• Previous Article
    A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents
  • DCDS-B Home
  • This Issue
  • Next Article
    Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions
doi: 10.3934/dcdsb.2021152

Wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

2. 

Center for Mathematical Sciences, China University of Geosciences, Wuhan, 430074, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

4. 

School of Mathematics and Big Data, Foshan University, Foshan 528000, China

* Corresponding author: Chufen Wu

Received  February 2021 Revised  April 2021 Published  May 2021

This paper is concerned with the wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse. We first show the well-posedness of solutions for such a problem. Then, in terms of the basic reproduction number and the wave speed, we establish a threshold result which reveals the existence and non-existence of the strong traveling waves accounting for phase transitions between the disease-free equilibrium and the endemic steady state. Further, we clarify and characterize the minimal wave speed of traveling waves. Finally, numerical simulations and discussions are also given to illustrate the analytical results. Our result indicates that the relapse can encourage the spread of the disease.

Citation: Jia-Bing Wang, Shao-Xia Qiao, Chufen Wu. Wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021152
References:
[1]

Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[2]

S. Blower, Modelling the genital herpes epidemic, Herpes 11, 3 (2004), 138A–146A. Google Scholar

[3]

S. M. BlowerT. C. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in HSV-2, Nat. Med., 4 (1998), 673-678.  doi: 10.1038/nm0698-673.  Google Scholar

[4]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[5]

J. Coville and L. Dupaigne, On a non-local eqution arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A., 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[6]

H. Cox, Tuberculosis recurrence and mortality after successful treatment: Impact of drug resistance, PLoS Med., 3 (2006), 1836-1843.  doi: 10.1371/journal.pmed.0030384.  Google Scholar

[7]

O. Diekman, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109–130. doi: 10.1007/BF02450783.  Google Scholar

[8]

A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.  Google Scholar

[9]

A. DucrotP. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.  Google Scholar

[10]

S. C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069.  Google Scholar

[11]

P. GuoX. S. Yang and Z. C. Yang, Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Adv. Difference Equ., 2014 (2014), 164-181.  doi: 10.1186/1687-1847-2014-164.  Google Scholar

[12]

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507.  doi: 10.1016/j.amc.2013.02.044.  Google Scholar

[13]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.  doi: 10.1007/s00285-006-0050-x.  Google Scholar

[14]

G. HuangY. TakeuchiW. Ma and D. 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.  Google Scholar

[15]

W. Huang and C. Wu, Non-monotone waves of a stage-structured SLIRM epidemic model with latent period, Proc. Roy. Soc. Edinburgh Sect. A. doi: 10.1017/prm.2020.65.  Google Scholar

[16]

V. HutsonS. MartinezK. Mischailow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[17]

C. Y. KaoY. Lou and W. Shen, Random diseprsal vs nonlocal dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[18]

M. Kermack and A. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721.   Google Scholar

[19]

T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960.  doi: 10.1080/00036811.2016.1199796.  Google Scholar

[20]

W. T. LiJ. B. Wang and X.-Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2.  Google Scholar

[21]

W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.  Google Scholar

[22]

Y. LiW. T. Li and F. Y. Yang, Traveling waves for nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[23]

J. MartinsA. Pinto and N. Stollenwerkc, A scaling analysis in the SIRI epidemiological model, J. Biol. Dyn., 3 (2009), 479-496.  doi: 10.1080/17513750802601058.  Google Scholar

[24]

H. N. Moreira and Y. Wang, Global stability in an $S\rightarrow I\rightarrow R\rightarrow I$ model, SIAM Rev., 39 (1997), 496-502.  doi: 10.1137/S0036144595295879.  Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[26]

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.  doi: 10.1137/1032003.  Google Scholar

[27]

P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.  doi: 10.1016/j.mbs.2006.09.017.  Google Scholar

[28]

C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstr. Appl., 9 (2013), 50-61.   Google Scholar

[29]

J. B. WangW. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.  Google Scholar

[30]

J. B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[31]

X. WangH. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[32]

Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[33] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, CRC Press, 1985.   Google Scholar
[34]

P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[35] D. V. Widder, Laplace Transform, Princeton University Press, Princeton, NJ, 1941.   Google Scholar
[36]

