# American Institute of Mathematical Sciences

June  2022, 27(6): 3241-3259. doi: 10.3934/dcdsb.2021183

## 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 Published  June 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (6) : 3241-3259. 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. [2] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109-130.  doi: 10.1007/BF02450783. [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. [4] A. Ducrot, P. 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. [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. [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. [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. [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. [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. [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. [11] A. N. Kolmogorov, I. 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. [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. [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. [14] G. Lin, S. 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. [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. [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. [17] J. D. Murray, Mathematical Biology, I. An Introduction, , Third edition, Interdisciplinary Applied Mathematics 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868. [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. [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. [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. [21] H. Shu, X. Pan, X.-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. [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. [23] X.-S. Wang, H. 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. [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. [25] H. F. Weinberger, M. 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. [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. [27] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 2011. [28] T. Zhang, W. 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.

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. [2] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69 (1978), 109-130.  doi: 10.1007/BF02450783. [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. [4] A. Ducrot, P. 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. [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. [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. [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. [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. [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. [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. [11] A. N. Kolmogorov, I. 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. [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. [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. [14] G. Lin, S. 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. [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. [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. [17] J. D. Murray, Mathematical Biology, I. An Introduction, , Third edition, Interdisciplinary Applied Mathematics 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868. [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. [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. [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. [21] H. Shu, X. Pan, X.-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. [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. [23] X.-S. Wang, H. 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. [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. [25] H. F. Weinberger, M. 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. [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. [27] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 2011. [28] T. Zhang, W. 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.
Spatial-temporal plots of (15)
Spatial plots at $t = 80,t = 100$
Spatial-temporal plots of (17) when $t\in [60,100]$
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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