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doi: 10.3934/dcdsb.2021170
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The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model

1. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

2. 

Department of Mathematics and Statistics, Arab American University, 240 Jenin 13, Zababdeh, Palestine

* Corresponding author

Received  January 2021 Revised  May 2021 Early access June 2021

Fund Project: Our research were supported by National Natural Science Foundation of China (12071151) and Natural Science Foundation of Guangdong Province (2021A1515010052)

In this paper, we study the influence of spatial-dependent variables on the basic reproduction ratio ($ \mathcal{R}_0 $) for a scalar reaction-diffusion equation model. We first investigate the principal eigenvalue of a weighted eigenvalue problem and show the influence of spatial variables. We then apply these results to study the effect of spatial heterogeneity and dimension on the basic reproduction ratio for a spatial model of rabies. Numerical simulations also reveal the complicated effects of the spatial variables on $ \mathcal{R}_0 $ in two dimensions.

Citation: Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021170
References:
[1]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522.  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. Dynam. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

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N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

[4]

S. Chen and J. Shi, Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment, SIAM J. Appl. Math., 80 (2020), 1247-1271.  doi: 10.1137/19M1289078.  Google Scholar

[5]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[6]

O. DiekmannJ. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[7]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[8]

D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79 (2019), 1581-1606.  doi: 10.1137/18M1211957.  Google Scholar

[9]

D. Gao and C. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc., 148 (2020), 1709-1722.  doi: 10.1090/proc/14868.  Google Scholar

[10]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.  doi: 10.1007/s00285-011-0463-z.  Google Scholar

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D. JiangZ. Wang and L. Zhang, A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4557-4578.  doi: 10.3934/dcdsb.2018176.  Google Scholar

[12]

D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3$^{rd}$ edition, Brooks/Cole, Pacific Grove, CA, 2002.  Google Scholar

[13]

X. LiangL. Zhang and X. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease), J. Dynam. Differential Equations, 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.  Google Scholar

[14]

P. MagalG. F. Webb and Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, SIAM J. Appl. Math., 79 (2019), 284-304.  doi: 10.1137/18M1182243.  Google Scholar

[15]

J. D. MurrayE. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. Royal Soc. London Ser. B, 229 (1986), 111-150.   Google Scholar

[16]

D. Pang and Y. Xiao, The SIS model with diffusion of virus in the environment, Math. Biosci. Eng., 16 (2019), 2852-2874.  doi: 10.3934/mbe.2019141.  Google Scholar

[17]

R. Peng and X. Zhao, A reactionn-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[18]

D. Posny and J. Wang, Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments, Appl. Math. Comput., 242 (2014), 473-490.  doi: 10.1016/j.amc.2014.05.079.  Google Scholar

[19]

P. SongY. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.  Google Scholar

[20]

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

[21]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[22]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.  Google Scholar

[23]

F. YangW. Li and S. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differential Equations, 267 (2019), 2011-2051.  doi: 10.1016/j.jde.2019.03.001.  Google Scholar

[24]

T. Yang and L. Zhang, Remarks on basic reproduction ratios for periodic abstract functional differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6771-6782.  doi: 10.3934/dcdsb.2019166.  Google Scholar

[25]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar

show all references

References:
[1]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.  doi: 10.1137/060672522.  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. Dynam. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[3]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

[4]

S. Chen and J. Shi, Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment, SIAM J. Appl. Math., 80 (2020), 1247-1271.  doi: 10.1137/19M1289078.  Google Scholar

[5]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[6]

O. DiekmannJ. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[7]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[8]

D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79 (2019), 1581-1606.  doi: 10.1137/18M1211957.  Google Scholar

[9]

D. Gao and C. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc., 148 (2020), 1709-1722.  doi: 10.1090/proc/14868.  Google Scholar

[10]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.  doi: 10.1007/s00285-011-0463-z.  Google Scholar

[11]

D. JiangZ. Wang and L. Zhang, A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4557-4578.  doi: 10.3934/dcdsb.2018176.  Google Scholar

[12]

D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3$^{rd}$ edition, Brooks/Cole, Pacific Grove, CA, 2002.  Google Scholar

[13]

X. LiangL. Zhang and X. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease), J. Dynam. Differential Equations, 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.  Google Scholar

[14]

P. MagalG. F. Webb and Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, SIAM J. Appl. Math., 79 (2019), 284-304.  doi: 10.1137/18M1182243.  Google Scholar

[15]

J. D. MurrayE. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. Royal Soc. London Ser. B, 229 (1986), 111-150.   Google Scholar

[16]

D. Pang and Y. Xiao, The SIS model with diffusion of virus in the environment, Math. Biosci. Eng., 16 (2019), 2852-2874.  doi: 10.3934/mbe.2019141.  Google Scholar

[17]

R. Peng and X. Zhao, A reactionn-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[18]

D. Posny and J. Wang, Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments, Appl. Math. Comput., 242 (2014), 473-490.  doi: 10.1016/j.amc.2014.05.079.  Google Scholar

[19]

P. SongY. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations, 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.  Google Scholar

[20]

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

[21]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[22]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.  Google Scholar

[23]

F. YangW. Li and S. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differential Equations, 267 (2019), 2011-2051.  doi: 10.1016/j.jde.2019.03.001.  Google Scholar

[24]

T. Yang and L. Zhang, Remarks on basic reproduction ratios for periodic abstract functional differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6771-6782.  doi: 10.3934/dcdsb.2019166.  Google Scholar

[25]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar

Figure 1.  The relation of $ \mathcal{R}_0 $ with different disease transmission $ \beta $ in $ 2D $. (a-c) $ \mathcal{R}_0 $ in $ 2D $ versus $ c_1 $ and $ c_2 $, where $ \beta = 0.2192(1+c_{1}\cos(\pi x_1))(1+c_{2}\cos(\pi x_2)) $ in (a-c), $ \alpha = 0.2 $ in (a), $ \alpha(x_1,x_2) = 0.2(1+0.5\cos(\pi x_1))(1+0.5\cos(\pi x_2)) $ in (b), and $ \alpha(x_1,x_2) = 0.2(1+\cos(\pi x_1))(1+\cos(\pi x_2)) $ in (c). (d) Comparison between $ \mathcal{R}_0 $ versus $ c_1 $ in $ 1D $ and that in $ 2D $, where $ \beta = 0.2192(1+c_{1}\cos(\pi x_1))(1+c_{2}\cos(\pi x_2)) $ for the $ 2D $ case, $ 0\leq c_{i} \leq 1 $, $ i = 1,2 $, and $ \beta = 0.2192(1+c_{1}\cos(\pi x_1)) $ for the $ 1D $ case (that is, $ \beta $ is independent of $ x_2 $), and $ \alpha = 0.2 $ in both dimensions
Figure 2.  The optimal vaccine strategy over a two-dimensional environment. (a) $ \mathcal{R}_0 $ versus $ L_1 $ and $ L_2 $ under $ c_0 = 0.61 $. (b) The lowest $ \mathcal{R}_0 $ versus $ L_1 $ under $ c_0 = 0.61 $. (c-d) The disease transmission $ \beta $ before and after taking the optimal vaccine strategy, respectively. The boundary of the vaccination region is highlighted (bold red lines).
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