P. WildyH. J. Field and A. A. Nash, Classical herpes latency revisited, Virus Persistence Symposium, 33 (1982), 133-167.   Google Scholar

[37]

C. WuY. YangQ. ZhaoY. Tian and Z. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068.  Google Scholar

[38]

C. WuY. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019.  Google Scholar

[39]

C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differential Equations, 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022.  Google Scholar

[40]

F. Y. YangY. LiW. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[41]

F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.  Google Scholar

[42]

G. B. ZhangW. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.  Google Scholar

[43]

C. C. ZhuW. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73 (2017), 1707-1723.  doi: 10.1016/j.camwa.2017.02.014.  Google Scholar

show all references

References:
[1]

Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[2]

S. Blower, Modelling the genital herpes epidemic, Herpes 11, 3 (2004), 138A–146A. Google Scholar

[3]

S. M. BlowerT. C. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in HSV-2, Nat. Med., 4 (1998), 673-678.  doi: 10.1038/nm0698-673.  Google Scholar

[4]

X. Chen and J. S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0.  Google Scholar

[5]

J. Coville and L. Dupaigne, On a non-local eqution arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A., 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[6]

H. Cox, Tuberculosis recurrence and mortality after successful treatment: Impact of drug resistance, PLoS Med., 3 (2006), 1836-1843.  doi: 10.1371/journal.pmed.0030384.  Google Scholar

[7]

O. Diekman, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109–130. doi: 10.1007/BF02450783.  Google Scholar

[8]

A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.  Google Scholar

[9]

A. DucrotP. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.  Google Scholar

[10]

S. C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069.  Google Scholar

[11]

P. GuoX. S. Yang and Z. C. Yang, Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Adv. Difference Equ., 2014 (2014), 164-181.  doi: 10.1186/1687-1847-2014-164.  Google Scholar

[12]

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507.  doi: 10.1016/j.amc.2013.02.044.  Google Scholar

[13]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.  doi: 10.1007/s00285-006-0050-x.  Google Scholar

[14]

G. HuangY. TakeuchiW. Ma and D. 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.  Google Scholar

[15]

W. Huang and C. Wu, Non-monotone waves of a stage-structured SLIRM epidemic model with latent period, Proc. Roy. Soc. Edinburgh Sect. A. doi: 10.1017/prm.2020.65.  Google Scholar

[16]

V. HutsonS. MartinezK. Mischailow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[17]

C. Y. KaoY. Lou and W. Shen, Random diseprsal vs nonlocal dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[18]

M. Kermack and A. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721.   Google Scholar

[19]

T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960.  doi: 10.1080/00036811.2016.1199796.  Google Scholar

[20]

W. T. LiJ. B. Wang and X.-Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2.  Google Scholar

[21]

W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.  Google Scholar

[22]

Y. LiW. T. Li and F. Y. Yang, Traveling waves for nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[23]

J. MartinsA. Pinto and N. Stollenwerkc, A scaling analysis in the SIRI epidemiological model, J. Biol. Dyn., 3 (2009), 479-496.  doi: 10.1080/17513750802601058.  Google Scholar

[24]

H. N. Moreira and Y. Wang, Global stability in an $S\rightarrow I\rightarrow R\rightarrow I$ model, SIAM Rev., 39 (1997), 496-502.  doi: 10.1137/S0036144595295879.  Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[26]

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.  doi: 10.1137/1032003.  Google Scholar

[27]

P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.  doi: 10.1016/j.mbs.2006.09.017.  Google Scholar

[28]

C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstr. Appl., 9 (2013), 50-61.   Google Scholar

[29]

J. B. WangW. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.  Google Scholar

[30]

J. B. Wang and C. Wu, Forced waves and gap formations for a Lotka-Volterra competition model with nonlocal dispersal and shifting habitats, Nonlinear Anal. Real World Appl., 58 (2021), 103208. doi: 10.1016/j.nonrwa.2020.103208.  Google Scholar

[31]

X. WangH. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[32]

Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[33] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, CRC Press, 1985.   Google Scholar
[34]

P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[35] D. V. Widder, Laplace Transform, Princeton University Press, Princeton, NJ, 1941.   Google Scholar
[36]

P. WildyH. J. Field and A. A. Nash, Classical herpes latency revisited, Virus Persistence Symposium, 33 (1982), 133-167.   Google Scholar

[37]

