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Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity

  • * Corresponding author: Yuxiang Zhang

    * Corresponding author: Yuxiang Zhang
X. Zhang is partially supported by the NSF of China (No. 11571200,11425105), and Y. Zhang is supported in part by the NSF of China (No. 11701415,11601386).
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  • This work is devoted to study the spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. In the case of the spatial domain is bounded and heterogeneous, we assume some key parameters in the model explicitly depend on spatial location. We first define the basic reproduction number $\mathcal{R}_0$ for the disease transmission, which generalizes the existing definition of $\mathcal{R}_0$ for the system in spatially homogeneous environment. Then we establish a threshold type result for the disease eradication ($\mathcal{R}_0 <1$) or uniform persistence ($\mathcal{R}_0>1)$. In the case of the domain is linear, unbounded, and spatially homogenerous, we further establish the existence of traveling wave solutions and the minimum wave speed $c^*$ for the disease transmission. At the end of this work, we characteristic the minimum wave speed $c^*$ and provide a method for the calculation of $c^*$.

    Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 37N25, 92D25.

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  • Table 1.  Biological interpretations for parameters in model (2)

    SymbolsInterpretations
    $N_0$Total population size at time $t=0$
    $d_i$Diffusion coefficients for $i=1, 2, 3, 4$
    $\mu$Birth/death rate
    $\sigma$Recovery rate
    $\mu_B$Loss rate of bacteria
    $\pi_B$Growth rate of bacteria
    $\beta(x)$Contact rate with contaminated water at location $x$
    $e(x)$Contribution of each infected person to the population of V. cholerae
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