Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media

Liangliang Deng; Zhi-Cheng Wang

doi:10.3934/dcdsb.2020313

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2021, Volume 26, Issue 9: 4789-4814. Doi: 10.3934/dcdsb.2020313

This issue Previous Article Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases Next Article Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics

Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media

Liangliang Deng and
Zhi-Cheng Wang^,

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

^* Corresponding author: Zhi-Cheng Wang
^* Corresponding author: Zhi-Cheng Wang

Received: April 30, 2020

Revised: July 31, 2020

Early access: October 2020

Published: September 2021

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Abstract

This paper is concerned with propagation phenomena for an epidemic model describing the circulation of a disease within two populations or two subgroups in periodic media, where the susceptible individuals are assumed to be motionless. The spatial dynamics for the cooperative system obtained by a classical transformation are investigated, including spatially periodic steady state, spreading speeds and pulsating travelling fronts. It is proved that the minimal wave speed is linearly determined and given by a variational formula involving linear eigenvalue problem. Further, we prove that the existence and non-existence of travelling wave solutions of the model are entirely determined by the basic reproduction ratio $ \mathcal{R}_{0} $. As an application, we prove that if the localized amount of infectious individuals are introduced at the beginning, then the solution of such a system has an asymptotic spreading speed in large time and that is exactly coincident with the minimal wave speed.

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Mathematics Subject Classification: Primary: 35K55, 35C07, 35B40; Secondary: 92D30.

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References

[1]	K. M. Alanaz, Z. Jackiewicz and H. R. Thieme, Spreading speeds of rabies with territorial and diffusing rabid foxes, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2143-2183. doi: 10.3934/dcdsb.2019222.
[2]	L. J. S. Allen, B. M. Bolker, Y. 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.
[3]	B. Ambrosio, A. Ducrot and S. Ruan, Generalized traveling waves for time-dependent reaction-diffusion systems, Math. Ann., (2020). doi: 10.1007/s00208-020-01998-3.
[4]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetic, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.
[5]	C. Beaumont, J.-B. Burie, A. Ducrot and P. Zongo, Propagation of salmonella within an industrial hen house, SIAM J. Appl. Math., 72 (2012), 1113-1148. doi: 10.1137/110822967.
[6]	H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022.
[7]	H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.
[8]	H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅱ: Genaral domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X.
[9]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅰ-Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3.
[10]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅱ-biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.
[11]	H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Math. Pura Appl., 186 (2007), 469-507. doi: 10.1007/s10231-006-0015-0.
[12]	A. Ducrot and T. Giletti, Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552. doi: 10.1007/s00285-013-0713-3.
[13]	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.
[14]	J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.1137/140953939.
[15]	R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.
[16]	W. E. Fitzgibbon, C. B. Martin and J. J. Morgan, A diffusive epidemic model with criss-cross dynamics, J. Math. Anal. Appl., 184 (1994), 399-414. doi: 10.1006/jmaa.1994.1209.
[17]	W. E. Fitzgibbon, J. J. Morgan and G. F. Webb, An outbreak vector-host epidemic model with spatial structure: The 2015-2016 Zika outbreak in Rio De Janeiro, Theor. Biol. Med. Modell., 14 (2017), 7.
[18]	T. Giletti, Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems, Calc. Var. Partial Differ. Equ., 51 (2014), 265-289. doi: 10.1007/s00526-013-0674-9.
[19]	A. Källén, P. Arcuri and J. D. Murray, A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393. doi: 10.1016/S0022-5193(85)80276-9.
[20]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118.
[21]	A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'État à Moscou(Bjul. Moskowskogo Gos. Univ.), Série internationale A, 1 (1937), 1–26.
[22]	K.-Y. Lam and Y. Lou, Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dyn. Differ. Equ., 28 (2016), 29-48. doi: 10.1007/s10884-015-9504-4.
[23]	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; Comm. Pure Appl. Math., 61 (2008), 137–138 (Erratum). doi: 10.1002/cpa.20221.
[24]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018.
[25]	Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8.
[26]	P. Magal and C. McCluskey, Two group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.
[27]	R. H. Martin Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York, 1976.
[28]	J. D. Murray, Mathematical Biology I: An Introduction and II: Spatial Models and Biomedical Applications, 3rd ed., Springer, New York, 2002. doi: 10.1007/b98868.
[29]	G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185. doi: 10.1017/S0956792511000027.
[30]	G. Nadin, The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406. doi: 10.1137/080743597.
[31]	N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, Oxford, 1997. doi: 10.2307/6013.
[32]	N. Shigesada, K. Kawasaki and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Population Biol., 30 (1986), 143-160. doi: 10.1016/0040-5809(86)90029-8.
[33]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. AMS, Providence, RI, 1995. doi: 10.1090/surv/041.
[34]	D. L. Smith, J. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biol., 2 (2004), 1957-1964. doi: 10.1371/journal.pbio.0020368.
[35]	G. Sweers, Strong positivity in $C(\overline{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271. doi: 10.1007/BF02570833.
[36]	H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.
[37]	W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.
[38]	X.-S. Wang and X.-Q. Zhao, Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differ. Equ., 259 (2015), 7238-7259.
[39]	Z.-C. Wang, L. Zhang and X.-Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dyn. Differ. Equ., 30 (2018), 379-403. doi: 10.1007/s10884-016-9546-2.
[40]	H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.
[41]	P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst., 29 (2011), 343-366. doi: 10.3934/dcds.2011.29.343.
[42]	C. Wu, D. Xiao and X.-Q. Zhao, Spreading speeds of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differ. Equ., 255 (2013), 3983-4011. doi: 10.1016/j.jde.2013.07.058.
[43]	J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.
[44]	X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dyn. Differ. Equ., 29 (2017), 41-66. doi: 10.1007/s10884-015-9426-1.
[45]	L. Zhang, Z.-C. Wang and X.-Q. Zhao, Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality, J. Evol. Equ., 20 (2020), 1029-1059. doi: 10.1007/s00028-019-00544-2.
[46]	G. Zhao and S. Ruan, Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78 (2018), 1954-1980. doi: 10.1137/17M1144106.
[47]	L. Zhao, Z.-C. Wang and S. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325. doi: 10.1088/1361-6544/aa59ae.
[48]	L. Zhao, Z.-C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915. doi: 10.1007/s00285-018-1227-9.
[49]	X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics (Ouvrages de Mathématiques de la SMC), 2$^{nd}$ edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.