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doi: 10.3934/dcdsb.2020313

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

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

* Corresponding author: Zhi-Cheng Wang

Received  May 2020 Revised  August 2020 Published  October 2020

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.

Citation: Liangliang Deng, Zhi-Cheng Wang. Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020313
References:
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[2]

L. J. S. AllenB. M. BolkerY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. BeaumontJ.-B. BurieA. Ducrot and P. Zongo, Propagation of salmonella within an industrial hen house, SIAM J. Appl. Math., 72 (2012), 1113-1148.  doi: 10.1137/110822967.  Google Scholar

[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.  Google Scholar

[7]

H. BerestyckiF. 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.  Google Scholar

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H. BerestyckiF. 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.  Google Scholar

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H. BerestyckiF. 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.  Google Scholar

[10]

H. BerestyckiF. 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.  Google Scholar

[11]

H. BerestyckiF. 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.  Google Scholar

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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.  Google Scholar

[13]

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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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.  Google Scholar

[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.  Google Scholar

[16]

W. E. FitzgibbonC. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

A. KällénP. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. H. Martin Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

N. ShigesadaK. 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.  Google Scholar

[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.  Google Scholar

[34]

D. L. SmithJ. 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.  Google Scholar

[35]

G. Sweers, Strong positivity in $C(\overline{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.  doi: 10.1007/BF02570833.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

Z.-C. WangL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

C. WuD. 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.  Google Scholar

[43]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[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.  Google Scholar

[45]

L. ZhangZ.-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.  Google Scholar

[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.  Google Scholar

[47]

L. ZhaoZ.-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.  Google Scholar

[48]

L. ZhaoZ.-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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. M. AlanazZ. 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.  Google Scholar

[2]

L. J. S. AllenB. M. BolkerY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. BeaumontJ.-B. BurieA. Ducrot and P. Zongo, Propagation of salmonella within an industrial hen house, SIAM J. Appl. Math., 72 (2012), 1113-1148.  doi: 10.1137/110822967.  Google Scholar

[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.  Google Scholar

[7]

H. BerestyckiF. 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.  Google Scholar

[8]

H. BerestyckiF. 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.  Google Scholar

[9]

H. BerestyckiF. 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.  Google Scholar

[10]

H. BerestyckiF. 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.  Google Scholar

[11]

H. BerestyckiF. 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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

W. E. FitzgibbonC. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

A. KällénP. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. H. Martin Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

N. ShigesadaK. 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.  Google Scholar

[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.  Google Scholar

[34]

D. L. SmithJ. 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.  Google Scholar

[35]

G. Sweers, Strong positivity in $C(\overline{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.  doi: 10.1007/BF02570833.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

Z.-C. WangL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

C. WuD. 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.  Google Scholar

[43]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[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.  Google Scholar

[45]

L. ZhangZ.-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.  Google Scholar

[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.  Google Scholar

[47]

L. ZhaoZ.-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.  Google Scholar

[48]

L. ZhaoZ.-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.  Google Scholar

[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.  Google Scholar

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