C. WuY. YangQ. ZhaoY. Tian and Z. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068.  Google Scholar

[38]

C. WuY. Wang and X. Zou, Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment, J. Differential Equations, 267 (2019), 4890-4921.  doi: 10.1016/j.jde.2019.05.019.  Google Scholar

[39]

C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differential Equations, 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022.  Google Scholar

[40]

F. Y. YangY. LiW. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[41]

F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.  Google Scholar

[42]

G. B. ZhangW. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.  Google Scholar

[43]

C. C. ZhuW. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73 (2017), 1707-1723.  doi: 10.1016/j.camwa.2017.02.014.  Google Scholar

Figure 1.  The profile of $ (S,I,R) $ connecting $ (\frac{7}{3},0,0) $ and $ (\frac{11}{6},\frac{3}{22},\frac{1}{11}) $ for $ c>c^* $
Figure 2.  The profile of $ S $ connecting $ \frac{7}{3} $ and $ \frac{11}{6} $ for $ c>c^* $ at time steps $ t = 5,12,20 $
Figure 3.  The profile of $ I $ connecting $ 0 $ and $ \frac{3}{22} $ for $ c>c^* $ at time steps $ t = 5,12,20 $
Figure 4.  The profile of $ R $ connecting $ 0 $ and $ \frac{1}{11} $ for $ c>c^* $ at time steps $ t = 5,12,20 $
Figure 5.  The profile of $ (S,I,R) $ connecting $ (\frac{7}{3},0,0) $ and $ (\frac{11}{6},\frac{3}{22},\frac{1}{11}) $ for $ c = c^* $
Figure 6.  The profile of $ S $ connecting $ \frac{7}{3} $ and $ \frac{11}{6} $ for $ c = c^* $ at time steps $ t = 5,12,20 $
Figure 7.  The profile of $ I $ connecting $ 0 $ and $ \frac{3}{22} $ for $ c = c^* $ at time steps $ t = 5,12,20 $
Figure 8.  The profile of $ R $ connecting $ 0 $ and $ \frac{1}{11} $ for $ c = c^* $ at time steps $ t = 5,12,20 $
Figure 9.  The profile of $ (S,I,R) $ connecting $ (5,0,0) $ and $ (\frac{24}{5},\frac{1}{24},\frac{5}{48}) $ for $ c<c^* $
Figure 10.  The profile of $ S $ connecting $ 5 $ and $ \frac{24}{5} $ for $ c<c^* $ at time steps $ t = 5,12,20 $
Figure 11.  The profile of $ I $ connecting $ 0 $ and $ \frac{1}{24} $ for $ c<c^* $ at time steps $ t = 5,12,20 $
Figure 12.  The profile of $ R $ connecting $ 0 $ and $ \frac{5}{48} $ for $ c<c^* $ at time steps $ t = 5,12,20 $
Figure 13.  The minimal wave speed $ c^* $ with respect to the relapse rate $ \delta $
[1]

Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure & Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037

[2]

Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969

[3]

Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125

[4]

Tomás Caraballo, Mohamed El Fatini, Roger Pettersson, Regragui Taki. A stochastic SIRI epidemic model with relapse and media coverage. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3483-3501. doi: 10.3934/dcdsb.2018250

[5]

Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014

[6]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057

[7]

Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107

[8]

Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236

[9]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699

[10]

Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001

[11]

Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020340

[12]

Xiaoli Wang, Guohong Zhang. Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4459-4477. doi: 10.3934/dcdsb.2020295

[13]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[14]

Zhiting Xu, Yiyi Zhang. Traveling wave phenomena of a diffusive and vector-bias malaria model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 923-940. doi: 10.3934/cpaa.2015.14.923

[15]

Meng Zhao, Wantong Li, Yihong Du. The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4599-4620. doi: 10.3934/cpaa.2020208

[16]

Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291

[17]

Li Ma, Youquan Luo. Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2555-2582. doi: 10.3934/dcdsb.2020022

[18]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006

[19]

Vando Narciso. On a Kirchhoff wave model with nonlocal nonlinear damping. Evolution Equations & Control Theory, 2020, 9 (2) : 487-508. doi: 10.3934/eect.2020021

[20]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (57)
  • HTML views (52)
  • Cited by (0)

Other articles
by authors

[Back to Top]