doi: 10.3934/dcdsb.2021183
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Propagation dynamics in a diffusive SIQR model for childhood diseases

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

* Corresponding author: Guo Lin

Received  December 2020 Revised  April 2021 Early access July 2021

This paper is concerned with the propagation dynamics in a diffusive susceptible-infective nonisolated-isolated-removed model that describes the recurrent outbreaks of childhood diseases. To model the spatial-temporal modes on disease spreading, we study the traveling wave solutions and the initial value problem with special decay condition. When the basic reproduction ratio of the corresponding kinetic system is larger than one, we define a threshold that is the minimal wave speed of traveling wave solutions as well as the spreading speed of some components. From the viewpoint of mathematical epidemiology, the threshold is monotone decreasing in the rate at which individuals leave the infective and enter the isolated classes.

Citation: Shuo Zhang, Guo Lin. Propagation dynamics in a diffusive SIQR model for childhood diseases. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021183
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, J. A. Goldstein (Ed. ), Lecture Notes in Math., 446, Springer, Berlin, 1975, 5-49. doi: 10.1007/BFb0070595.  Google Scholar

[2]

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

[3]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations., 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[4]

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

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Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation, Math. Biosci., 128 (1995), 93-130.  doi: 10.1016/0025-5564(94)00069-C.  Google Scholar

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R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

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S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner, X. Q. Zhao and X. Zou), Fields Inst. Commun., 48, AMS, Providence, RI, 2006, pp. 137-200.  Google Scholar

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J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

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Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods. Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.  Google Scholar

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W. O. Kermack and A. G. McKendrik, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A., 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.  Google Scholar

[11]

A. N. KolmogorovI. G. Petrovskii and N. S. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A: Mat. Mekh., 1 (1937), 1-26.   Google Scholar

[12]

X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar

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X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[14]

G. LinS. Pan and X.-P. Yan, Spreading speeds of epidemic models with nonlocal delays, Math. Biosci. Eng., 16 (2019), 7562-7588.  doi: 10.3934/mbe.2019380.  Google Scholar

[15]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differential Equations., 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.  Google Scholar

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R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[17]

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J. D. Murray, Mathematical Biology, II. Spatial Models and Biomedical Applications, , Third edition, Interdisciplinary Applied Mathematics 18, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

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S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

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S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, New York, 2007, 97-122.  Google Scholar

[21]

H. ShuX. PanX.-S. Wang and J. Wu, Traveling waves in epidemic models: Non-monotone diffusive systems with non-monotone incidence rates, J. Dynam. Differential Equations., 31 (2019), 883-901.  doi: 10.1007/s10884-018-9683-x.  Google Scholar

[22]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121.  doi: 10.1515/crll.1979.306.94.  Google Scholar

[23]

X.-S. WangH. Wang and J. Wu, Traveling 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

[24]

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 Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[25]

H. F. WeinbergerM. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[26]

L.-I. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differential Equations., 168 (2000), 150-167.  doi: 10.1006/jdeq.2000.3882.  Google Scholar

[27] Q. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 2011.   Google Scholar
[28]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, J. A. Goldstein (Ed. ), Lecture Notes in Math., 446, Springer, Berlin, 1975, 5-49. doi: 10.1007/BFb0070595.  Google Scholar

[2]

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

[3]

A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations., 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar

[4]

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

[5]

Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation, Math. Biosci., 128 (1995), 93-130.  doi: 10.1016/0025-5564(94)00069-C.  Google Scholar

[6]

R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[7]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations (eds. H. Brunner, X. Q. Zhao and X. Zou), Fields Inst. Commun., 48, AMS, Providence, RI, 2006, pp. 137-200.  Google Scholar

[8]

J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[9]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods. Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.  Google Scholar

[10]

W. O. Kermack and A. G. McKendrik, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A., 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.  Google Scholar

[11]

A. N. KolmogorovI. G. Petrovskii and N. S. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A: Mat. Mekh., 1 (1937), 1-26.   Google Scholar

[12]

X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar

[13]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[14]

G. LinS. Pan and X.-P. Yan, Spreading speeds of epidemic models with nonlocal delays, Math. Biosci. Eng., 16 (2019), 7562-7588.  doi: 10.3934/mbe.2019380.  Google Scholar

[15]

G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to diffusive Lotka-Volterra competition models with distributed delays, J. Dynam. Differential Equations., 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.  Google Scholar

[16]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, I. An Introduction, , Third edition, Interdisciplinary Applied Mathematics 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[18]

J. D. Murray, Mathematical Biology, II. Spatial Models and Biomedical Applications, , Third edition, Interdisciplinary Applied Mathematics 18, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[19]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[20]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, New York, 2007, 97-122.  Google Scholar

[21]

H. ShuX. PanX.-S. Wang and J. Wu, Traveling waves in epidemic models: Non-monotone diffusive systems with non-monotone incidence rates, J. Dynam. Differential Equations., 31 (2019), 883-901.  doi: 10.1007/s10884-018-9683-x.  Google Scholar

[22]

H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121.  doi: 10.1515/crll.1979.306.94.  Google Scholar

[23]

X.-S. WangH. Wang and J. Wu, Traveling 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

[24]

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 Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[25]

H. F. WeinbergerM. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

[26]

L.-I. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differential Equations., 168 (2000), 150-167.  doi: 10.1006/jdeq.2000.3882.  Google Scholar

[27] Q. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 2011.   Google Scholar
[28]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differential Equations., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.  Google Scholar

Figure 1.  Spatial-temporal plots of (15)
Figure 2.  Spatial plots at $ t = 80,t = 100 $
Figure 3.  Spatial-temporal plots of (17) when $ t\in [60,100] $
Table 1.  Approximate level sets in Figures 2
Level sets $ L^{I}_t(0.1) $ $ L^{Q}_t(0.02) $ $ L^{R}_t(0.02) $
$ t=80 $ -105 -107.2 -102.2
$ t=100 $ -133.8 -134.6 -130.2
Averaging moving speed of level sets 1.42 1.37 1.40
Level sets $ L^{I}_t(0.1) $ $ L^{Q}_t(0.02) $ $ L^{R}_t(0.02) $
$ t=80 $ -105 -107.2 -102.2
$ t=100 $ -133.8 -134.6 -130.2
Averaging moving speed of level sets 1.42 1.37 1.40
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