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

Raimund BÜrger; Gerardo Chowell; Elvis GavilÁn; Pep Mulet; Luis M. Villada

doi:10.3934/mbe.2018004

Article Contents

2018, Volume 15, Issue 1: 95-123. Doi: 10.3934/mbe.2018004

This issue Previous Article Mathematical analysis of a weather-driven model for the population ecology of mosquitoes Next Article Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis

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

1.
CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2.
School of Public Health, Georgia State University, Atlanta, Georgia, USA
3.
Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85287, USA
4.
Division of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
5.
Departament de Matemàtiques, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
6.
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
7.
CI²MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Received: September 29, 2016

Accepted: January 14, 2017

Published: February 2018

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Abstract

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.

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

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

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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

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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

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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

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

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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

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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

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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

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

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

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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

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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

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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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