10.1137/S0036144500371907.

[8]

A. L. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Infectious disease modeling of social contagion in networks, PLoS Comput. Biol., 6 (2010), e1000968, 15 pp. doi: 10.1371/journal.pcbi.1000968.

[9]

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

[10]

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.

[11]

B. Li, H. C. Li and Y. C. 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.

[12]

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.

[13]

Y. J. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Bio., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[14]

L. Nirenberg, Topic in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.

[15]

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.

[16]

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.

[17]

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, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[18]

R. PengJ. P. Shi and M. X. 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.

[19]

Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Analysis: RWA, 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.

[20]

H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8 (1975), 295-310. 

[21]

Y. X. Wu and X. F. 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.

[1]

Lian Duan, Lihong Huang, Chuangxia Huang. Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3539-3560. doi: 10.3934/cpaa.2021120

[2]

Yuan-Hang Su, Wan-Tong Li, Fei-Ying Yang. Effects of nonlocal dispersal and spatial heterogeneity on total biomass. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4929-4936. doi: 10.3934/dcdsb.2019038

[3]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433

[4]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35

[5]

Yu-Xia Wang, Wan-Tong Li. Combined effects of the spatial heterogeneity and the functional response. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 19-39. doi: 10.3934/dcds.2019002

[6]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124

[7]

Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025

[8]

Jian-Wen Sun, Wan-Tong Li, Zhi-Cheng Wang. A nonlocal dispersal logistic equation with spatial degeneracy. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3217-3238. doi: 10.3934/dcds.2015.35.3217

[9]

Roger M. Nisbet, Kurt E. Anderson, Edward McCauley, Mark A. Lewis. Response of equilibrium states to spatial environmental heterogeneity in advective systems. Mathematical Biosciences & Engineering, 2007, 4 (1) : 1-13. doi: 10.3934/mbe.2007.4.1

[10]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

[11]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237

[12]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619

[13]

Luis Barreira, Claudia Valls. Quadratic Lyapunov sequences and arbitrary growth rates. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 63-74. doi: 10.3934/dcds.2010.26.63

[14]

Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for virus-immune models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3093-3114. doi: 10.3934/dcdsb.2015.20.3093

[15]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[16]

Rongsong Liu, Jiangping Shuai, Jianhong Wu, Huaiping Zhu. Modeling spatial spread of west nile virus and impact of directional dispersal of birds. Mathematical Biosciences & Engineering, 2006, 3 (1) : 145-160. doi: 10.3934/mbe.2006.3.145

[17]

Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145

[18]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[19]

Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597

[20]

Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (459)
  • HTML views (204)
  • Cited by (0)

Other articles
by authors

[Back to Top]