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Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents

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  • In this article we describe the transmission dynamics of hantavirus in rodents using a spatio-temporal susceptible-exposed-infective-recovered (SEIR) compartmental model that distinguishes between male and female subpopulations [L.J.S. Allen, R.K. McCormack and C.B. Jonsson, Bull. Math. Biol. 68 (2006), 511-524]. Both subpopulations are assumed to differ in their movement with respect to local variations in the densities of their own and the opposite gender group. Three alternative models for the movement of the male individuals are examined. In some cases the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution of density values as proposed, in another context, in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369-400]. An efficient numerical method for the resulting convection-diffusion-reaction system of partial differential equations is proposed. This method involves techniques of weighted essentially non-oscillatory (WENO) reconstructions in combination with implicit-explicit Runge-Kutta (IMEX-RK) methods for time stepping. The numerical results demonstrate significant differences in the spatio-temporal behavior predicted by the different models, which suggest future research directions.

    Mathematics Subject Classification: Primary: 92D30; Secondary: 97M60, 65L06, 65M20.

    Citation:

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  • Figure 1.  Numerical solution of the ODE version of (2.1), Model 0, for the initial data (4.1)

    Figure 2.  Case 1 (Model 1, Scenario 1): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 3.  Case 2 (Model 2 with $K=1000$, Scenario 1): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 4.  Case 3 (Model 3, Scenario 1): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 5.  Cases 1-3: integral quantities $\mathcal{I} (X)$ defined by (4.3) for each compartment obtained by evaluating numerical solutions

    Figure 6.  Case 4 (Model 1, Scenario 2): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 7.  Case 5 (Model 2 with $K=1000$, Scenario 2): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 8.  Case 6 (Model 3, Scenario 2): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 9.  Cases 4-6: integral quantities $\mathcal{I} (X)$ defined by (4.3) for each compartment obtained by evaluating numerical solutions

    Figure 10.  Cases 4-6: Moran's index $\mathsf{I}$ defined by (4.4), (4.5) for each compartment obtained by evaluating numerical solutions

    Figure 11.  Case 8 (Model 3, Scenario 2, periodic variation of parameters): numerical solution for $N_{\rm{m}}$, $N_{\rm{f}}$, $I_{\rm{m}}$ and $I_{\rm{f}}$ at the indicated times

    Figure 12.  Case 8: (Model 3, Scenario 2, periodic variation of parameters): solutions of Model 0 (left column) and integral quantities $\mathcal{I} (X)$ of Model 3 (right column) defined by (4.3) for each compartment obtained by evaluating numerical solutions

    Table 1.  Case 7 (Model 1, order test with smooth solution): approximate total $L^1$ errors $\smash{e_M^{\smash{tot}}}$ and observed convergence rates $\theta_M$

    $M$ 8 16 32 64 128 256
    $e_M^{\smash{tot}}\times 10^{3}$ 368.90 383.19 379.01 153.73 34.70 9.14
    $\theta_M$ -0.05 0.02 1.30 2.15 1.92 -
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