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June  2022, 27(6): 3005-3017. doi: 10.3934/dcdsb.2021170

The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model

Tianhui Yang 1, , Ammar Qarariyah 2, and  Qigui Yang 1,,

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 Published  June 2022 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.

Keywords: basic reproduction ratio, reaction-diffusion equation, spatial heterogeneity, two-dimensions, epidemic model.
Mathematics Subject Classification: Primary: 92D30, 35K57.
Citation: Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
References:
[1]

L. J. S. Allen, B. M. Bolker, Y. 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.

[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. Dynam. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[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.

[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.

[5]

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

[6]

O. Diekmann, J. 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.

[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.

[8]

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

[9]

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

[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.

[11]

D. Jiang, Z. 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.

[12]

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

[13]

X. Liang, L. 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.

[14]

P. Magal, G. 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.

[15]

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

[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.

[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.

[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.

[19]

P. Song, Y. 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.

[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.

[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.

[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.

[23]

F. Yang, W. 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.

[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.

[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.

show all references

References:
[1]

L. J. S. Allen, B. M. Bolker, Y. 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.

[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. Dynam. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[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.

[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.

[5]

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

[6]

O. Diekmann, J. 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.

[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.

[8]

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

[9]

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

[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.

[11]

D. Jiang, Z. 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.

[12]

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

[13]

X. Liang, L. 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.

[14]

P. Magal, G. 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.

[15]

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

[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.

[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.

[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.

[19]

P. Song, Y. 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.

[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.

[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.

[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.

[23]

F. Yang, W. 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.

[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.

[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.

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
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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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