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

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

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  • 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.

    Mathematics Subject Classification: Primary: 92D30, 35K57.


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  • 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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